Onset of Rayleigh-Bénard-Marangoni Instability: Comparison between Theory and Experiment

The onset of convection is reexamined for the Benard problem, using a nondimensional model which includes both buoyancy and surface-tension driven mechanisms, as well as the explicit consideration of the dynamics of the upper phase and the deviation of the fluid-fluid interface from its mean planar position. The critical conditions for the onset of convection, assumed to occur from a stationary state, and the critical wavelengths characteristic of the resulting flow are determined numerically after a normal mode-type linear stability analysis. These theoretical results differ from existing numerical calculations for water-benzene layers but are in close quantitative agreement with relevant experimental findings for silicone oil-air layers. When compared with previous linear stability results, it is found that the explicit consideration of the dynamics of the gas layer has a stabilizing effect on a model which assumes a passive gas phase and this stabilizing influence tends to increase as the gas layer becomes thinner. Introduction The classical subject of linear stability theory analyzes disturbances to an equilibrium state by neglecting all terms except those which are linear in the perturbation quantities. The resulting class of eigenvalue problems has been under intensive study for many years. Physically, the main fruits of linear theory are the determination of the critical conditions for the onset of instability to initially infinitesimal disturbances and the determination of the critical wavelength of those disturbances which are most likely to grow first. Perhaps the bulk of this research has been directed toward understanding certain prototype problems in hydrodynamics [1]. One of these is the Benard problem of natural convection in a thin liquid layer heated uniformly from below and cooled from above by means of a gas layer [2]. The work of Koschmieder [3] and Palmer and Berg [4] represent two series of carefully controlled experiments for this phenomenon. Koschmieder [3] performed his experiments on a shallow layer of silicone oil lying on a plane, circular copper plate 0340-0204/82/0007-0169 $02.00 © Copyright by Walter de Gruyter & Co. · Berlin New York 17o E. N. Perm, D. J. Wollkind which was maintained at a uniform temperature while being heaied from below. The ambient air layer on the surface of the oil was cooled from above by means of a horizontal glass plate which was maintained at a fixed uniform temperature. Motions were observed to begin in the form of concentric circular rolls which, after the center ring formed, broke down into a hexagonal cellular pattern. The appearance of the concentric rings prior to the breakdown into hexagons permitted an accurate determination of the wavelength characteristic of such convective motion at various depths of the oil layer. Palmer and Berg [4] also performed their experiments on silicone oil-air layers but used a different experimental apparatus in which both the lower and upper horizontal plates were aluminium. They were primarily concerned with determining the critical temperature difference between these plates at which the onset of convection would first occur. By precisely measuring the temperature drop and the heat transfer across the whole two-fluid layer and then plotting these against each other, it was possible to determine this critical value accurately upon identifying the break point of that curve with the onset of instability. Whereas the distance between the plates was fixed in Koschmieder's experiment and only the depth of the oil layer was varied, the gap width setting for the plates was variable as well in this set of experiments. In order to compare these experimentally determined critical temperature differences between the plates at which the onset of convection occurs and critical wavelengths characteristic of the resulting flow with the predictions of linear stability theory, it is essential to employ a model including all the effects present in the experiments just described. The first such analysis was developed by Zeren and Reynolds [5]. Although previous researchers had incorporated one or more of these features into the'ir analyses (an excellent survey of which is presented by Zeren and Reynolds [5]), no single model for the Benard problem until that of Zeren and Reynolds [5] simultaneously took into account the dynamics of both fluid layers, the buoyancy effect due to the expansion of heated fluid, the surface tension effect due to interfacial tractions induced by temperature variations on the fluid-fluid interface, and the deflexion of that interface itself. Zeren and Reynolds [5] numerically determined the critical conditions for the onset of convection, assumed to occur from a stationary state, after performing a linear stability analysis on the appropriate exact solution to their governing system of equations and boundary conditions using a normal mode technique. These numerical calculations were made for water-benzene layers with a fixed total depth and various water depth fractions and then compared with relevant experiments conducted by Zeren and Reynolds [5] for both the cases of heating from above and below. We wish to use theoretical results analogous to those determined by Zeren and Reynolds [5] but appropriate for silicone oil-air layers instead of water-benzene ones in order to compare the predictions of linear stability theory with the experimental evidence of Koschmieder [3] and Palmer and Berg [4]. Our main purpose in making such a comparison is to explain the onset of Benard convection more fully. Given the manner in which numerical calculations are reported, it is thus necessary for us to reexamine Zeren and Reynolds' [5] theoretical treatment of the Benard problem. Toward that end, we redo their linear stability analysis and calcuate the critical conditions for the onset of convection in silicone oil-air layers numerically. As a byJ. Non-Equilib. Thermodyn. , Vol. 7, 1982, No. 3 Onset of Rayleigh-Benard-Marangoni instability 171 product of that numerical procedure we recalculate the critical conditions for the onset of convection in the water-benzene layers of Zeren and Reynolds [5] and find that our predictions differ from theirs in a quantitative manner. Hence in what follows we present a slightly more detailed development of our theoretical results than we would have normally done otherwise. 1. Content We begin in Sect. 2 with a formulation of the problem which is a nondimensionalized version of the governing equations used by Zeren and Reynolds [5]. In Sect. 3, we present a sketch of our linear stability analysis of the relevant steady-state motionless stratified planar interface solution to that basic system of equations and numerically calculate the critical conditions for the stationary onset of convective instability in silicone oil-air layers. Our normal mode approach explicitly represents the eigenvectors of linear theory in a real form as opposed to the complex one employed by Zeren and Reynolds [5]. We begin Sect. 4 by demonstrating that our numerical procedure yields quantitatively different results than those obtained by Zeren and Reynolds [5] when it is applied to their laboratory water-benzene configuration for the case of heating from below. We next turn to a detailed comparison of our theoretical predictions with the experiments of Koschmieder [3] and Palmer and Berg [4]. In particular we show that there is close quantitative agreement between our critical wavelengths and the wavelengths of the rolls observed by Koschmieder [3] and our critical temperature differences and the temperature differences at the onset of convection measured by Palmer and Berg [4]. We then discuss our findings in the context of the recent work of Davis and Homsy [6] which is both a linear and energy stability analysis of a model differing from ours only in that the dynamics of the gas layer is not considered explicitly. 2. Nondimensionalized formulation of the problem In order to motivate our choice of nondimensi nal variables, we first consider the physical situation involving dimensional variables depicted in Fig. 1. With no loss of generality [6], we employ a two-dimensional coordinate system (x, z) to describe two immiscible viscous Newtonian fluids bounded by rigid infinite horizontal plates located at z = d"" and z = — d respectively and maintained at uniform temperatures T* = T! and T = T0 where T0 > Tx. In the above and what follows a plus superscript is used to distinguish a quantity defined in the upper gas phase from the corresponding property defined in the lower liquid phase. The interface separating the two is assumed to be a material surface of discontinuity possessing negligible mass and for which surface viscosities are neglected. This interface satisfies the equation æ = ζ (÷, Ô) where À= time. It has a mean position of æ = 0, coinciding with the ÷ axis, and a mean interfacial temperature Tr = Ô where Tj < Ô < T0. We next examine the thermal properties of this system for a motionless (velocity -\ components u = w = 0), steady-state (— = 0), stratified (T"" and T dependent only ot J. Non-Equilib. Thermodyn., Vol. 7, 1982, No. 3 172 E. N. Perm, D. J. Wollkind Fig. l: A diagram employing dimensional variables which depicts the liquid-gas interface and the thermal properties of our system. on z) planar interface (J = 0) situation. Since we then have [5] = 0 for 0