Final Report 

NASA Grant NAG 3-393 

NASA Lewis Research Center 

entitled 

Investigation of Surface Tension Driven 

Convection as a Feasibility Study for a 

Micro-Gravity Experiment 

E. L, Koschmieder 

submitted to 

Grant Office 

Nasa Lewis Research Center 

Cleveland, Ohio 

March 1988 

(EASA-CR- 182504) 11VEStIGA!IICY 01 SUBPACB 188-18739 3EYSICll P%ZYEH CCWVECTXICII AS I EPASIBILXTY 

STUDY PCB A HICEC-GBAVfSI BIEUILELT fiaal 

IiepoEt (%exas Univ.) 64 F CSCL 226. Onclas 

63/29 0124767 
c 

I. Introduction 

We shall provide first a review of the basic problem 

that was addressed in the work done for the feasibility 

study of a micro-gravity surface tension driven convection 

experiment. The origin of the surface tension driven convection problem is a classic paper of Benard (1900) about 

cellular convection in a shallow fluid layer. The fundamental discovery of this study were the hexagonal convection 

cells, which are now referred to as the Benard cells. A 

sample of hexagonal cells is shown in Fig. 1. 

Hexagonal Benard cells. After Koschmieder (1974) 

The pioneering theoretical work dealing (supposedly) 

with the explanation of the Benard cells was Rayleigh's 

(1916) theory about the stability of a fluid layer heated 

uniformly from below. A truely classical paper. It concerns the most simple stability problem in Fluid Mechanics, 

namely the stability of a fluid layer which is initially at 

rest. The investigation of Benard convection and RayleigFs 

theory were believed to deal with the same problem until 

1956, wk-m in a one page paper Block showed experimentally 

that surface tension effects must play a fundamental role in 

the formation of the Benard cells. Independently it was 

shown theoretically by Pearson (1958) that surface tension 

gradients alone are able to create an instability in a resting fluid, and that thereby hexagonal cells can be formed. 
2 

The most interesting feature of Pearson's theory in the context of our feasibility study is that Pearson postulated 

that gravity is - zero. Nevertheless the hexagonal cells are 

a solution of his theory, which is linear, hence mathematically unambiguous but not strictly verifiable in a laboratory on Earth. It is, however, nowadays accepted as a fact 

that surface tension is the driving force in the formation 

of the hexagonal convectibn cells, and not buoyancy, as it 

originally followed from Rayleigh's theory. Since surface 

tension can be easily eliminated in a convection experiment 

by putting a lid into contact with the fluid, the Rayleigh 

problem has remained a fundamental problem in Hydrodynamic 

Stability, it has now,however, a parallel, Pearsons's 

problem. 

The amount of experimental work on surface tension 

driven Benard convection (Pearson's problem) was very 

limited when our feasibility study began. There was one 

paper dealing with the size (or the so-called wavelength) of 

the hexagonal cells by Koschmieder (1967) and one paper by 

Palmer and Berg (1971) on the measurement of the critical 

Marangoni number required to be reached for onset of surface 

tension driven convection. There was also an attempt to 

observe the onset of surface tension driven convection in 

microgravity on board of the Apollo 14 and 17 space flights, 

( Grodzka and Bannister 1972, 1975). I have served as a 

scientific consultant for these experiments. In spite of 

the determined and admirable efforts of the astronauts these 

experiments are considered to be inconclusive. 

2. The feasibility studies. 

When I made my first proposal to NASA Lewis concerning 

a microgravity surface tension driven convection experiment 

two items were foremost on my mind. First that we had to 

make first of all a good laboratory test of the consequences 

of surface tension gradients on the onset of convection in a 

laboratory on Earth, and second that we had to learn about 

surface tension driven convection in small containers, 

because this configuration was likely to be the way a convection experiment in space had to be made. 

Concerning item #1 the following: It was known from a 

theoretical study of Nield (1968) how gravity interacts with 

surface tension to cause the onset of convection in shallow 

fluid layers on Earth. It was argued by some that the 

results of Nield's theory permitted experiments in a laboratory on Earth which would approximate with sufficient accuracy the results of Pearson's theory, if only the fluid 

layer was sufficiently thin. Although this argument was not 

convincing, the only way to settle this matter was by an 

experiment. Our first effort was therefore the investigation of the onset of convection in thin fluid layers of 

large lateral extent (or aspect ratio). The results of our 
3 

experiments are described in the paper of Koschmieder and 

Biggerstaff (1986), see appendix 1. It would be senseless 

to repeat here the details. I will, however, state our 

basic result. We have learned that it is - not possible to 

approximate the results of Pearson's theory by running 

convection experiments in very thin fluid layers in the 

presence of gravity. The reason for our finding is the 

occurrence of a subcritical instability. This means that in 

very thin fluid layers of depth of about 1 mm (or less) 

motions form in the fluid at Marangoni numbers below the 

critical Marangoni number predicted by Pearson. The thinner 

the layer is, the earlier the subcritical motions appear, 

contradicting flatly the expected behavior. That is not a 

fault of Pearson's theory, rather a consequence of the 

attempt to create (via a thin fluid layer) conditions which 

approximate Pearsons's theory. There are not shortcuts, if 

one wants to verify Pearsons's theory unambiguously one has 

to do that with zero g (or its approximation microgravity). 

In microgravity there is no need to use thin fluid layers to 

approximate Pearsons's theory, actually the experimental 

difficulties increase as the fluid depth decreases. 

The subcritical instability that we discovered in our 

experiments was theoretically unexpected and not welcome 

with some of my colleagues, although the evidence is quite 

clear, the subcritical motions can be seen with the naked 

eye, see figures 2a-c in appendix 1. We were already 

engaged in experiments with small containers when it 

occurred to me that there is a way to run a critical test on 

the reality of the subcritical instability. This can be done 

by checking for subcritical motions in thin fluid layers in 

buoyancy driven convection, or Rayleigh-Benard convection as 

it is called. We have, therefore, interrupted our experiments with small containers and made a crash effort with 

Rayleigh-Benard convection in thin fluid layer, a topic 

never investigated before. The results of this effort are 

described in appendix 2. This manuscript has be submitted 

for publication to the Journal of Fluid Mechanics. We will 

not present here the details of this paper either. The 

basic result is quite simple. A subcritical instability 

occurs in Rayleigh-Benard convection as well, if the fluid 

layer is sufficiently thin. As it turned out, the subcritical instability is a principal feature of onset of convection in very thin fluid layers, regardless whether the 

instability is surface-tension driven or buoyancy driven. 

If one does not work with thin fluid layers the subcritical 

instability does not occur, buoyancy driven experiments 

provide correct results, and surface tension driven experiments should provide correct results, if the acceleration of 

gravity is eliminated. 

Having completed this study we returned to surface tension driven convection in small containers. This problem 

had been studied theoretically by Rosenblat, Davis, and Homsy 

(1982 a,b). The patterns found in our study for various 

aspect ratios in one 
and the same either circular or square container are shown 

in figures 2 and 3. 

While the patterns in the circular container conform with 

expectations, the patterns found in the square container are 

quite original. The fluid has found surprising ways to pack 

either two or three square cells or e.g. six square cells. 

It is, of course, natural to pack a square container with 

either one or four square cells, and the fluid readily does 

so. The patterns shown here are startling examples of the 

effect lateral boundaries can have on pattern formation. 

doing such experiments in microgravity will only enhance the 

clarity of the results, because microgravity will reduce the 

thermal effects the lateral boundaries might have on the out 

come of these experiments. The last experiment on this 

topic WLS made on Aug 20, 1987, our financial support ended 

on February 28, 1987. Due to other pressing commitments the 

paper describing these experiments has not yet been written, 

but is on my agenda for this spring semester. 
Y 

5 

. 

I would finally, like to express my sincere appreciation to NASA for the interest in and support of this work. 

It was as a a matter of fact, on several occasions exciting 

to pursue these investigations. 
Onset of surface-tension-driven Bknard convection 

By E. L. KOSCHMIEDER AND M. I. BIGGERSTAFF 

('olli~pe of' Ihgintwring and 1. I'ripoginr Center for Statisticid ,\lwhanic*s. 

The 11niversity of Trsas. Austin. TX 78712. 1'SA 

(Itecrived 4 April I!H5 and in revised form 20 Sovemher 198.5) 

An cbxj)cvimcmtal investigation of the onset of convection in shallow fluid layers heated 

ut~iforn~ly from bclow and coolcd from above by an air layer has been made. If the 

dvpth of thc silic*onc layer is smaller than 2 mm the onset of convection takes place 

in two stages. Thcrc is first a weak pattern, which is characterized by its appearance 

at ever smallcr temperature gradients as the depth of the fluid is decreased. When 

tht: temperature difference across the fluid is increased a second strong pattern forms 

ncw the predicted critical Marangoni number. The cells in this pattern are hexagonal 

and swm to be what one has always referred to as BBnard cells. The temperature 

gradient at which this pattern appears increases with decreased depth. The heat 

transfer through the fluid has been measured. The critical tcmpcrature gradient for 

the formation of the hexagonal pattern has been determined from the break of the 

heat transfer curve. 

1. Introduction 

'I'hc: investigations of &nard (1900) marked the beginning of the study of fluid 

motions in a shallow fluid layer heated uniformly from below. The most notable 

observation of Bdnard was the hexagonal convection cells, now commonly referred 

to as 136nard cells. Today, 86 years later, we still do not possess a convincing 

cxplanation for the formation of the hexagonal cells, and for the preference for 

hexagonal (.ells over the other possible cellular patterns. 

Rayleigh's (1916) pioneering theoretical analysis of the stability of a layer of fluid 

heated from below seemed to providc a basis for the understanding of the Bdnard 

cells. Itayleigh's most fundamental result however was the discovery of the cxistence 

of a mitical or minimal temperature gradient required for the onset of convection. 

Udnarcl had not noticed the existence of such a critical gradient in his experiments. 

A(*tu:dIy. in a later evaluation of his experiments (Bdnard 1930). he came to the 

c*onc.lusion that he had observed convection at temperature differences about 10P 

to times smaller than the critical temperature gradient predicted by Rayleigh. 

Bdnard's interpretation of his own experiments was ignored. We shall see in thc 

following that his interpretation of his experiments was correct. 

A ncw aspect was introduced into the explanation of the BCnard cells by Block 

(1956). He concluded from his experiments that surface tension must play a 

signitkant role in the formation of the hexagonal cells. He also pointed out that 

mnvcctive motions occurred at temperature differences of a fraction of the predictrd 

critical value. A short time thereafter the same points were established theoretically 

by l'carson (1958). I'earson, who investigated surface forces only, showed that thc 

onsct of surfac.e-tension-driven convection is determined by the critical value of a 

non-dimensional parameter, the Marangoni number. From the value of the critical 
MiwiLngoni niiitilwr. and from thc lincwr depcndencc of the Yarangoni number on the 

tl(*l~t h ol’thc fluitl it fi)llo\w that the onsct of convection in shallow fluid layers should 

oi*viir iii (+ritii*;d tvinperibture differrnces much smaller than those predicted by 

Itiiylc4gli’s tlivory. An (ax planation for the discrepancies of the cxpcrimental and 

1 Iicv)r~i ic*iLl viIIiws of thc witical tcniperature difference had apparently been found. 

I Iouc~vc.i*. NT will show in thc folloi\ ing that the onset of surface-tcnsion-drivcn 

c*onvwt ion owurs at temperaturc differences significantly below those predicted by 

t IN. carit ical XliLranponi number. 

I’cwson’s thcwr?, has bccn augmented by the studies of Sield (1964), and Scriven 

CV Stc.rn1ing ( 1!164). Zu’ield studied the conditions for the onset of surface-tension-driven 

c.oiivwtion in the presence of gravity. i.e. he dealt with the situation experienced 

iii t hi> lahoratory and obviously present in Benard’s experiments. He found that 

sirrfiLc.c.-tensioIi-tlriven convection and buoyancy-driven convection are coupled. In 

tho cas(’ of very shallow fluid layers the minimal temperature difference required for 

t hc onsct of convection in the presence of gravity is, according to Nield, determined 

I)y thc criti(d MIarangoni number. Scriven & Sternling neglect gravity, just as 

I’carson did, but take various other parameters into account, in particular the surface 

tvnsioii c.oefic+ient S, not only the variation of surface tension with temperature 

tlN/t17’, in which form surface tension appears in the Marangoni number. Smith (1966) 

cbxtcwtfcd these two studies by considering the effect of gravity waves. He found that 

surfacc waves are usually important only for very small wavenumbers. 

The results of Scriven & Sternling differ drastically from those of Pearson as well 

as Nield. They find that onset of convection in very shallow fluid layers can occur 

at very small temperature differences, the onset temperature difference decreasing 

with dccrcased fluid depth. This differs clearly from the result to be expected when 

vither the critical Jlarangoni number or the critical Rayleigh number determine the 

onsvt of motion. An onset of convection governed by, for example, the critical 

Marangoni number implies that the critical temperature gradient increases as the 

drpth of the fluid is decreased. As will be shown in the following, the onset of 

cwnwc-tion with a free (air) surface actually occurs at smaller temperature differences 

if thr fluid depth is dccrcased, provided that the depth of the fluid is smaller than 

;I wrtain value above which the conventional critical Marangoni number determines 

the onset of motion. 

‘I’hc stability of a non-deformable fluid layer subject to buoyancy and surface-tension 

forws has bccn investigated with the energy method by Davis (1969). Davis finds 

a suh*ritiwl instability in a small range of Marangoni numbers for sufficiently small 

1tayIc.igh numbrrs. These energy method studies have been confirmed by Davis & 

Hornsy (1980) with an investigation of the same problem but with a deformable 

surfiwc. Castillo 8: Vrlarde (198%) have studied, also with the energy method, the 

stiihility of two-c-omponent or one-component fluid layers with a deformable surface 

hcatcd from bclo\\ or from above, and found likewise this subcritical instability. 

Nonlincar thcorctical studies of surface-tension-driven convection have been made 

by Scanlon & Scgel(1967), Kraska 8: Sani (1979), and Cloot & Lebon (1984), all trying 

to cbxplain the preference for the hexagonal cella. Recently Rosenblat, Davis & Homsy 

(1 !)H2n, b) have investigated surface-tension-driven convection in bounded circular 

or rwtangular fluid layers of small aspect ratio. There are only two modern experimcvitnl investigations of surface-tension-driven B6nard convection. Koschmieder 

(1967) obscrvccl the formation of a regular hexagonal cell pattern and determined 

thc \vavelrngth of thr convective motions. Palmer & Berg (1971) determined the 

c.riti(d Miirangoni number from the break of the heat transfer curve which is caused 
, 

Surface-tension-drioen Be'nard convection 

ae,F.IGl"AL PAGE Is 

51 

FIGURE 1. Schematic section though the convection apparatus. 

by the onset of the convective motions. These two sets of experiments were made 

with fluid layers 2-5 mm deep. We will show in the following that one has to go to 

layers less than 2 mm deep in order to observe purely surface-tension-driven motions. 

BBnard's experiments were made with fluid layers of depths ranging from about 1 mm 

to about 0.5 mm. 

2. Description of the apparatus 

The apparatus used is a modified version of an apparatus used previously by 

Koschmieder & Pallas (1974~). A schematic diagram of the apparatus is shown in 

figure 1. The bottom of the fluid layer was a 5 em thick copper block, which was heated 

from below by an electric current going through a resistance wire. The wire was 

cemented in circular grooves in a brass plate. The excellent thermal conductivity and 

the thickness of the copper block in combination with poor thermal conduction on 

top assure that the temperature at the top of the copper plate is practically uniform, 

as is required for comparison with the theories describing BBnard convection. To 

avoid lateral heat losses the copper block was placed in a vacuum tank. The copper 

plate was covered by shallow layers of silicone oil, bounded laterally by a circular 

lucite rim of 13.55 cm diameter. With a fluid depth of say 1 mm the aspect ratio of 

the fluid, defined as the ratio of the horizontal extension of the fluid divided by the 

fluid depth, was 135. The fluid was cooled from above through a very thin layer of 

air between 0.3 and 0.5 mm deep. The air was bounded on top by a glass plate 2.3 mm 

thick. The glass plate was the bottom of a lid in which water was circulated at a rate 

of 80 em3 s-l in order to fix the temperature of the glass plate and thereby the 

temperature on top of the fluid. 

Most of the space above the glass plate was filled usually with the so-called heat 

sensor, a device to measure the increase of the temperature of the cooling water caused 

by the heat transferred through the fluid layer and the air above it. The heat sensor 

is essentially a thermopile of 10 thermocouples in series, measuring the temperature 

difference between a copper ring surrounding the water inlet and copper ring being 

in contact with the cooling water after it passed over the glass plate.The heat sensor 

t 

1. 
62 E. I,. Koschmieder and ill. 1. Biqqerrh'triff 

has been dcscribcd in detail elscwherc (Koschmieder & l'allas 19540). Tcmperaturc 

differences of 0.001 OC between the incoming and outgoing water can bc mcasured 

with the heat sensor. From the moasurcd temperature increase of the cooling water 

follows the heat flux through the fluid layer. as will be discussed later. 

The cooling water was taken from a 50 1 insulated tank filled with water. The 

temperature of the water in the insulated tank was controlled by a heat exchanger 

whose temperature was determined by water coming from a commereial regulated 

water bath. Cooling uater cannot be used from the water bath directly as the heat 

sensor is too sensitive. The heat sensor picks up the fairly regular fluctuations of the 

water temperature of the bath, which are caused by the heating and cooling of the 

temperature control. The enormous heat capacity of the water in thc insulated tank 

reduces the fluctuations of the water-bath temperature. A steady signal for the 

temperature difference between incoming and outgoing water was then obtained. 

3. The experiments 

3.1. Pattern formation 

The fluid motions were made visible by aluminum powder suspended in the fluid, 

which was either 100 cs silicone oil, or in some cases 50 cs silicone oil (Dow Corning 

200 fluid). The properties of both fluids are listed in table 1. All experiments were 

made in a nearly steady state, it took usually about eight hours to heat up the fluid 

from rest at zero temperature difference to the highest temperature difference applied, 

which was in the maximal case about 60 "C between the copper plate and the glass 

lid. The fluid depth in the experiment shown in figure 2 was 1.3i mm, the depth of 

the air gap on top of the fluid was 0.5 mm. 

When the fluid layer is heated up the first faint sign of motion is, besides a strong 

roll along the rim, a system of circular concentric rings, see figure 2(a). There are, 

in this picture, two stronger individual cells, which are caused by two impurities which 

have come into the fluid with the aluminum powder. The convecting fluid is seen on 

the shiny copper plate through the lid covering the fluid layer. The lid is filled with 

cooling water in order to maintain the temperature difference across the fluid. The 

heat sensor is taken out of the lid. Increasing the temperature difference causes the 

rings in figure 2 (a) to break up into cells, figure 2 (b). The cells clearly have a centre, 

the eel1 outline on the other hand is rather weak and not regular, certainly not of 

a regular hexagonal form. Increasing AT further intensifies the cells, figure 2 (c). The 

cell boundaries then become outlined by floating aluminum powder which is extruded 

from the fluid. In figure 2 a larger than usual amount of aluminum powder was added 

to the fluid in order to be able to see the flow on photographs. In figure 2(c) the 

extruded aluminurn powder can be seen floating on the surface of the fluid in the space 

between the outermost ring of cells and the clearly visible rim roll. 

When the temperature difference is increased further a very definite change in 

appearance of the cells occurs spontaneously at a certain reproducible (critical) 

temperature difference A%, as can be recognized clearly in figure 2 (d). The new type 

of cell is strikingly reminiscent of B6nard cells as we have seen them many times 

before, a good example of such a pattern can be found in Koschmieder (1974). As 

the hexagonal cclls form, the floating aluminum powder, which outlined the cells of 

the first pattern, disappears from the surface of the fluid. Some aluminum powder 

then settles at the bottom in the centre of the cells underneath the uprising fluid. 

The settled powder can be rccognizcd as little black spccks in figurc 2(d). The 

appearance of the hexagonal cclls is also accompanied by an increase of the heat flux 
FI(;I-RE 2. Visualization of the onset of convection in a layer of silic-one oil of 100 cs viscosity. 

1.37 mmd~ep,nithanairgapof0.51 mm.(a)AT = 0.57 AT,;(b)0.72 A7;;(r)0.H6AT2;(d)0.n.r, AC; 

(e) AT,; (f) 1.08 AT!. 

through the fluid layer, as will be shown later. A truly critical transition should, under 

ideal conditions, occur simultaneously over the entire area of the fluid. We have not 

ohscrved such a genuine transition because of experimental difficulties. In particular. 

we cannot keep the depth of the air layer uniform over the entire area of the fluid 

layer. The reason for this is that the glass plate of the lid has to be glued to the framc 

of the lid. We have not been able to do this with an accuracy better than (at best) 

,& mm. But & mm is a SI Wantial fraction of the gap between thc fluid and the 

gliiss, which ranges from 0.3 to 0.5 mm in different experimcmts. A large portion, in 

many CitSeS aotually the larger part, of the applicd tenipcmturc diffcrencc between 
54 

ORIGINAL' PAGE IS 

E. L. Koschmieder and 11.3. 1. BigyrrstaffOF POOR QUALITY 

V P a K S dS/dT 

(cm*/d (g/cm3) ("C-') (cm2/s) (dyne/cm) (dyne/cm "C) 

1 .OO 0.968 0.00096 0.001 095 19.36 -0.050 

0.50 0.960 0.001 04 0.001 025 18.35 -0.047 

TABLE 1. Properties of the fluids at 25 "C 

FIGURE 3. Shadowgraph of the hexagonal pattern at AT = 1 .I9 A?. Viscosity 100 cs, fluid 

depth 1.35 mm. 

the copper and the glass lid falls off in the air gap because of the very poor thermal 

conductivity of air. An air gap of unequal depth therefore causes a one-sided 

temperature distribution on top of the fluid. Hence the onset of the transition is 

one-sided. 

When the temperature difference is increased further the hexagonal cells spread 

over an increasing area of the fluid, see figure 2 (e). Finally figure 2 (f) shows hexagonal 

cells covering the entire surface of the fluid. In this photograph the lid has been 

removed, so that one can see the rim of the layer with the rim roll, which was obscured 

in figure 2 (a-e). There are about 200 regular hexagonal cells in the pattern in figure 

2 (f). When the slightly supercritical temperature difference is maintained another 

strange twist in the behaviour of the ahminum powder occurs. The small heaps of 

settled aluminum powder underneath the cell centres disappear, which means that 

the powder goes back into solution. This has never happened before in our 

experiments with deeper fluid layers. The peculiar behaviour of the aluminum powder 

does not seem to have fluid dynamics significance, but is an indication that surface 

forces play a role in the experiments with very shallow layers. 

i 

I 
Surface-tension-driven Be'nard convection 55 

. . . 

I 1 I * 

5 10 AT,("C) 

FIGURE 4. Heat transfer in the 1.81 mm deep fluid layer of 100 cs viscosity. 

Nu 

. . . 0 

. . . 

I I -- 

5 10 ATr ("C) 

FIGURE 5. Heat transfer in the 1.31 mm deep fluid layer of 100 cs viscosity. 

There is another way to visualize the fluid motions, namely the shadowgraph 

technique described by Silveston (1958). A parallel beam of white light is shone 

through the fluid layer where it is reflected from the bottom. The index of refraction 

in the fluid varies with temperature and consequently varies locally in the fluid with 

the temperature in the convective cells. The optical path of the light in the fluid layer 

is therefore different in different parts of a cell. Interference occurs therefore in the 

reflected light beam, outlining the cell pattern. We were not able to obtain a 

shadowgraph from the first pattern when the applied temperature difference was 

substantially below AT2, regardless of whether we used the copper plate or a glass 

mirror placed on the copper plate as the bottom of the fluid. The surface of the 

copper block produces a shadowgraph of the second, hexagonal pattern, with minor 

imperfections resulting from the remaining uneveness of the copper surface caused 

by the machining. Shadowgraphs of the hexagonal cells obtained with the mirror 

were nearly perfect, see figure 3. One has to keep in mind that the mirror changes 

the thermal boundary condition at the bottom of the fluid, rcplacing the excellent 

DIIIGINAL PAGE IS 

D!E POOR QIJ ILTTY 
56 

1.1 - . . 

5 10 ATr ("C) 

FIGURE 6.Heat transfer in the 0.93 mm deep fluid layer of 100 cs viscosity. 

1.1 

...e 

.. .. :.*'* 

.. . ....' l.o~avl .* . .. ... .... 

.. 

5 10 ATr ("C) 

FIGIJRE 7.Heat transfer in the 0.72 mm deep fluid layer of 100 cs viscosity. 

thermal conductivity of copper by the poor thermal conductivity of glass. The 

cmvcctive flow visualized by shadowgraphs with either the copper bottom or the 

mirror was nevertheless the same as the flow visualized with the aluminum powder. 

3.2. Heat flux measurements 

The usual procedure to determine the critical temperature gradient for the onset of 

convection is via the break of the heat transfcr curve brought about by the additional 

heat transfer of the convective motions, a method introduced by Schmidt & Milverton 

(1935). The results of our heat transfer measurements are shown in figure 4-7. The 

heat flux when plotted in absolute units, or, for example, as microvolts measured by 

the heat sensor, has such a small change of slope with the onset of convection that 

the critical temperature difference cannot be determined accurately from the break 

of the curve. The heat flux in figures 4-7 is therefore plotted in units of the Susselt 

number, which is the ratio of the actual heat flux divided by the heat flus caused 

. . 1 *. 
by the thermal conductivity of the fluid only. On a plot of Nusselt numbcr versus 

applied temperature difference the break of the heat transfer curve is much more 

easily rccognizablc. The value of the Susselt number before onset of convection 

is 1. 

The abscissa in figures 4-7 is the temperature difference AT, applied to the fluid. 

, This temperature difference cannot be measured directly, because there is no space 

in the air gap above the fluid layer to mount thermocouples. The value of AT, has 

to be calculated from the temperature difference AT between the copper plate and 

the cooling water. For the heat flux measurements we did not use the glass plate on 

top of the air layer, but a copper plate of 2 mm thickness. This was done to avoid 

additional corrections for the temperature drop in the glass. The value of ATf is given 

by 

(1) 

AT 

AT, = (1 +!e!%) 

where A, is the mean thermal conductivity in the fluid layer, 2, the mean thermal 

conductivity in the air layer, AZ, the depth of the air layer and AZf the depth of 

the fluid layer. AT can be measured with an accuracy better than 1 yo. Both thermal 

conductivities are known only with about 1 % accuracy. The depth of the fluid is 

known to about 1 Yo, while the depth of the air layer, which is of order 0.5 mm or 

less, is known at best with only 2 yo accuracy. So the cumulative uncertainty in the 

value of AT, is at least 5%. Since the Nusselt number is proportional to AT,, the 

experimental values of Nu can differ from 1 by f 5 yo or more. It is likely that the 

uncertainty in the value of Nu will increase with decreased AZa and AZ,, because in 

particular the uncertainty in AZ, becomes increasingly large. All measurements of 

Nu give the mean value of the heat flux at a given AT, determined by four 

experiments. The standard deviation of the four measurements determines the 

experimental error of the heat flux measurement. Since so many error bars would 

confuse the graphs, only the average value of the error bars is indicated in the figures. 

We will discuss the heat flux curve of the fluid layer with 1.81 mm depth (figure 4) 

first. In this case it appeared from corresponding visual observations that at 

AT, x 6.6 "C a hexagonal pattern had established without a prior first pattern. 

There is, of course, an uncertainty in the visual determination of AT2, because the 

pattern does not form spontaneously over the entire fluid. This uncertainty is of the 

order of 5 yo of AT,. As can be seen in figure 4 the heat flux is, within the experimental 

error, about Nu = 1.03 for AT, up to 5 "C, then it seems to increase prior to the 

observed Aq = 6.68 OC, and increases steadily after AT,. As we will see in the data 

evaluation it appears that with this fluid depth the formation of the first pattern has 

just taken place prior to the formation of the hexagonal cells. This is also indicated 

by the increase in the measured Nu before AT,. The increase of Nu after AT2 is in 

agreement with expectations. 

With a fluid layer of 1.31 mm (figure 5) we observed visually the formation of the 

first pattern at Aq x 3.7 "C. Our heat flux measurements indicate an increase of the 

Nusselt number for values of AT, < 3.7 "C. This is caused by an erroneous signal of 

the heat sensor. At very small values of AT, the heat flux through the fluid layer is 

very small. The temperature increase in the coolant measured by the heat sensor is 

then determined mainly by heat generated by dissipation in the cool: ig fluid, as it 

moves from the inlet over the lid to the outlet. The contribution of dissipation to 

the heat sensor signal decreases as the heat flux to the lid from underneath increases. 
58 id. I,. Koschniied4r mid AI. I. lliqyrrstuff 

In order to obtain a signal from the heat sensor which is less affected by dissipation 

in the coolant, we have, in all heat flux measurements, reduced the flow rate of the 

cooling water to 20 em3 s-l. For values of AI'; between AT; and AT, at z 7.9 "C the 

heat flux in figure 5 incrcascs on average just a little, then increases steadily and at 

a faster rate after AI';. There is, of course, no abrupt change in the slope of the heat 

transfer curve at AT, because we are dealing here with the gradual transformation 

of a pattern, as shown in figure 2 (d-e). 

Figure 6 shows the heat flux for a fluid layer of0.93 mm depth, and figure 7 for 

a depth of 0.72 mm. It can be seen on these figures that the critical temperature 

difference AT, for the formation of the hexagonal cells increases with decreased depth, 

while on the other hand the temperature difference AT, for the formation of the first 

pattern decreases with decreased depth. The values of these temperature differences 

for the different fluid layers are listed later in table 2. It is already apparent in figure 

6 and seems to be certain in figure 7 that the heat flux through the fluid layer is no 

longer constant after the formation of the first pattern, but increases with increased 

AT,. This means that the motions in the cells of the first pattern are sufficiently fast 

to transfer an additional amount of heat. The formation of the hexagonal cells is still 

marked by a clear break in the heat flux curve in figure 6, at the temperature 

difference at which we visually observed the transition. In figure 7 however the break 

in the heat flux curve is barely recognizable. This probably means that near A% the 

motions in the first pattern have become so vigorous that they hardly differ from 

the velocity of the flow in the hexagonal cells. 

We have also measured the heat flux for two fluid layers of 50 cs silicone oil. Since 

there is no additional significant feature in these measurements their curves will not 

be shown. 

3.3. Data evaluation 

In order to understand the meaning of our observations the various experimental data 

have to be expressed in terms of the non-dimensional parameters involved. Buoyancy 

driven convection is governed by the Rayleigh number, which is 

agATd3 R=-, 

VK 

with the volume expansion coefficient a, the acceleration of gravity g, the depth of 

the fluid d, the kinematic viscosity v and the thermal diffusivity K. The critical 

Rayleigh number R, for onset of convection in a fluid layer with a free surface is 

R, = 1100.6. This value of R, applies when the top free surface is an excellent thermal 

conductor. Air on top of the fluid however approximates an insulating top boundary. 

The critical Rayleigh number in this case is then R, = 669 (Nield 1964). 

Surface-tension-driven convection is governed by the Marangoni number, which 

is 

dS ATd 

dT ~VK ' 

Ma=-- (3) 

where dS/dT is the variation of the surface-tension coefficient with temperature, d 

is again the depth of the fluid, and p is the density of the fluid. The critical Marangoni 

number Ma, with an insulating top boundary is Ma, = 79.6 (Nield 1964). 

Table 2 gives, for different fluid depths, the values of the temperature difference 

for the appearance of the first pattern AT,, the temperature difference for the 

appearance of the hexagonal cell pattern AT,, and the corresponding Rayleigh and 

I 

i ! 

I 

I 
*If, 14 *!I, 

d AY; AT, 

so. (in m ) (Y1) ("e) 4 

( 100 CH) 

I 2.57 - 5.98 - - 91.5 76.8 

2 1.81 6.68 6.68 35.9 61 .o 35.9 61.0 

3 1.31 4.12 7.92 8.21 26.4 16.4 52.9 

4 O.!)S 3.86 11.3 2.73 17.5 8.73 56.2 

5 0.72 2.0!) 12.4 0.67 7.17 4.49 48.2 

(50 CS) 

6 1.34 2.88 5.16 13.8 37.0 25.4 68.0 

7 (1.93 2.74 6.82 4.40 24.5 11.5 63.8 

'I'ABI.~: 2. Temperature difference and non-dimensional numbers of the pattern formation 

Marangoni numbers. The lower part of table 2 gives the same parameters for the 

experiments with the 50 cs oil. 

In agreement with our visual observations and with the heat transfer measurements 

(figure 4) we have set the temperature difference for the onset of the first pattern in 

the 1.81 mm deep layer equal to the temperature difference for the appearance of the 

hexagonal pattern. Actually, the first pattern forms, with this fluid depth, just before 

thc hexagonal pattern, but within the experimental error the temperature differences 

for the formation of both patterns coincide. As can be seen in table 2, the temperature 

difference at which the first pattern forms decreases systematically with decreased 

fluid depth, while the temperature difference for the formation of the hexagonal 

pattern increases with decreased depth. In the 0.72 mm deep layer the Rayleigh 

number for onset of the first pattern is < 1, the Rayleigh number for onset of the 

hexagons is only 4.5. The Marangoni number for onset of the first pattern in the 

0.72 mm deep layer is only 7, i.e. much smaller than the critical Marangoni number. 

On the othcr hand the Marangoni numbers for onset of the hexagonal pattern are 

near the theoretical critical Marangoni number. 

The values of the Rayleigh number as well as the Marangoni number for the onset 

of both patterns in the 50 cs oil are in each case larger than the corresponding values 

for the 100 cs oil. The differences are too large to be explained by the experimental 

uncertainties. There are, on the other hand, not enough data to decide whether or 

not there is a systematic shift to higher non-dimensional numbers with the 50 cs oil. 

We emphasize that the visual observations with both oils gave completely analogous 

rcsul ts. 

Wc note that the uncertainty in the values of the Itaylcigh numbers is of the order 

of 10(Yo. because AT is known only to about +5%, as discussed before. The depth 

d is known to about 1 so d3 is known to about 3 76, v and K are both known to 1 "/. 

The uncertainty in the values of the Marangoni numbers is also of the order of 10 %. 

We have measured the value of dS/dT. The tensiometers used to determine 8 are 

accurate to only 1 %. On the other hand, dS/d9' is quite small, so the error in thc 

detcrminat ion oftfS/dTis much larger. We have measured dS/dT before (Koschmiedcr 

1967) and found for the 100 cs silicone oil dS/dT = 0.058 dync/cm OC. Our present 

valuc. dS/dT = 0.050 dyne/cm "C is slightly smaller, but the diffcrcncc is probably 

primarily the systematic uncertainty in determining tlS/dT. We do not claim an 

*-, 

t ,I 

PLY 167 :1 
60 

0. I 

5 

0 

01 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

1 

-2 I 

03 

03 04 

,s 04 05 -b 

MIM, 

FIGURE 8. Comparison of the normalized Rayleigh and Marangoni number for the onset of 

convection of the two patterns. Open circles indicate first pattern, full circles indicate hexagonal 

pattern. The numbers refer to the different layers listed in table 2. The dashed line shows the 

expected onset of surface-tension-driven convection. 

accuracy better than f 5 yo. The uncertainty in the Marangoni number is of the order 

of 10 yo, because 5 yo comes from dS/dT, 1 06 each from v and K and 5 "A from AT. 

For comparison with theory, in particular with figure 1 in Nield (1964, we divide 

the Rayleigh numbers and Marangoni numbers in table 2 by the corresponding 

critical numbers. However, the critical values of the numbers for the insulating free 

surface cannot be used now. The upper thermal boundary condition is taken into 

account by Nield via the parameter L, which is given here by L = (dNu/dT) ATp. The 

values of dNu/dTfollow from figures 4-7. The value of L for the five fluid layers listed 

in table 2 is I, = 0.19f0.02. The critical Marangoni number with I, = 0.19 is, 

according to Nield, Mc = 86.8, the critical Rayleigh number is then R, = 693.5. Using 

these critical numbers we arrive at the points shown in figure 8. The dashed line in 

figure 8 shows the curve for the onset of convection in the presence of buoyancy and 

surface-tension effects. The critical curve can be written as R/Rc + M/Xc x 1. As we 

see in figure 8 the measured points for the onset of the hexagonal pattern are near 

the critical line. The points deviate from the expected value to lower values of M/Mc 

for the fluid layers with smaller depth. This seems to be a systematic effect. We 

speculate that this is caused by finite amplitude effects of the flow in the first pattern. 

Before the formation of the hexagons the motion in the first pattern seems to become 

so vigorous that the hexagons form prematurely, before one would expect the 

hexagons to form in a resting fluid layer. The vigorous motion in the first pattern 

is clearly documented by the increase of the convective heat transfer before the 

formation of thc hexagons (figures 4-7). 

The points in figure 8 which mark the onset of the first pattern are clearly 

subcritical from the point of view of the theory of convection caused by the 

surface-tension gradient dS/d'l'. As the thickness of the fluid layer increases the 
5 

Surfuce-tension-drilen NP'nard convection 61 

ditli~rcnw in thv Marangoni number of the onset of the two patterns decreases, until 

at rirounrl 2 mm depth (with this fluid) the hexagonal pattern becomes the only 

pal 1c.m to appcw at all. On the other hand, the tendcncy of the first pattern to appear 

at (w(*I' srnallcr trmp<iraturc, gradients, or Rayleigh and Marangoni numbers, is quite 

ohvious. 

4. Discussion 

'I'hc principal result of these experimcnts is the discovery of the unexpected first 

piit tcrn. In ordcr to observe this pattern it is necessary to work with fluid layers of 

smid1 depth, in our case smaller than 2 mm. Small depth was involved in &nard's 

obscv-vation that his cells formed at extraordinarily small Rayleigh numbers and is 

also a rcquiremcnt for surface-tension forces to have their full impact. 

It has been suggested that there may be three reasons for the theoretically 

uncbxpcctcd formation of the first pattern, namely 

(u) an experimental imperfection; 

(b) a subcritical instability; 

(c) an instability not contained in the conventional theories. 

An experimental imperfection would be a deviation of the experimental set-up from 

the case studied in conventional theories, that is from the case of a plane motionless 

fluid on an infinite horizontal plate heated uniformly from below and cooled 

unifbrmly from above through the upper surface which is subject to surface tension 

effects. 

In our apparatus such an imperfection is the meniscus of the fluid layer at the wall, 

whicsh is not considered in the theories. The meniscus is a natural phenomenon 

occurring whcnever a fluid layer is in a container. In reality the meniscus seems to 

poso more of a problem than one would expect. We have observed that the silicone 

oil moves up along the rim as a thin film to a height of over 1 cm if the applied 

tcmpature difference is large, probably a consequence of the vertical temperature 

gradicbnt in the rim. However, the meniscus cannot be the cause of the first pattern 

berausc one ran arrange a fluid layer without a meniscus and still observe the first 

pattern. The meniscus was eliminated by containing the fluid with a thin lucite ring 

(of 102 mm diameter and about 1.4 mm depth). Fluid was put into the bowl formed 

by the copper bottom plate and the lucite ring so that the level of the fluid was plane 

with the top of the ring. The first pattern formed as usual. 

In connection with the first occurrence of circular cells in our experiments it has 

been suggestcd that the first pattern may be caused by lateral heat loss or other radial 

tcmprature gradients, such as those caused by the misalignment of the top glass plate 

or by the non-uniformity of the temperature of the top glass plate caused by the 

tempwaturc increase of the cooling water. Lateral heat loss in our experimcnts has 

prac.tically been eliminated by surrounding the rim of the fluid with vacuum as 

indicated in figurc 1. But this arrangement makcs the vertical temperature gradient 

in thv rim smaller than the temperature gradient in the fluid because the cooling at 

the rim is applied at a greater height above the copper plate than for the glass plate 

over the fluid layer (figurc 1). At the levcl of the fluid the rim will be warmer than 

the fluid, i.c. there is a horizontal temperature gradient. However, the lurite rim is 

a vc~y poor thermal conductor, while the copper on the bottom of the very thin 

siliconc-oil layer is an excellent conductor. Even from the top of the fluid layer the 

uniform tcrnperature of the copper plate is only about 1 mm away, while the rim is, 

for iI large part of thc fluid, some centimetres away. Furthcrmorc, the thermal 

32 
ti2 E. L. Koschnzieder and M. I. Hippr.sl(i<rf 

cwndrrctivity of wpper is about 2500 times better than the thcrmal c*onduc*tivjty of 

silicwie oil. Thus it would require very large temperature differences bet\\ wn the rim 

and the fluid in order to create an effective horizontal temperature gradieiit in the 

interior of the fluid. Also, there is no mechanism known by which small-scaIt. (.ells 

would be created via a horizontal temperature gradient in a fluid. Finally the 

cxpriment with the fluid contained by a small lucite ring does not support the 

ronwpt of the significance of the lateral temperature gradient either. bccause in that 

set-up the Huid layer is separated from the rim of the apparatus by an air gap of 12 mm 

width. Air is an excellent thermal insulator which should shield the fluid effectively 

from the temperature distribution in the rim. 

To investigate the non-uniformity of the temperature of the glass lid caused by 

the increase of the temperature of the cooling water, we measure this temperature 

increase with the heat sensor. For the extreme ease of the lowest fluid depth (0.73 mm) 

and the largest applied temperature difference across the oil, the temperature increase 

of the cooling water amounted to 3% of the temperature difference across the oil. 

However, the temperature difference in the cooling fluid extends from the centre of 

the lid to the rim of the lid. If one compares the temperature gradients then the ratio 

of the temperature gradient across the lid to the vertical temperature gradient across 

the convecting fluid is about 2 x In order to see whether this has a noticeable 

effect on the formation of the pattern, we increased the flow rate of the cooling water 

from the 20 em3 s-l used when we measured with the heat sensor, to the maximal 

flow rate with this pump which was about 80 em3 s-*. The ratio of the radial to the 

vertical temperature gradient was then 5 x No noticeable consequence of the 

increased flow rate was observed. We do not, therefore, believe that the radial 

temperature gradient in the glass lid is significant for the formation of the first 

pattern. 

The misalignment of the plane of the glass lid with respect to the plane of the fluid 

surface discussed in $3.1 has a potentially greater influence on the uniformity of 

the top temperature of the fluid, than the non-uniformity of the temperature caused 

by the cooling. The consequences of the misalignment of the lid on the onset of the 

hexagonal pattern were obvious in all experiments, see e.g. figure 2 (d). All efforts to 

improve on this situation had little or no success. Misalignment of the lid seemed to 

have little if any effect on the formation of the first pattern, which in all experiments, 

seemed to form essentially in an axisymmetric and not one-sided way. Figure 2 (a, d) 

are from the same experiment and show that there is no apparent relation between 

the symmetry of the formation of the first pattern and the asymmetry of the 

formation of the hexagonal pattern. 

Motion of the air between the fluid and the glass lid might also be the cause for the 

formation of the first pattern. However, the Rayleigh numbers of the air layer are. 

even with the highest temperatures applied, smaller than R,. The prime reason 

for this is the very small depth of the air layer (x 0.5 mm) and the dependence of 

the Rayleigh number on the third power of the depth. It seems to be most unlikely 

that the first pattern is induced by motion of the air layer. 

Finally, another possible experimental imperfection might be that the aluminum 

powder added to the fluid causes the formation of the first pattern. The aluminum 

powder settles gradually and causes a compensating upward motion of the fluid. 

However, one would expect any possible effect of the settling of the powder to be 

smaller in the shallow layers of fluid than in deep layers, because the aluminum 

powder will sc>ttle sooner in the shallow layers and so it seems to be unlike1.v that 

the first pattern would appear preferentially in shallow fluid laycrs. One should also 
SurfutcP-trrision-rr~i?~n Hinard coiir‘ection 63 

expoct an effwt of the concentration of the aluminum particles. The concentration 

of thc particles was varied substantially (wecstimati! by a factor ofat least lo), mainly 

bcc*;iuse of our efforts to take photographs of the flow which requires substantially 

bigger concentrations of thc aluminum powder, but we did not notice any indication 

that this had influenced the appearance of the first pattern. We know of no theory 

thikt would prcdict the formation of a cellular pattcrn on the basis of thc settling 

motion of an impurity in a fluid. Such a settling motion is not related to the diffusive 

two-component Benard convection problem which has been reviewed by Schechter, 

Vclarde & Platten (1974). 

To summarize, wc do not see a convincing reason for believing that the first pattern 

is caused by an imperfection of the experimental set-up. 

Concerning the question of the possibility of a subcritical instability we note that 

a subcritical instability was predicted by Davis (1969). Later, Davis & Homsy (1980) 

and Castillo & Velarde (1982) elaborated on this topic. Of importance in connection 

with this problem is the value of the crispation number C, which is a characteristic 

parameter for the importance of the surface tension coefficient S. C is defined as 

C = ,uu/Sd, wherep is the dynamic viscosity of the Auid. In our experiments the value 

of C is about 5.5 x so the deformability of the surface is very small. This means 

that the basic results concerning the subcritical instability are contained in Davis’ 

(1969) paper. From figure 1 therein, it follows for L = 0 (we have L = 0.2) that the 

subcritical instability ranges from lcla = 57 to Ma = 80 at R = 0. However, in our 

experiments with the 0.72 mm deep layer the first pattern appears according to 

table 2 at Ma = 7.2 (with R = 0.7), which is deep in the stable region of figure 1 of 

Davis. So, although according to the energy method a range of instability is predicted 

which is at a lower Marangoni number than the critical value of the instability 

predicted by linear theory, the instability apparent in the first pattern that we 

observe is deep in the stable region according to the energy method. There is also the 

problem that the subcritical instability is probably associated with the appearance 

An explanation of the first pattern can be sought in Scriven & Sternling’s (1964) 

paper. The onset of convection driven by surface tension at ever smaller temperature 

gradients is actually predicted there. There are, however, difficulties. Scriven & 

Sternling predict that the small onset temperature gradients are accompanied by 

increased wavelengths or cell sizes. However, we did not observe in any of our 

experiments a noticeable difference in the size of the cells of the first pattern and the 

hexagonal pattern. The wavelength of the hexagonal pattern is determined by Nield’s 

theory and does not change with decreased depth of the Auid. It can be seen in figure 

2 (d) that there is no significant difference in the cell sizes of the first pattern and the 

hexagonal pattern. \Ve did not make a systematic survey of the cell sizes. but we 

believe that within 10 76 accuracy the wavelength of the first pattern and the 

experimental observation and this aspect of Scriven & Sternling’s results. Also, it 

follows from Smith (1966) and Davis & Homsy (1980) that only very small shifts in 

the critical Marangoni number are expected if the crispation number is as small as 

it is in our experiments. 

Our experimental results for the onset of the hexagonal pattern, are in quite good 

agrecrnent wit,h the basic results of Pearson (1958) and Nield (1964). Xicld’s study 

seems to be particularly relevant bccausc it incorporates gravity, which is present 

in thc laboratory and is of such magnitude that it cannot easily be neglected. Thc 

fact that thc hexagonal pattern appears in thin layers at subcritical Marangoni 

I of the hexagonal pattern, and not a pattern of a different form as we observed. 

i 

I hexagonal pattern are the same. So there seems to be a discrepancy betueen the 

ONGINAL PAGE IS 

OE POOR QUALITY 
64 E. L. Koschmieder and M. 1. Biggrrstafl 

numbers scems to be understandable as a finite-amplitude effect of the first pattern. 

It appears that Pcarson’s and Kield’s theories correctly predict the consequences of 

the presence of the surface tension gradient dS/dT on convcctiw motions. So the 

formation of the first pattern, which appears neither in Pearson’s nor in Sield’s study, 

is not caused by the variation of surface tension with tcmperature. 

Help and comments from Professor R. S. Schechter and Professor S. H. Davis are 

gratefully acknowledged. Mr S. Prahl has helped with the visualization experiments. 

lClr J. Gardner has measured the surface-tension coefficients. Support of this work 

through the National Aeronautics and Space Agency and through the Sational 

Science Foundation is also gratefully acknowledged. 

REFERENCES 

BENARD, H. 1900 Rev. Gen. Sci. Pure Appl. 1, 1261-1271, 1309-1328. 

BENARD, H. 1930 Proc. 3rd. Znt. Congr. Appl. iMech. 1, 120. 

BLOCK, M. J. 1956 Nature 178, 650-651. 

CASTILLO, J. L. &. VELARDE, M. G. 1982 J. Fluid Mech. 125, 463474. 

CLOOT, A. & LEBON, G. 1984 J. Fluid iMech. 145,447-469. 

DAVIS, S. H. 1969 J. Fluid Mech. 39. 347-359. 

DAVIS, S. H. & HOMSY, G. M. 1980 J. Fluid Mech. 98, 527-553. 

KOSCHMIEDER, E. L. 1967 J. Fluid Mech. 30, S15. 

KOSCHMIEDER, E. L. 1974 Adv. Chem. Phys. 26, 177-212. 

KOSCHMIEDER, E. L. & PALLAS, S. G. 1974a htl J. Heat Mass Transfer 17, 991-1002. 

KOSCHMIEDER, E. L. & PALLAS, s. G. 1974b Rev. sei. Instrum. 45, 1164-1 165. 

KRASKA, J. R. & SANI, R. L. 1979 Intl J. Heat Mass Transfer 22, 535-546. 

KIELD, D. A. 1964 J. Fluid Mech. 19, 341-352. 

PALMER, H. J. & BERG, J. C. 1971 J. Fluid Mech. 97, 779-787. 

PEARSON, J. R. 1958 J. Fluid Mech. 4. 489-500. 

RAYLEIGH. LORD 1916 Phil. Mag. 32. 529-546. 

ROSENBLAT, S., DAVIS, S. H. & Hows~, G. hI. 1982a J. Fluid Mech. 120, 91-122. 

ROSENBLAT, S., HOMSY, G. M. & DAVIS. S. H. 19826 J. Fluid Mech. 120, 123-138. 

SCANLON, J. W. & SEGEL, L. A. 1967 J. Fluid Merh. 30, 149-162. 

scHEcHTER, R. s., VELARDE, M. G. & PLATTEX, J. K. 1974 Adc. Chem. Phys. 26, 265-301 

SCHMIDT, R. J. & MILVERTON, S. W. 1935 Proc. R. Soc. A152, 58G594. 

SCRIVEN, L. E. & STERNJJNQ, C. V. 1964 J. Fluid Xech. 19, 321-340. 

SILVESTON, P. L. 1958 Forsch. Ing. Il’es. 24, 2S32, 59-69. 

SMITH, K. A. 1966 J. Fluid Mech. 24. 401414. 
1 

Onset of Rayleigh-Benard Convection in Thin Fluid Layers 

E. L. Koschmieder and J. R. Campbell 

College of Engineering and I. Prigogine Center 

for Statistical Mechanics 

The University of Texas, Austin, Texas 78712, U.S.A. 

The onset of Rayleigh-Benard convection in thin fluid layers 

heated uniformly from below has been studied experimentally. Subcritical motions have been found to occur if the depth of the fluid was 

smaller than 2 mm. The thinner the fluid layer was, the larger was 

the range in subcritical temperature differences over which the subcritical motions could be observed. Large temperature gradients are 

required to reach the critical temperature difference in the thin 

fluid layers. The variation of viscosity with temperature then make 

the fluid layer non-Boussinesq. The onset of convection in nonBoussinesq layers was found to take place in the form of hexagonal 

cells which transform at slightly supercritical temperature gradients 

into rolls, in agreement with theoretical predictions. Onset of 

Rayleigh-Benard convection under non-Boussinesq conditions was found 

to occur at Rayleigh numbers larger than the conventional critical 

Rayleigh number Rc = 1708. 

1. Introduction 

In an earlier investigation of convection in thin fluid layers 

under an air surface (Koschmieder and Biggerstaff, 1986) subcritical 

convective motions appeared prior to the formation of a hexagonal cell 

pattern at the critical Marangoni number, if the depth of the fluid 
2 

layer was less than 2 mm. This observation posed the question for the 

cause of these subcritical motions. This is an important question 

because it is one of the elementary tenets of the theory of 

hydrodynamic stability that the fluid remains completely at rest 

before the critical condition for onset of convection is reached. 

Experiments with convection under an air surface introduce a number of 

difficulties and uncertainties about the conditions and surface 

tension gradients at the fluid surface. It might have been one of 

those difficulties which caused the premature motions in the fluid. 

However, these uncertainties can be eliminated by putting a lid into 

contact with the fluid, thereby changing the experiment from a surface 

tension driven Benard convection experiment to a Rayleigh-Benard 

convection experiment in which the driving force is not surface 

tension gradients but buoyancy. We will see that, contrary to 

expectations, subcritical convective motions appear also in RayleighBenard convection, provided that the fluid layer is sufficiently thin. 

Working with thin fluid layers in Rayleigh-Benard convection requires large vertical temperature gradients since the critical temperature gradient is inversely proportional to the third power of the 

depth of the fluid. Large temperature gradients tend to make the 

problem non-Boussinesq, because of the variation of the viscosity of 

the fluid with temperature. The consequences of the variation of viscosity with temperature on the onset of Rayleigh-Benard convection 

have been studied in a number of papers, beginning with Palm (1960). 

Palm predicted the formation of a hexagonal pattern at a temperature 

difference smaller than the critical temperature difference of the 

Boussinesq case. The theoretical work on the consequences of the 
3 

variation of viscosity with temperature was continued by Stuart and 

Segel (1962), Palm, Ellingsen and Gjevik (19671, Busse (1967) and 

Davis and Segel (1968). These studies revealed that convection in 

variable viscosity fluids should start at subcritical values of the 

Rayleigh number with a hexagonal pattern which should, however, change 

to a roll pattern at slightly supercritical Rayleigh numbers. Part of 

this prediction has been confirmed experimentally by Hoard, Robertson 

and Acrivos (1970). They showed, that in a fluid in which the viscosity varied by a factor of about 10 between the top and bottom boundaries, the pattern of the flow at onset of convection was hexagonal 

and appeared at a temperature gradient smaller than the critical temperature gradient for the Boussinesq case. However, no subsequent 

transformation of the hexagonal pattern to a roll pattern was observed. All of the above mentioned studies were concerned with a 

small perturbation of the Boussinesq approximation by the temperature 

dependence of viscosity. 

The onset of convection under highly non-Boussinesq conditions 

with very large variations of the viscosity has been studied by 

Stengel, Oliver and Booker (19821, and Oliver and Booker (1983). 

Using glycerol in their study the viscosity on top of the fluid layer 

differs by a factor of up to 3000 from the viscosity of the fluid at 

the bottom of the layer. Stengel et al. predicted that in the highly 

non-Boussinesq case the Rayleigh number for onset of convection is 

larqer The 

experiments described in Stengel et al. seem to confirm the increase 

of the critical Rayleigh number with increased viscosity ratio, 

although there are substantial experimental uncertainties in their 

than the critical Rayleigh number in the Boussinesq case. 
4 

data, because the critical Rayleigh numbers are not determined from 

Nusselt number measurements. The flow pattern in such experiments is 

described in Oliver and Booker ( 1983 1. There are rolls if the 

viscosity variation is small, and hexagons with large viscosity 

variation. The hexagons change to square cells at higher supercritical 

Rayleigh numbers. 

In the following we will describe experiments studying the onset 

of convection in a conventional fluid (silicone oil), in which the 

viscosity variation is comparatively small. The viscosity variation 

and its consequences are only a secondary topic which appears in our 

experiments as a necessary consequence of the prime objective, which 

is the study of the onset of convection in thin fluid layers. 

2. The Apparatus 

The apparatus was the same apparatus that was used in Koschmieder 

and Biggerstaff (19861, where a sketch of the equipment is shown. In 

- all experiments described in the present paper the lid was in contact 

with the fluid layer. We can list here only the most elementary 

features. The bottom of the fluid layer is a 5 cm thick circular 

copper block, heated electrically from below. The lateral confinement 

of the fluid is a lucite ring of 13.55 cm inner diameter. The fluid 

layer was cooled from above by water from a constant temperature bath. 

The water flowed either over a sapphire crystal for the visualization 

experiments or over a copper plate for the heat flux measurements. 

The sapphire is the sane as the one used in the experiment of 

Koschmieder and Pallas (1974a), where such a sapphire was used for the 

first time in a convection experiment. The advantage of the sapphire 

is its good thermal conductivity in spite of its transparency. The 
5 

sapphire makes it possible to visualize the entire fluid layer and yet 

to realize closely the (perfect) thermal conduction condition assumed 

in theory at the horizontal boundaries. The sapphire has a diameter 

of 13.335 cm and a thickness of 0.508 cm, its top and bottom surfaces 

are polished. 

The lid, either the sapphire or the copper plate, were supported 

from below by lucite rings of various thickness, all with an inner diameter of 12.38 cm. Since the fluid must be provided with space to 

expand when the fluid is warmed up, the lucite rings were separated 

from the lid by three spacers 0.0025 cm thick. So the fluid had at 

its outside around the circumference a gap of 0.0025 cm depth into 

which the fluid could expand. The axisymmetry of the setup was preserved in this way. For the heat flux experiments the space on top of 

the lid was filled with the heat sensor, a device to measure the increase of the temperature of the cooling water as it passed over the 

lid covering the heated silicone oil layer. From the temperature increase of the cooling water follows the heat flux through the silicone 

oil. The heat sensor is described in detail in Koschmieder and Pallas 

(1974b). 

Flow visualization was accomplished by the shadowgraph technique, 

introduced by Silveston (1958). A beam of white light is shone from 

above through the oil layer, reflected on the bottom of the layer and 

experiences interferences on a screen a distance behind the apparatus. 

The interferences are caused by the different path length of the light 

in the fluid, which vary because of the variation of the index of refraction with temperature. The interference pattern correspond to the 

horizontal variation of the temperature field in the fluid, which is 
6 

caused by the pattern of motion in the fluid. The interpretation of 

shadowgraph patterns has recently been discussed by Jenkins (1987). 

For the shadowgraph technique to work, one needs a reflecting bottom 

of the fluid. but is not 

good enough to serve as a mirror for a shadowgraph. We have therefore 

placed either a chrome plated stainless steel mirror of 0.8 mm thickness or a glass mirror of 3.2 rnm thickness on the bottom of the fluid 

in the visualization experiments. It would have been too expensive tc 

buy a metal mirror of this size with an optical surface. 

3. The Experiments 

3.1 Pattern Formation 

The copper block has a near mirror finish, 

It is necessary to verify the correct functioning of the apparatus under conventional conditions before we go on to study the existence of subcritical motions in Rayleigh-Benard convection. In order 

to prove that the apparatus works properly we have studied the onset 

of convection in a 5 mm deep layer of silicone oil of 1 cm2/sec viscosity. the critical 

temperature difference being 14.5"C and the viscosity ratio vmax/vmin 

= 1.15. But we have observed before (Koschmieder, 1965) that under 

these conditions a pattern of circular concentric rolls formed in a 

circular container, just as the same pattern formed in a 1 cm deep 

layer of the same oil, when the critical temperature difference was 

only 2OC and the conditions were certainly Boussinesq . 

The conditions are then not strictly Boussinesq, 

I? the experiment with the 5 mm deep layer the fluid was heated 

for 4 hours to the critical theoretical temperature difference AT^ = 

14.5OC. At that temperature difference a weak pattern appeared on the 

shadowgraph, on which prior to that AT no pattern was visible at all. 
7 

The pattern at ATc consisted of circular concentric rolls with some 

weak cellular divisions. Thirty minutes later, at a temperature difference of 16.OoC or 1.10 ATc an intense pattern of regular circular 

concentric rolls had formed, as is shown in Fig. la. We have observed 

such circular pattern before in Koschmieder (1965) and Koschmieder and 

Pallas (1974a). It is worth noting that the configuration of the circular lateral confinement of the fluid was different in the case here 

as well as in the two other cases, the differences of the wall configurations being necessitated by the particular purpose of each of these 

experiments. Nevertheless, we have observed perfect axisymmetric patterns in each of these experiments. The bright circles in Fig. la indicate the location of descending fluid motions. There is one additional bright ring just outside of the field of view of the photograph. There are therefore 12 circular concentric rolls in the fluid 

layer in agreement with the aspect ratio of the layer, which is r = 

2r/d = 24.7. When the temperature difference across the fluid was 

slowly increased to higher supercritical values, the diameter of the 

center roll shrunk (Fig. lb), the width of the bright rings increased, 

and wall effects in the form of rolls oriented perpendicular to the 

wall became apparent. Ten hours after the experiment had started and 

at AT = 2.33 ATc the center roll had been reduced to a very bright 

spot (Fig. IC), indicating the increase of the wavelength of the convective motions with increased supercritical Rayleigh-number, which we 

had first noted in Koschmieder (1965). Having confirmed the essential 

features of Rayleigh-Benard convection in the Boussinesq case in this 

apparatus, we could proceed to experiments with non-Boussinesq 

conditions. 
8 

With a fluid layer of 4 mm depth and the silicone oil of 1 

cm2/sec viscosity the onset of convection is expected at AT = 25.OoC. 

The viscosity ratio is then Vmax/Vmin = 1.29, meaning that the 

conditions are no longer Boussinesq. Onset of convection in the 

experiment occurred at AT = 26.25"C in the form of a circular pattern 

with a superposed cellular structure, see Fig. 2a. Prior to this AT 

no pattern was visible on the shadowgraph. Increasing the temperature 

difference by 1°C to AT = 1.05 ATc eliminated the cellular structure 

completely (Fig. 2b). The events were qualitatively the same as with 

the 5 mm deep layer only that the cellular pattern at onset of 

convection was much more conspicuous and a little more persistent. In 

Fig. 2b there are more bright rings than in Fig. la because the aspect 

ratio of the fluid layer is different, being r = 30.95 with the 4 mm 

deep layer. There is then space for 15 concentric rolls with sinking 

motion in the center of the layer. The sinking motion there appears 

as the bright center point in Fig. 2b. The rays which go out to the 

right and left from the fluid in the photograph Fig. 2b originate from 

an optical distortion caused by the lucite wall of the lid. The 

sequence of events shown in Fig. 2a, 2b is in qualitative agreement 

with the theoretical prediction that in a slightly non-Boussinesq 

fluid the onset of convection occurs in the form of hexagonal cells 

which transform to a roll pattern if the temperature gradient is 

increased slightly. The occurrence of this transformation has, to our 

knowledge, not been shown before. We will discuss the quantitative 

aspects of this problem when we come to the heat flux measurements. 

The experiments of Koschmieder and Biggerstaff (1986) showed that 

subcritical motions in surface tension driven convection appear only 
9 

if the layer is sufficiently thin, and that the range of subcritical 

flow increases with decreased depth of the fluid. We looked therefore 

for the appearance of subcritical flow in Rayleigh-Benard convection 

in fluid layers of about 2 mm depth or less. It is then necessary to 

use oil of 0.10 cm2/sec viscosity, because with the 1 cm2/sec silicone 

oil the critical temperature difference would be about 200°C, which is 

impractical. With a 2.10 mm deep layer, the 0.10 cm2/sec silicone 

oil, a glass mirror at the bottom, and the sapphire lid, a weak subcritical pattern of ill defined small scale rolls appeared on the 

shadowgraph at a temperature difference of 15OC. When the temperature 

difference was increased the pattern became a little more intense, and 

transformed at 18.75OC rapidly into a very intense cellular pattern. 

This marked the critical condition. We will show later that the 

appearance of the strong cellular pattern coincides with the break in 

the heat flux curve, which has traditionally been considered the proof 

for onset of convection. Since the range of subcritical flow is 

small, the results of this experiment may be inconclusive and we will 

proceed to the experiments with the smallest fluid depth tried. 

First, however, we add a comment concerning the use of the glass 

mirror in these experiments. The glass mirror is undesirable because 

of the thickness of the glass (3.2 mm) and its comparatively poor 

thermal conductivity whose accurate value we do not know. The glass 

mirror has, however, the advantage of a far more uniform reflectivity 

than the chrome plated steel mirror. Looking at a shadowgraph of a 

fluid layer at rest above the steel mirror one sees a grainy (black 

and white) texture; while the glass mirror under the same conditions 

shows a uniform bright image. The texture on the image of the steel 
10 

mirror makes the steel mirror unsuited for the observation of subcritical motions whose shadowgraph image is so weak that it cannot be distinguished from the texture of the steel mirror. If, on the other 

hand, the flow is critical or supercritical, then the interference 

pattern of the shadowgraph is so strong that the texture of the mirror 

cannot be seen anymore, as is, for example, the case in Figs. la-c and 

2a-b. We have, however, made certain that the subcritical motions to 

be discussed in the following are not a consequence of the glass mirror. One can indeed see subcritical motion on the steel mirror, but 

its pattern is superposed on the image of the texture of the mirror, 

and this combination of two images is useless for the determination of 

the form of the subcritical flow. Pictures of the shadowgraph of subcritical flow can be taken only after all cooling fluid on top of the 

sapphire has been removed, because even a few millimeters of water on 

top of the lid wash out the interference of the light beam. It takes 

about half a minute to remove the heat sensor on top of the sapphire 

as well as all the cooling fluid, take a photograph and putting the 

heat sensor back in place. If the fluid underneath the lid is 

critical or supercritical water on top of the lid does not interfere 

with the shadowgraph, although fringe effects, such as the rays on 

Fig. 2b, increase. 

With a fluid depth of 1.63 mm, the 0.10 cm2/sec silicone oil, the 

glass mirror on the bottom of the fluid, and the sapphire lid, subcritical flow appeared at a temperature difference of 17.5OC = 0.45 

ATc. The flow was in the form of a very weak pattern of small-scale 

rolls. The shadowgraph was so weak that a photograph could not be 

taken. We note that the subcritical motion appeared at a much lower 

, 
11 

value of AT/ATc than the value at which they had appeared with the 

2.10 mm deep layer (AT/ATc = 0.8). This is analogous to our observations with surface tension driven convection (Koschmieder and Biggerstaff, 1986). We also note that the appearance of the subcritical 

flow does not seem to be a critical phenomenon. With increased AT 

subcritical flow apparently develops continuously from an imperceptible to a barely perceptible flow. 

Increasing the temperature difference across the fluid increased 

the intensity of the shadowgraph a little. At AT = 28.75OC = 0.74 AT^ 

the first photograph was taken (Fig. 3a). One sees very weak, 

somewhat irregular short rolls which cover the entire plate. 

Increasing the temperature difference further intensified the subcritical motions only slightly, at AT = 32.5OC = 0.84 AT, the flow looked 

like Fig. 3b. The pattern had changed very little as compared with 

Fig. 3a. At AT = 35.OoC = 0.90 ATc the pattern (Fig. 3c) had a much 

more cellular structure than before, but was still subcritical. At AT 

= 38.75OC ATc a very strong intensification of the image of the 

shadowgraph occurred, see Fig. 3d. The pattern consisted then of a 

very large number of hexagonal cells with one small patch of forming 

rolls. This pattern marked the critical condition. The temperature 

difference AT = 38.75OC is the temperature difference between the 

copper block and the cooling fluid on top of the sapphire. The 

temperature drop on the glass plate is therefore included in the value 

of AT. The heat flux measurements show that the critical temperature 

difference with copper-copper boundaries of the fluid is actually 

30.8OC. The viscosity variation of the fluid at this temperature 

difference is Vmax/vmin = 1.68, so the conditions are non-Boussinesq. 
12 

Onset of convection in the form of hexagonal cells fits the 

theoretical description of the consequences of weak non-Boussinesq 

conditions, as well as our observations with the 4 mm deep fluid 

layer. 

Increasing the temperature difference across the fluid to supermitical values brought about a rapid transformation of the hexagonal 

cell pattern to a pattern of rolls. At AT = 4OoC = 1.03 ATc the patch 

of forming rolls had become unquestionable rolls (Fig. 3e), and at AT 

= 42.5OC = 1.1 AT, (Fig. 3f) the rolls had spread over most of the 

layer. Out of concern for the sapphire the temperature difference was 

then not increased further. The quite obvious transformation of the 

hexagonal cells to rolls is again in agreement with theoretical 

expectations for the weakly non-Boussinesq case. The non-Boussinesq 

effects are now so strong as to overcome the, by design, weak effects 

of the lateral wall. So in spite of being contained by a circular 

boundary the pattern in Fig. 3f is not circular and concentric as it 

was in Fig. 1 and Fig. 2. 

The prime result of this experiment is the observation of the existence of subcritical flow in Rayleigh-Benard convection over a large 

range of subcritical Rayleigh numbers. As a by-product of this experiment we have learned again that the onset of Rayleigh-Benard convection in a non-Boussinesq fluid layer takes place in the form of a 

pattern of hexagonal cells which change into rolls at slightly supercritical Rayleigh numbers. 

3.2 Heat Flux Measurements 

Since the experiments of Schmidt and Milverton (1935) heat transfer measurements have frequently been used in order to determine quan- 
13 

titatively the onset of convection. The onset of convection follows 

from the so-called "break" in the heat transfer curve. The break of 

the heat transfer curves occurs at that AT at which thermal conduction 

through the fluid layer by the molecular conductivity of the fluid is 

supplemented by heat transferred through the motion of the fluid after 

onset of convection. The heat transfer curves are usually plotted 

using the Nusselt number Nu as ordinate. The Nusselt number is the 

ratio of the actual heat flux to the heat flux caused by thermal 

conduction only, in a formula 

Qconv + Qcond 

Nu = 

Qcond 

Before onset of convection we have Nu = 1 for all values of the temperature difference applied to the fluid layer. In the experiments 

described in the following the value of the Nusselt number was determined using the formula 

(ATr - COrr) A2 cP Nu = - 

A AT A (1 + % AlucAluc/h) 

where Cp is the specific heat of the cooling fluid (water), m is the 

mass rate of flow of the cooling fluid, A is the area of the fluid 

layer,and % AIuc is the area of the lucite ring covered by the copper 

lid divided by the area of the fluid layer. ATr is the temperature 

difference between the inlet and outlet of the cooling fluid in the 

heat sensor, and Corr is a correction to ATr accounting for 

dissipation of the cooling fluid in the heat sensor. This correction 

is very small, namely of order of 2*10-30C, but quite important for 

the value of Nu if the temperature difference applied to the fluid is 

small and consequently ATr is small. The product Cp rn (ATr-COrr) is 
the measured value of Qconv+Qcond in formula (1). AT in equation (2) 

is the temperature difference across the fluid layer, A the thermal 

conductivity of the fluid, Az the depth of the fluid layer and A the 

area of the fluid layer. The product XATA/AZ is Qcond of the fluid 

layer. The term in parenthesis in the denominator of equation (2) is 

a correction necessitated by the heat flux through the lucite ring on 

which the copper lid rests. This correction is of order of % Aluc = 

525, because the area covered by the copper lid is 5% of the area of 

the fluid layer. The thermal conductivities of the silicone oil and 

the lucite are fairly similar, AIuc being larger than the X of 

silicone oil. We use AI,, = 4.17 cal/gm cm sec, which is certainly 

not more accurate than + 10%. We have no information about the 

variation of AlUc with temperature. The values of the thermal 

conductivities of the two silicone oils is given later on in equations 

5a-b. The parenthesis in the denominator of (2) has then the value 

1.056 for the experiments with the 1.0 cm2/sec silicone oil, and the 

value 1.069 for the experiments with the 0.1 cm2/sec silicone oil. 

The sum of the uncertainties of all the terms in equation (2) determines the total uncertainty of the Nusselt number, which is about 

3% in our experiments, but can be much larger at small values of AT 

because of the correction to AT by the dissipation in the heat sensor. 

The main contributor to the 3% uncertainty is the experimental uncertainty in the determination of AT which has, around the critical 

tmperature difference, a standard deviation of - + 1%. Another percent 

of the principal uncertainty originated from the uncertainty of the 

value of the thermal conductivity A, which is known, at best, with 1% 

accuracy. 
15 

Of the heat flux curves determined we will show here those for 

the two deepest fluid layers (Fig. 4-51, and those for the two 

thinnest fluid layers (Fig. 6-7). The curves for the deep layers were 

made to verify the correct functioning of the apparatus. Fig. 4 shows 

a practically abrupt break of the heat flux in the 0.501 cm deep layer 

of 1 cm2/sec silicone oil at a critical temperature difference of 15.2 

- + 0.1OC. The same can be seen on Fig. 5 showing the heat transfer in 

the same oil but in a 0.400 cm deep layer. The rounding of the break 

at AT, = 26.2 2 0.2OC is a little more pronounced than in Fig. 4. 

Other than perhaps through the rounding of the Fig. 5 offers 

no clue that the motion of the fluid changed from a hexagonal pattern 

to a roll pattern near onset of convection. The breaks of the heat 

transfer curves on Figs. 4 , 5 occur at values of AT which agree with 

the critical temperature differences expected from the visual observations. Fig. la was taken at AT = 16.0°C and is already supercritical. 

The very first motions observed in this experiment on the shadowgraph 

occurred at AT = 14.SoC, slightly below ATc on Fig. 4. The critical 

temperature difference on Fig. 5 is ATc = 26.2OC, in good agreement 

vJith AT = 26.25OC at which Fig. 2a was taken. 

The Nusselt numbers below the critical temperature difference in 

Figs. 4,s are not equal to one, as one expects, but slightly smaller, 

increasing gradually from Nu = 0.96 to Nu = 1. This variation is either an artificial consequence of the value of the correction for the 

dissipation in the heat sensor, or a consequence of the initial 

thermal conditions of the apparatus. The correction for the 

dissipation in the heat sensor has been determined separately in long 

runs in which AT = 0 was approached very slowly. At AT = 0 the heat 
16 

sensor records the temperature difference generated by dissipation in 

the cooling fluid. This is a very small signal amounting to 

0.0O22S0C. This value fluctuates by up to 30%, the fluctuations being 

caused by fluctuations of the cooling water temperature and by 

electrical noise in the laboratory affecting the emf of the thermopile 

in the heat sensor. Choosing a different (smaller) value of Corr 

makes it possible to plot the points for AT < ATc on a line straddling 

the value Nu = 1. If dissipation in the heat sensor is not considered 

(Corr = 0), then the measured points approach Nu = 1 on a line which 

is above Nu = 1. For consistency we use in Figs. 4-7 the same 

correction (Corr = 0.00225) since the dissipation in the heat sensor 

should be independent of the type and depth of the oil in the fluid 

layer. 

Another point to be considered is whether or not the fluid layer 

was in thermal equilibrium. for the 

experiments with the 1 cm2/sec oil is about 250 sec, the relaxation 

time for the 0.1 cm2/sec oil is about 50 sec. The heating was increased all 15 minutes so that the temperature increase was 50pV, 

which corresponds to 1.25OC. So in the 1 cm2/sec oil more than three 

relaxation times, and in the 0.1 cm2/sec oil more than 10 relaxation 

times elapsed between successive heatings. This should assure thermal 

equilibrium. Actually since the 0.501 cm deep 1 cm2/sec silicone oil 

layer needed the most time for thermal relaxation, the time interval 

for heating was increased to 20 minutes in this case. 

The thermal relaxation time d2/, 

In the thin fluid layers with the 0.1 cm2/sec silicone oil the 

heat transfer curves (Fig. 6-7) have breaks at AT = 21.2 2 0.2OC (Fig. 

6) and at AT = 30.8O 2 0.2"C (Fig. 7). The curve in Fig. 7 corre- 
17 

sponds to the sequence of pictures shown in Fig. 3a-f, in which experiment, however, the measured temperature differences are larger than 

on Fig. 7 because of the glass mirror on the bottom of the fluid, 

while the AT'S in Fig. 7 have been determined with copper-copper 

boundaries. Other than the small rounding of the breaks there is 

again no clue in these two figures that the pattern changes from 

hexagons to rolls above hTc. Fig. 7 also provides no clue for the existence of the extended range of subcritical motion which we have 

shown in Figs. 3a-c. The very small increase of Nu between AT = 15°C 

and ATC in Fig. 7 seems to be well within the standard deviation of 

the measurements, as well as within the uncertainty of the slope of 

the subcritical section of the heat flux curve caused by the uncertainty of the value of the correction for the dissipation in the heat 

sensor. Without this correction the subcritical heat flux data are on 

a line approaching Nu = 1 from above. The failure to pick up the subcritical flow in the heat transfer measurements really does not come 

as a surprise since the motions were so weak that it was difficult to 

visualize them. The subcritical motions are obviously so slow that 

the amount of additional heat transferred by them is too small to be 

measured with our equipment. We have, likewise, not been able to pick 

up a signal for the increase of the heat transfer by subcritical rolls 

induced by the lateral wall in earlier Rayleigh-Benard convection experiments (Koschmieder and Pallas, 1974). 

3.3 Critical Rayleigh Numbers 

From the critical temperature differences determined by the heat 

flux measurements follow the critical Rayleigh numbers, using the formula 
18 

agATcd3 

- 

VK 

Rc - 

(3) 

where a is the volume expansion coefficient of the fluid, g the acceleration of gravity, ATC the critical temperature difference, d the 

depth of the fluid, v the kinematic viscosity, and K the thermal diffusivity of the fluid. The value of the critical Rayleigh number of a 

Boussinesq fluid layer on an infinite plane is Rc = 1708. Determining 

the value of Rc in our experiments is important so that we can compare 

our results with the theoretical predictions about the onset of 

convection in non-Boussinesq fluid layers. 

In order to use (3) one has to know the material properties a,, v 

and K of the fluid, and their dependence on temperature. The volume 

expansion coefficients of both fluids used (Dow Corning, 200 Fluid) 

are independent of temperature over the range of temperature 

considered, and have according to the manufacturer the values a = 

0.00096 ("C)"l for the 1 cm2/sec oil, and a = 0.00108 ("C)'l for the 

0.1 cm2/sec oil. The kinematic viscosity is the property that changes 

most with temperature. According to McGregor (1954) the kinematic 

viscosity of the silicone oils has the smallest temperature slope of 

all organic oils. The relationship for the decrease of v with 

temperature is 

log v = 722.5 + 0.00032 VI + 1.004 log VI - 2.447 

*m *m 

where Tm is the mean temperature of the fluid layer and VI the viscosity of the fluid at 25°C. For the 1 cm2/sec silicone oil the viscosity varies from 1.0 cm2/sec at 25°C to 0.66 cm2sec at 50°C mean temperature. For the 0.1 cmo/sec oil the viscosity varies from 0.1 

cm2/sec at 25°C to 0.062 cm2/sec at 50°C mean temperature. 
19 

The viscosity of the fluids which we used have been checked by us 

because the fluids had been stored in the laboratory for a couple of 

years. For the determination of the viscosity of the (nominal) 1.0 

cm2/sec oil we used a Contraves Low Shear viscometer, as well as a 

Brookf ield Syncroelectric viscometer. For the nominal 0.1 cm2/sec oil 

we used three viscometers, the Contraves, the Brookfield, and the capillary Cannon-Finske viscometer. The results of these viscosity measurements were disappointing, there were significant differences in 

the values of the viscosity of one and the same fluid at the same 

temperature when measured with different instruments. There was a 

deviation of about 10% between the lowest and highest value of the 

viscosity. To eliminate operator error the viscosity measurements 

have been made independently by two people, the consistency of their 

results was excellent being within less than 1%. A decision had to be 

made which of the measured viscosities was most likely correct. This 

decision was based on the assumption that the instrument that measured 

brand new silicone oil of nominal 1.0 cm2/sec viscosity at 1.00 5 0.01 

cm2/sec was indeed working correctly. The viscosity of the old 

nominal 1.0 cm2/sec oil was then determined to be v = 1.079 5 0.005 

cm2/sec at 2S°C, the error quoted being the standard deviation of the 

measurements, not the principal uncertainty. For the 0.1 cm2/sec oil 

the Contraves and Brookfield viscometers use spindles different from 

those used with 1.0 cm2/sec fluids, which make the consideration above 

irrelevant. For this viscosity measurement the capillary viscometer 

was used as the norm because a calibration fluid was available for 

this instrument. The reading of the instrument agreed within 1.5% 

with the viscosity of the calibration fluid. The viscosity of the old 
20 

0.1 cm2/sec (nominal) silicone oil was so determined to be 0.102 f 

0.001 cm2/sec at 25OC, the error being again the standard deviation of 

the measurements with this instrument. 

The thermal diffusivity K follows from the formula K = A/pcp 

where A is the thermal conductivity, p the density and cp the specific 

heat. Values of h have been given by the manufacturer at 50°C and 

8OoC, from which one has to extrapolate to room temperature. In the 

range of temperatures considered we have used linear approximations 

which are 

A = (4.325 - 0.013 Tm)=10-4 cal/cm sec OC (5a) 

and A = (3.2233 - 0.00166Tm)*10-4 cal/cm sec "C (5b) 

for the 1.0 cm2/sec oil and for the 0.1 cm2/sec oil respectively. 

There is certainly a principal error in these values of order of 1% or 

even more. We will see that the thermal conductivity can, however, be 

assessed from the subcritical part of the heat transfer curves, and 

that within about 1% the thermal conductivities given in (5a), (5b) 

are correct. The density p of the fluid varies linearly with the expansion coefficient, for the 1.0 cm2/sec oil at 25°C it is for example 

p = 0.960 gm/cm3, and for the 0.1 cm2/sec oil p at 25°C is p = 0.934. 

The specific heat cp is also a linear function of temperature. For 

the 1.0 cm2/sec oil cp is for example cp = 0.381 cal/gm at 25OC, and 

for the 0.1 cm2/sec oil it is cp = 0.392 cal/gm at 25°C. The value of 

K of the 1.0 cm2/sec oil is then K = 0.00109 cm2/sec at 25"C, and for 

the 0.1 cm2/sec oil K = 0.30087 cm2/sec at 25°C. 

The cumulative uncertainty in the experimental value of Rc is de- 

 AT^ is deterThe intersection of the subcritical 

termined mainly by the uncertainty of ATc, d3, v and K. 

mined from the heat flux curves. 
21 

part of the heat flux curves with the supercritical part of the heat 

flux curves is certainly not known with an error less than 5 1%, which 

is also the average magnitude of the standard deviation of the measurements of the Nusselt numbers of the points which make up the heat 

flux curves. It is therefore optimistic to ascribe to ATc an error of 

only 5 1%. We measure the depth of the fluid layer with an accuracy 

of about 10'2mm, so the uncertainty of d3 is on the average, about - +

1%. We believe that the uncertainty in v is of order of 2%, if not 

more. The uncertainty in the value of K, which involves the cumulative uncertainty of A, p,and cp, is also of order 2%. The cumulative 

uncertainty of Rc is then of order of fi 6%. 

The values of Rc determined from the measured AT^ are listed in 

Table 1, together with the depth of fluid layers and the critical temperature differences. The depth 0.210 cm is listed twice, and the 

second case is marked c. This refers to control experiments done at 

this depth with a brand new sample of silicone oil of 0.1 cm2/sec 

viscosity. These control experiments were done because of the 

uncertainty of the value of the viscosity of the fluids. Table 1 also 

has a column listing values of At/X, which column deals with the 

thermal conductivity of the fluids as it follows from the subcritical 

part of the heat flux curves. This item will be discussed first. 

As mentioned before in connection with equation (2) the heat conducted through the apparatus by the resting fluid is proportional to 

the term (1 + % Ahluc/A). This term is the ratio of the total thermal 

conductivity At of the fluid and the lucite rim to the conductivity A 

of the fluid only. The experimental value of At/A follows from the 

requirement that the Nusselt number should be equal to one in the sub- 
22 

critical part of the heat flux curve or at least equal to one just before ATc. The values of At/A necessary to satisfy this requirement 

are listed in Table 1. The following column lists the deviation A of 

the experimental values from the values to be expected from (1 + % 

AAluc/A). As mentioned in context with equation (2) the term At/h = 

(1 + % AAluc/A) is = 1.056 for the 1 cm2/sec oil, and = 1.069 for the 

0.1 cm2/sec oil. The actual value of At/A varies slightly because A 

is a function of temperature, as is Aluc; but this is of little 

significance. As can be seen in Table 1 the deviation A is in most 

cases positive. A positive deviation means that more heat is 

conducted than expected. This can be caused by a thermal conductivity 

of the fluid which is larger than the conductivity we have used in the 

calculation of Xt/A. Note, however, that then all layers with the 

same fluid are affected in the same way. A positive A can also be 

caused by additional heat being conducted to the copper lid by 

portions of the lucite rim not being covered by the lid, in other 

words, by an apparent increase of % A. A positive A can, furthermore, 

be caused by a fluid layer with a depth smaller than the depth used in 

the calculations; the depth of the layer is not known with an accuracy 

better than fi 1%. Negative deviations A mean that the heat flux is 

smaller than expected, which can be caused by a fluid depth larger 

than the depth we have used. This seems to be most applicable to the 

experiment with 0.163 cm depth. In all, the deviations of ht/A are 

not more than 5 2% and confirm with that accuracy the thermal 

conductivities given by equations 5 a-b. This is important because it 

means that the critical Rayleigh numbers cannot be affected by more 

than f. 2% by uncertainties of the value of the thermal conductivity. 
. 23 

Finally, we discuss the critical Rayleigh numbers. The column Rc 

in Table 1 lists the values of the Rc determined from our experiments. 

The errors listed are the 6% of the Rc which are the principal uncertainty of the measurements of Rc as discussed before. As is apparent 

the values of Rc fall into two categories, namely values of around 

1800 f 100 for the two deep fluid layers with the 1.0 cm*/sec oil, and 

values of around 2200 5 130 for the four thinner fluid layers with the 

0.1 cm2/sec oil. The column Rc/Rctheor shows how far the measured Rc 

differ from the theoretical value Rctheor = 1708. While the values of 

Rc for the two deep fluid layers are larger than RCtheor, they are 

within the experimental uncertainty still compatible with Rctheor. 

The conditions in the two deep layers are already non-Boussinesq, the 

degree of viscosity variation can be seen from the last column of 

Table 1, where the ratio of the maximal viscosity in the fluid layer, 

(namely in the fluid just under the lid), to the minimal viscosity in 

the fluid, (namely in the fluid at the bottom), is listed. Although 

the experimental values of Rc for the two deep layers are still 

compatible with Rctheor, they raise questions about the validity of 

the concept that the onset of convection in non-Boussinesq fluid 

layers takes place as a subcritical bifurcation at a value of R 

Smaller than Rctheor. 

The experimental critical Rayleigh numbers in the thin fluid layers are all clearly above 1708 by about 30%, with no recognizable 

trend with regard to either depth or viscosity ratio. In order to 

check on the startling value of Rc the experiments with the 0.210 cm 

deep layer have been repeated with brand new silicone oil of 0.1 

cm2/sec viscosity (nominal), in order to eliminate any possible conse- 
24 

quence originating from the use of one particular sample of the oil. 

As Table 1 shows the Rc measured with the new oil fits right in with 

the previous results, so there is no particular property of the old 

oil that affected the outcome of the experiments. A 30% deviation of 

the critical Rayleigh number from the value of 1708 appears to be 

outside of the experimental error. We have mentioned above that we 

have ascertained that the thermal conductivity of the fluid cannot 

vary more than t 2% from the values which follow from (5a,b). We have 

mentioned the possibility of a substantial uncertainty in the value of 

the kinematic viscosity. However, the value of v of the 0.1 cm2/sec 

oil is not only the value suggested by the manufacturer, but also the 

value of v determined with the capillary viscometer. This instrument 

is best suited for the measurement of small viscosities, and the 

accuracy of this viscometer has been verified with the calibration 

fluid. Assuming that the largest discrepancy of the viscosity 

measurements, which was + lo%, would actually be a correct viscosity, 

then that would still not reduce our experimental Rc to a value compatible with 1708. It does not appear that the other material properties or ATc contribute an error sufficiently large in order to explain our value of Rc. We conclude, therefore, that the experimental 

value of Rc is indeed above Rc = 1708, contradicting thereby theories 

that place the onset of convection in slightly non-Boussinesq fluid 

layers at values of R < 1708. We note, on the other hand, that the 

theory of Stengel, Oliver and Booker (1982) predicts that the onset of 

convection in strongly non-Boussinesq fluid layers occurs at R > 1708. 

Our results confirm this finding, although we work with viscosity 

variations not nearly as large as those considered by Stengel et al. 
25 

4. Conclusions 

Our experiments have two essential results. First, we have found 

that in thin fluid layers the onset of Rayleigh-Benard convection is 

preceded by subcritical convective motions. Secondly, we have found 

that the onset of Rayleigh-Benard convection in non-Boussinesq fluid 

layers takes place in the form of hexagonal cells at Rayleigh numbers 

larger than the critical Rayleigh number Rc = 1708 which determines 

the onset of convection in Boussinesq fluid layers. 

Concerning our first finding, namely the observation of the subcritical motions, these occur only if the fluid is sufficiently thin, 

of order of 2 mm or less. The thinner the fluid is the larger is the 

range of occurrence of subcritical motions. There is apparently no 

critical value of the temperature difference for the appearance of the 

subcritical flow, the motions form with barely observable amplitude 

which increases very slowly as the temperature difference is increased. This is followed by a spectacular spontaneous amplification 

of the amplitude of the flow at the genuine onset of convection, which 

becomes apparent as an enormous multiplication of the intensity of 

light on the shadowgraph. The search for subcritical motions was the 

motif for our experiments and originated from the unexpected appearance of subcritical motions in the Benard convection experiments 

of Koschmieder and Biggerstaff (1986). Observing subcritical motions 

also in Rayleigh-Benard convection confirms what we found in the 

Benard convection experiments, in which surface tension gradients seem 

to amplify the subcritical motions, they were much more easily observable under an air surface than in the Rayleigh-Benard experiments. 

Our observations in the current experiments have shifted the search 
26 

for the explanation of the subcritical flow in Benard convection to a 

search for an explanation for the subcritical motions in RayleighBenard convection. We will not speculate here about the cause of the 

subcritical motions. 

It appears that subcritical motions in thin fluid layers heated 

uniformly from below are a principal feature, not an avoidable 

consequence of an experimental deficiency. We are led to believe this 

by the significant differences in the horizontal boundary conditions 

of the fluid under which we have observed subcritical flow. We have 

observed subcritical flow with either an air surface in Koschmieder 

and Biggerstaff (1986) or here with the sapphire on top of the fluid. 

This involves the differences between near slip and no slip 

conditions, as well as the change from very poor thermal conduction to 

very good thermal conduction. On the bottom of the fluid we have had 

the copper block, or the steel mirror or the glass mirror. That again 

involves the very large changes in thermal conductivity between the 

copper block and the glass mirror, as well as changes in the 

smoothness of the bottom, being only good with the copper but 

excellent with the glass mirror; while the smoothness of the top 

boundary of the fluid was excellent with either the air surface or the 

sapphire. We believe that influences of the lateral wall are minimal 

because the lateral wall is made of a very poor thermal conductor and 

the aspect ratio of the fluid layer is very large (r = 60). We note, 

furthermore, that the range of subcritical motions increases as the 

influence of the lateral wall decreases with ever thinner fluid layer 

or increased aspect ratio. The decisive feature for the occurence of 

subcritical flow seems to be the small depth of the fluid layer. 
27 

Concerning the question of the onset of Rayleigh-Benard convection in non-Boussinesq fluid layers, we are faced there with a 

dilemma. We can confirm quite clearly the long-standing theoretical 

predictions that the onset of convection in this case should take 

place in the form of hexagonal cells which transform into rolls at 

slightly higher Rayleigh numbers. While the theories predict the pattern and the change of the pattern correctly we come, however, to a 

contradiction with the same theories when we look at the Rayleigh number at which the onset of convection occurs. We find that onset of 

convection takes place at Rayleigh numbers larger than R, = 1708, 

while the theories predict that the hexagonal cells form at Rayleigh 

numbers smaller than Rc = 1708. Only the most recent theoretical 

investigation of convection in non-Boussinesq fluid layers (Stengel, 

Oliver and Booker, 1982) places the onset of convection at values of 

R>Rc, provided that the viscosity variation in the fluid layer is very 

large. It is necessary to first clarify the theory with regard to the 

point whether or not the onset of convection in non-Boussinesq fluid 

layers occurs at Rayleigh numbers larger or smaller than Rc = 1708 before a conclusion can be drawn as to whether our experimental findings 

are in agreement with theory. 

Help of Professor R. S. Schechter is gratefully acknowledged. We 

thank Mr. T. H. Koschmieder for programming the data evaluation. 

Support of this work through the National Aeronautics and Space Agency 

is also gratefully acknowledged. 
A 

% 

Table 1 

RC 

0.501 15.3 1.082 +2.5 1819+109 1.065 

0.400 26.2 1.073 +1.6 177 6z10 7 1.040 

0.221 14.2 1.081 +1.2 2 196513 2 1.286 

0.210 16.6 1.058 -1.0 22842137 1.337 

0.210~ 15.9 1.082 +1.3 22582136 1.322 

0.187 21.6 1.077 +0.8 2175+130 1.274 

0.163 30.8 1.045 -2.2 2240213 4 1.311 

1.31 

1.57 

1.29 

1.34 

1.33 

1.46 

1.69 
References 

Busse, F. H. 1967 J. Fluid Mech. - 30, 625-649. 

Davis, S. H. & Segel, L. A. 1968 Phys. Fluids ll, 470-476. 

Hoard, C. Q., Robertson, C. R. b Acrivos, A. 1970 Intl. J. Heat Mass 

Transfer l3, 849-856. 

Jenkins, D. R. 1986 J. Fluid Mech. Submitted. 

Koschmieder, E. L. 1966 Beitr. Phys. Atmos. 39, 1-11. 

Koschmieder, E. L. & Pallas, S. G. 1974 Intl. J. Heat Mass Transfer 

- 17, 991-1002. 

Koschmieder, E. L. & Pallas, S. G. Rev. Sci. Instrum. 45, 1164-1165. 

Koschmieder, E. L. & Biggerstaff, M. I. 1986 J. Fluid Mech. 167, 

49-64. 

McGregor, R. R. 1954 Silicones and Their Uses, McGraw-Hill, New 

York. 

Oliver, D. S. & Booker, J. R. 1983 Geophys. Astrophys. Fluid 

Dynamics 27, 73-85. 

Palm E. 1960 J. Fluid Mech. 8, 183-192. 

Palm E., Ellingsen, T. & Gjevik, B. 1967 J. Fluid Mech. 30, 651- 

661. 

Schmidt, R. J. & Milverton, S. W. 1935 Proc. R. SOC. A 152, 586- 

594. 

Silveston, P. L. 1958 Forsch. Ing. Wes. 24, 29-32, 59-69. 

Stengel, K.C., Oliver, D. S. & Booker, J. R. 1982 J. Fluid Mech. 

- 120, 411-431. 

Segel, L. A. & Stuart, J. T. 1962 J. Fluid Mech. l3, 289-306. 
Figure Captions 

Fig. la. Shadowgraph pictures of convection in the form of circular 

Fig. 

Fig. 

Fig. 

Fig. 

Fig. 

Fig. 

Fig. 

Fig. 

concentric rolls at AT = 16.0"C = 1.10 AT, in the 5.01 mm deep 

layer of silicone oil of 1.0 cm2/sec viscosity. The steel mirror 

is at the bottom, the sapphire on top of the fluid. 

lb. Convection in circular concentric rolls in the 5.01 mm deep 

layer at AT = 1.64 ATc. The diameter of the innermost ring has 

shrunk. 

IC. Convection in circular concentric rolls in the 5.01 mm deep 

layer at AT = 2.33 AT,. The original innermost ring has been 

reduced to a point by the increase of the wavelength of the 

motion. 

2a. Onset of convection in the 4.00 mm deep layer of silicone oil 

of 1.0 cm2/sec viscosity at AT = 26.25OC = ATc. The pattern is 

cellular in a circular arrangement. Steel mirror - sapphire 

boundaries. 

2b. Circular concentric rolls in the 4.00 mm deep layer at AT = 

1.05 ATc. The cells have disappeared. 

3a. Shadowgraph picture of subcritical convective motions in the 

1.63 mm deep layer of 0.10 cm2/sec viscosity at AT = 28.75 OC = 

0.74 ATc. A glass mirror is at the bottom, the sapphire on top 

of the fluid. 

3b. Subcritical flow in the 1.63 mm deep layer at AT = 0.84 ATC. 

3c. Subcritical flow in the 1.63 mm deep layer at AT = 0.90 AT,. 

3d. Critical cellular flow in the 1.63 mm deep layer at AT = 

38.75OC = AT,. 
Fig. 3e. Beginning of the transformation of the cellular pattern into 

a roll pattern. 1.63 mm deep layer, AT = 1.03 ATc. 

Fig. 3f. Supercritical flow in the 1.63 mm deep layer. The 

transformation of the pattern into rolls is nearly completed. 

Fig. 4. Heat transfer in the 5.01 mm deep layer of silicone oil of 1 

cm2/sec viscosity. Copper - copper boundaries. Only three 

characteristic error bars have been plotted. 

Fig. 5. Heat transfer in the 4.00 mm deep layer of silicone oil of 1 

cm2/sec viscosity. Copper - copper boundaries. 

Fig. 6. Heat transfer in the 1.87 mm deep layer of silicone oil of 

0.10 crn2/sec viscosity. Copper-copper boundaries. 

Fig. 7. Heat transfer in the 1.63 mm deep layer of silicone oil of 

0.10 cm2/sec viscosity. Copper - copper boundaries. 

Table 1. Critical temperature differences, thermal conductivities, 

Rayleigh numbers and viscosity variations for the different fluid 

layers. 
Table 1 

0.501 

0.400 

0.221 

0.210 

0.210c 

0.187 

0.163 

15.3 

26.2 

14.2 

16.6 

15.9 

21.6 

30.8 

1.082 

1.073 

1.081 

1.058 

1.082 

1.077 

1.045 

A 

% 

RC %&theor Vmax/vmin 

+2.5 1819+109 1.065 1.31 

+1.6 17 7 6T107 1.040 1.57 

+1.2 21 9 6113 2 1.286 1.29 

-1.0 2284+137 1.337 1.34 

+s. 3 22 58713 6 1.322 1.33 

+0.8 2175T13 0 1.274 1.46 

-2.2 224 0213 4 1.311 1.69 
I 
i 

! 
0KIG:NAL PAGE IS 

OF POOR QUALITY 
'ORIGINAL PAGE IS 

OF POOR QUALITY 

C 

ORIGINAL PAGE IS 

OF POOR QUALITX 
Nu 

1.4 

1.3 

1.2 

1.1 

1 .o 

Fq. 4 

I 1 I b 

10 15 20 AT (OC) 
Nu 

1.4 

1.3 

1.2 

1.1 

1 .o 

.L 

1 1 + 

10 20 30 AT (OC) 
Nu 

1 .d 

1 .: 

1 .z 

1.1 

1 .o 
: 4 

6 

Nu 

1.3 

1.2 

1.1 

1 .C 

1 1 1 1 + 

10 20 30 40 AT(OC)