# On the Asymptotic Approach to Thermosolutal Convection in Heated Slow Reactive Boundary Layer Flows

Authors: Stanford Shateyi; Precious Sibanda; Sandile S. Motsa
Journal: Journal of Applied Mathematics
 (2008)

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## Abstract

The study sought to investigate thermosolutal convection and stability of two dimensional disturbances
imposed on a heated boundary layer flow over a semi-infinite horizontal plate composed of a
chemical species using a self-consistent asymptotic method. The chemical species reacts as it diffuses
into the nearby fluid causing density stratification and inducing a buoyancy force. The existence of
significant temperature gradients near the plate surface results in additional buoyancy and decrease in
viscosity. We derive the linear neutral results by analyzing asymptotically the multideck structure of
the perturbed flow in the limit of large Reynolds numbers. The study shows that for small Damkohler
numbers, increasing buoyancy has a destabilizing effect on the upper branch Tollmien-Schlichting (TS)
instability waves. Similarly, increasing the Damkohler numbers (which corresponds to increasing the
reaction rate) has a destabilizing effect on the TS wave modes. However, for small Damkohler numbers,
negative buoyancy stabilizes the boundary layer flow.


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## Body

## 1. Introduction

Convection  in which the buoyancy forces are due to
both temperature and chemical concentration gradients are referred to as
thermosolutal or double diffusive convection. Ostrach [1] pointed out that different modes
of such convection exist depending on how the temperature and concentration
gradients are oriented relative to one another. Some natural convection flows
in the atmosphere and micrometeorological phenomena are often thermosolutal.
The heating of the earth by the sun causes atmospheric thermal convection which
is usually modified by the presence of moisture evaporated from the ground.In lakes and oceans, thermosolutal convection is
caused by stable vertical concentration distribution with heating from the side
or from the top. The stability theory has been used to explain the occurrence
of layered structures observed in oceans as explained by Turner [2].Studies on natural convection flows caused by the
simultaneous diffusion of thermal energy and chemical species were carried out
by Gebhart and Pera [3]. They considered small species concentration levels
and showed that the species Boussinesq approximation led to similarity
solutions similar in form to those for single buoyancy mechanism flows.The effect of heat transfer on the upper-branch
stability of Tollmien-Schlichting instabilities (TSIs) in accelerating boundary
layer over a rigid surface in incompressible flows was investigated by Mureithi
et al. [4]. The study
indicated that buoyancy has destabilizing effect on rigid bodies. Their
analysis also showed that in the presence of strong buoyancy forces, the
five-zone asymptotic structure alters but for moderate buoyancy, the five-zone
structure of Smith and Bodonyi [5] remains with very few alterations.Shateyi et al. [6] considered the effect of fluid buoyancy and chemical
reaction between the chemical species and the fluid on the linear stability of
two dimensional disturbances wave modes. They extended the theory of boundary
layer flows over horizontal surfaces to include a chemical species and the
effect of the Damkohler number. Results showed that when the wave number and
speed number are varied against the scaled Damkohler number, the effect of
increasing buoyancy was destabilizing in agreement with assertions by Motsa et
al. [7]. It was shown
that increasing reaction kinematics (thus, reducing the density and viscosity)
has destabilizing effects on the TS waves.Pons and Le Quéré [8] showed that Boussinesq equations do not exactly
represent buoyancy—induced natural convection. The study observed that
thermodynamic consistency is retrieved when both the work due pressure forces
and the heat generated by viscous friction are accounted for in the heat
equation. The study recommended that any theoretical study of buoyancy-induced
natural convection should be done with the thermodynamic Boussinesq model and
not with the usual Boussinesq approximations. However, in spite of this
weakness, the usual Boussinesq equations are still very useful (see, e.g.,
Azizi et al. [9]). To that end in this study, we will use the
Boussinesq approximations.The present work presents an asymptotic analysis of
the flow induced by buoyancy effects due to both temperature and chemical
concentration gradients near the flat surface. This paper is a direct extension
of the earlier work in Shateyi et al. [6] to include the effects of temperature differences
within the boundary-layer. The flow has uniform surface conditions with the
buoyancy effects primarily away from the surface. Our analysis is limited to
processes which occur at low concentration gradients. We give an asymptotic
investigation of the interactions between the reaction kinematics and the fluid
hydrodynamics with the Damkohler number (the ratio of the hydrodynamic time
scale to the reaction time scale) as the parameter of primary interest. In the
limit of large Damkohler numbers, the reaction kinematics proceed at a much
faster rate compared to the fluid hydrodynamics. If the Damkohler number is
close to zero, the chemical reactions are slow compared to the motion of the
fluid. In this case, a nonreactive fluid can be assumed. The greatest
interaction between reaction kinematics and fluid dynamics occurs when the
Damkohler is of O(1).In the study, we focus attention on the case when the
Damkohler number is small (that is, less than
unity). The case for large Damkohler numbers was investigated
by Shateyi et al. [6]. The aim is to determine the influence of small
Damkohler numbers on the stability characteristics of the upper-branch TS
instability waves. The major difference between the current work and that of
Mureithi et al. [4]
arises from the introduction of reaction kinematics. The absence of wall
compliance and the presence of a chemical species makes
the current work different from that of Motsa et al. [7]. The presence of wall
heating or cooling and small Damkohler numbers makes the current work different
from Shateyi et al. [6]. The current approach provides asymptotic solutions
to the linearised Navier-Stokes rather the numerical solution of the
Orr-Sommerfeld equation.


## 2. Mathematical Formulation

We consider a
two-dimensional, incompressible fluid flow over a heated horizontal plate which
is composed of a chemical species maintained at a fixed concentration. The
chemical species diffuses into the nearby fluid inducing a buoyancy force. A
change in the temperature of the fluid near the plate surface due to exogenous
heating effects also results in additional buoyancy.The governing nondimensional unsteady Navier-Stokes
equations for an incompressible fluid under a Boussinesq-type approximation are
as follows:(2.1)∂u∂x+∂v∂y=0,(2.2)∂u∂t+u∂u∂x+v∂u∂y=−∂p∂x+1Re(∂2u∂x2+∂2u∂y2),(2.3)∂v∂t+u∂v∂x+v∂v∂y=−∂p∂y+1Re(∂2v∂x2+∂2v∂y2)+GtT+GcC,(2.4)∂C∂t+u∂C∂x+v∂C∂y=1ReSc(∂2C∂x2+∂2C∂y2)−DaC,(2.5)∂T∂t+u∂T∂x+v∂T∂y=1PrRe(∂2T∂x2+∂2T∂y2). However, (2.1)–(2.5) have been
nondimensionalised such that the space coordinates are given by (x*,y*)=L(x,y),
the velocity components are (u*,v*)=U∞(u,v),
the pressure is p*=ρU∞p,
and the time is t*=(L/U∞)t,
where L is the characteristic length scale (e.g., the
distance measured from the leading edge of the plate). The species chemical
concentration and the temperature are, respectively, (2.6)C=C*−C∞Cw−C∞,T=T*−T∞Tw−T∞, where the asterisks in these
relations denote dimensional quantities and the subscripts “w" and “∞" refer to the conditions at the wall and the
free stream values, respectively. For purposes of this study, the important
parameters are the Damkohler number Da that is defined as the ratio of the flow time
scale to the chemical time scale and the buoyancy terms Gc=Grc/Re2 and Gt=Grt/Re2,
where Grc=βcgL3(Cw−C∞)/ν2 and Grt=βtgL3(Tw−T∞) are, respectively, the Grashof numbers for
mass and thermal diffusion, βc is the coefficient of expansion with respect
to mass transfer, and βt is the volumetric coefficient of thermal
expansionThe other parameters and variables in (2.1)–(2.5) are
the Prandtl number Pr,
the flow Reynolds number Re,
and the Schmidt number Sc(=μ/Dρ),
where D is the binary diffusion coefficient, μ is the coefficient of dynamic viscosity, ρ is the density of the fluid, ν is the coefficient of kinematic viscosity, and g is the acceleration due to gravity.The  momentum boundary conditions are the no-slip
conditions:(2.7)u=v=0aty=0. We assume that the horizontal plate is maintained at a
prescribed temperature TBw and uniform concentration CBw.
In the far field, we assume that the fluid temperature, species chemical
concentration, and the fluid velocity approach their free stream values.The basic boundary-layer flow is given
by(2.8)(u,v,p,T,C)=(UB,Re−1/2VB,pB,TB,CB)(x,Y), where Y=Re1/2y is the boundary coordinate. For general
pressure gradient boundary layers, the following properties
hold: (2.9)UB∼λ1Y+λ2Y2+⋯asY→0,TB∼γ1+γ2Y+⋯asY→0,UB∼μ1+μ2Y+⋯asY→0,UB→1,TB→0,CB→0asY→∞. The coefficients λ1=UBy|y=0>0 and λ2=UByy|y=0<0 are, respectively, the skin friction and
curvature of the basic flow profile. The coefficients γ1 and γ2 are the heat transfer coefficients and μ1 and μ2 are the concentration transfer coefficients.Disturbances to the basic flow of amplitude factorδ are introduced and these spread through the
boundary layer. The Reynolds number is large while the amplitude of the
spatially growing disturbances is assumed to be sufficiently small for linear
theory to hold. The TSI grows by extracting energy from the mean flow to the
disturbance within the boundary layer (see, e.g., Carpenter and Gajjar
[10]).The asymptotic structure of the boundary layer is now
well known and described (e.g., Smith and Bodonyi [5] and Mureithi et al.
[4]). It consists of
the five regions (see Figure 1) where R1 is the main part of the boundary layer with
thickness O(Re−(1/2)), R2 is a thinner inviscid adjustment region of
thickness O(Re−(7/12)) containing the viscous critical layer R3, the wall layer R4 of thickness O(Re−(2/3)),
and the outer potential flow R5 of thickness O(Re−(5/12)).
Consistent with earlier work on linear analysis, the viscous critical layer is
ignored in the present theory except in so far as it produces a phase shift in
the boundary layer pressure and velocity.Figure 1
Schematic sketch of flow structure
showing the multilayered nature of the boundary layer and the relative
positioning of the five regions.The disturbances are taken to be in the form of a
modulated wave train periodic inX where, for the upper branch of the neutral
stability curve, the scaled streamwise and temporal variables are x=ϵ5X, t=ϵ4τ,
and ϵ=Re−(1/12) is a small parameter. The neutral wave number α and phase speed c of the disturbances are  O(ϵ−5) and O(ϵ−4),
respectively. We thus expand α and c as (2.10)α=ϵ−5α0+⋯,c=−ϵ−4c0+⋯, where α0 and c0 are the scaled real wave number and real phase
speed of the travelling wave disturbances. The derivatives ∂/∂x and ∂/∂t are then replaced by ϵ−5α0∂/∂X and −ϵ−4ω0∂/∂τ,
respectively. The other important scalings (see also Shateyi et al. [6]) are Gc/Gt=O(ϵ−5) and Da∼O(ϵ−4) when the TS eigenrelation is significantly
altered for the first time by the effects of fluid buoyancy.


## 3. Stability Analysis

Information on
the disturbance expansions relevant to the upper-branch stability of boundary
layers is now well documented in the literature (see, e.g., Gajjar and Smith
[11], Motsa et al.
[7], and the
references therein). Only the details necessary to obtain the linear dispersion
relations will be given. In the main part of the boundary layer, region R1,
we define the coordinate y=ϵ6Y,
where Y=O(1),
and introduce a small disturbance of size δ into the basic flow. The expand disturbance
quantities are then expanded as (3.1)u=UB+δu0+ϵδu1+⋯,(3.2)v=ϵδv0+ϵ2δv1+⋯,(3.3)p=PB+δϵp0+ϵ2δp1+⋯,(3.4)T=TB+δT0+ϵδT1+⋯,(3.5)C=CB+δC0+ϵδC1+⋯, where ui,vi,
and so  forth are functions of the boundary layer variable Y and  of the scaled streamwise variable X,
and δ is the amplitude of the disturbance which is
very much smaller than unity so that terms quadratic in δ are ignored thereby restricting the analysis
to linear  stability theory. Substituting (3.1)–(3.5) into the governing equations and
solving the resulting system of leading-order equations yields the following
first-order solutions: (3.6)u0=A0UBY,v0=−α0A0XUB,T0=A0TBY,C0=A0CBY,p0=P0+G0tA0(TB−γ1)+G0cA0(CB−μ1). At the next order, O(δ2), we obtain the solutions (3.7)v1=α0c0A0X+α0UB∫Y02P0XUB2dY+α0UBG0tA0X∫Y0Y(TB−γ1)UB2dY−α1A0X+α0UBG0cA0X∫Y0Y(CB−μ1)UB2dY−α0A1XUB,p1=P1−α02A0∫Y0YUB2dY−D0A0G0tα0∫Y0YTBYUBdY−G0cA0D0α0∫Y0YCBYUBdY+A1(G0tTB+G0cCB)−G0t∫Y1YTBY(∫0Y1[P0+G0tA0(TB−γ1)UB2]dY1)dY−−Gc∫Y1YCBY(∫0Y1[P0+G0cA0(CB−μ1)UB2]dY1)dY−A0G0tG0c{∫0YTBY(∫Y0Y1(cB−μ1)UB2dY1)dY+∫0YCBY[∫Y0Y(TB−γ1)UB2dY1]dY}, where Ai=Ai(X) and Pi=Pi(X)(where,i=0,1) are unknown functions representing the
displacement and the pressure amplitudes. In the results above, we set Ai=A¯ieiX+c.c,Pi=P¯ieiX+c.c (where c.c denotes the complex conjugate). The lower
limit of the integrals, Y0,
is a non-zero constant introduced for convenience, whose value does not alter
the eventual results for wave numbers and frequencies.In regionR2,
we define y=ϵ7Y¯ with Y¯=O(1) and the expansions follow from R1: (3.8)u=λ1ϵY¯+ϵ2λ2Y¯2+δ(u(0)+ϵu(1)+⋯),v=δϵ(ϵv(0)+ϵ2v(1)+⋯),p=pB+δ(ϵp(0)+ϵ2p(1)+⋯),T=γ1+ϵγ2Y¯+ϵ2γ3Y¯2+δ(T(0)+ϵT(1)+⋯),C=μ1+ϵμ2Y¯+ϵ2μ3Y¯2+δ(C(0)+ϵC(1)+⋯). Substituting these equations
into the governing equations and solving the resulting equations yield the
following first-order solutions: (3.9)u(0)=λ1A0,v(0)=−α0pX(0)λ1−α0A0Xλ1ξ,p(0)=P(0)=c0λ1A0,C(0)=μ2λ12(A0λ12ξ+P(0))(ξ−iD0α0λ1)−1,T(0)=γ2(A0+p(0)λ12ξ), where ξ=Y−(c0/λ1).
At the next order, the velocity and pressure terms are (3.10)v(1)=−1λ1α0Px(1)+(γ2G0t+μ2G0c)λ1α0A0x(ξ{ln|ξ|+ϕ±})−λ1α0ξA¯1x−α0λ2A0x(ξ2+2c0λ1ξ[ln|ξ|+ϕ±]−c20λ21)−(γ2G0t+μ2G0cλ12)α0c0A0x−(γ2G0t+μ2G0c)2λ1A0x(α0c0+iD0)ln|ξ2+D02α02λ12|−(γ2G0t+μ2G0c)D0λ1α0A0x(α0c0+iD0)ξtan−1(α0λ1ξD0)−i(γ2G0t+μ2G0c)2λ1D0α0A0x(α0c0+iD0)ξ(ln|D02+α02λ12ξ2α02λ12ξ2|+ϕ±)+i(γ2G0t+μ2G0cλ12)A0x(α0c0+iD0)[tan−1(D0α0λ1ξ)+ϕ±],p(1)=P(1)+(γ2G0t+μ2G0c)A0Y+(γ2G0t+μ2G0c)(α0c0+iD0)A0(ln|ξ2+D02α02λ12|)−i(γ2G0t+μ2Gc)α0λ1(α0c0+iD0)A0[tan−1(D0α0λ1ξ)+ϕ±]. The solutions in this region
possess both logarithmic and algebraic singularities as ξ→0.
These singularities are smoothed out by the introduction of the critical layer
consisting of a thin viscous region situated in the neighbourhood of the
critical level ξ=0 where ϕ± and ϕp± are the phase-shift terms introduced to
connect the solutions in the normal velocity and pressure, respectively, on
either side of the critical layer.Solutions in the other regions (namely, the wall layerR4 and the outer potential-flow layer R5) follow in a straightforward manner, and the important solutions of the wall layer are given by (3.11)v˜0=iα02P˜0mω0(1−mZ−e−mZ),p˜0=P˜0(X,Z), where m=(α0c0)1/2e−iπ/4, y=ϵ8Z,
and Z is an O(1) coordinate.In regionR5,
we set y=ϵ5y^,
where y^∼O(1).
The leading-order solutions are given by(3.12)u^0=−P^0e−α0y^,v^0=−iP^0e−γ0y^,p^0=P^0e−γ0y^, where P^0 is an unknown function which describes the
disturbance pressure at the outer extreme of the boundary layer. At the next
order, the important solutions are (3.13)v^1=−i[P^1−ω0α0P^0]e−α0y^,p^1=P^1e−α0y^, where P^1 is an unknown function which describes the
disturbance pressure at the outer extreme of the boundary layer.


## 4. Linear Neutral Results and Eigenrelations

In this
section, we asymptotically match the solutions in their respective overlap
regimes. We will be matching the normal velocities and pressures of the same
orders in these respective overlapping regions. The first eigenrelation is
found by matching the first-order solutions across the entire boundary layer flow
regime to be(4.1)c0λ1=G0tγ1+G0cμ1+α0. The matching of second-order
pressure components between R1 (as Y→∞)  and R5 (as y^→0) yields (4.2)P^1=P1−α02A0J0−G0tA0(J3+D0α0J1)−G0cA0(J4+D0α0J2)+G0tA1TB∞+G0cA1CB∞−A0G0tG0cI1, where TB∞=limY→∞TB and CB∞=limY→∞CB.
The constants Ii′s and Ji′s for i=1,…,4 are defined in the appendix.Matching the pressure terms acrossR2 (as Y¯→∞) and R1 (as Y→0) gives (4.3)P1=P(1)−i(γ2G0t+μ2G0c)α0λ1(α0c0+iD0)ϕ+−A1G0tTB0−A1G0cCB0, where TB0=limY→0TB and CB0=limY→0CB.
Matching the pressure terms across R4 (as Y¯→∞) and R2 (as Y→0) gives (4.4)p˜1=P(1)+(γ2G0t+μ2G0c)2α0λ1(α0c0+iD0)ln|α02c02+D02α02λ12|−i(γ2G0t+μ2G0c)α0λ1A0tan−1(D0α0c0)−i(γ2G0t+μ2G0c)α0λ1(α0c0+iD0)ϕ−. Matching the normal velocity
components between R1 (as Y→∞) and R5 (as y^→0) at the second order gives (4.5)P^1=−P^0c0+α0A1UB∞−α0c0A0−α0UB∞P0I2−α0UB∞G0tA0H1−α0UB∞G0cA0H2+α1A0. Matching the normal velocity
across regions R2 and R1 yields (4.6)B0A0X−α0λ1A1X+L0G0tA0X+L1GcA0X=(γ2G0t+μ2G0c)λ1α0A0Xϕ+−λ1α0A¯1X−2α0λ2c0A0Xλ1ϕ+−iα0(γ0G0t+μ2G0c)2λ1D0(α0c0+iD0)A0Xϕ+, where the constants B0 and Li′s for i=0,1 are defined in the appendix. Lastly, a
matching of the normal velocity between R2 and R4 gives (4.7)iα0P¯0mc0=−α0PX(1)λ1+(B1+E0λ1+ω1−α1c0)A0X+α0c0A¯1X−2α0λ2c02λ1A0Xϕ−−c0α0(γ2G0t+μ2G0c)λ12A0Xϕ−−(γ2G0t+μ2G0c)2λ12A0X(α0c0+iD0)ln|c02λ12+D02α02λ12|−(γ2G0t+μ2G0c)D0λ12c0α0A0X(α0c0+iD0)tan−1(α0c0D0)×iα0c0(γ2G0t+μ2G0c)2λ12D0(α0c0+iD0)ln|D02+α02c02α2c02|+iα0c0(γ2G0t+μ2G0c)2λ12D0A0X(α0c0+iD0)ϕ−−i(γ2G0t+μ2G0c)λ12A0X(α0c0+iD0)tan−1(D0α0c0)+i(γ2G0t+μ2Gc)λ12A0X(α0c0+iD0)ϕ− The relations (4.1)–(4.7)
above may be used to eliminate A¯1, P(1), P0, P^0 to obtain a relation which determines the
higher-harmonic components of A1.
If we restrict our attention to the eiX components, then, after some algebra,
(4.1)–(4.7) lead to (4.8)(γ2G0t+μ2G0c)λ1α0c0A0X(ϕ+−ϕ−)−2c02α0λ2λ1A0X(ϕ+−ϕ−)−i(γ2G0t+μ2G0c)2λ1D0A0xα0c0(α0c0+iD0)(ϕ+−ϕ−)−i(γ2G0t+μ2G0c)λ1A0X(α0c0+iD0)(ϕ+−ϕ−)−(γ2G0t+μ2G0c)2λ1A0X(α0c0+iD0)ln|c02α02+D02α02λ12|−(γ2G0t+μ2G0c)D0λ1A0Xα0c0(α0c0+iD0)tan−1(α0c0D0)+iα0c0(γ2G0t+μ2G0c)2λ1D0(α0c0+iD0)ln|D02+α02c02α02c02|A0x−i(γ2G0t+μ2G0c)λ1A0X(α0c0+iD0)tan−1(α0c0D0)−B3A0X+B4A1X=0, where B3 and B4 are defined in the appendix. The results for
linear theory are now well known and are derived by taking the jump across the
critical layer, ϕ to be equal to iπ.
Taking the real parts of equation (4.8) then gives (4.9)α0λ122m˜−2α0λ2c02λ1π+2α0c0(γ2G0t+μ2G0c)π+(γ2G0t+μ2G0c)λ1D0π+(γ2G0t+μ2G0c)2λ1ln|c02α02+D02α02λ12|−α02c02(γ2G0t+μ2G0c)2λ1D0ln|c02α02+D02α02c02|=0, where m¯=α0c0.
However, (4.1) and (4.9) are the crucial eigenvalue relations which fix the
neutral wave number to the neutral wave speed.


## 5. Results and Discussion

To obtain a
clearer understanding of the effects of fluid buoyancy and the chemical
reaction on the linear stability of the two-dimensional disturbance wave modes,
we consider a number of limiting cases when the fluid buoyancy parameters are
either large and small and the Damkohler numbers are small. This allows for a
detailed examination of the linear eigenrelations (4.1) and (4.9).We first investigate the limiting behaviour of the
neutral eigenrelations as the buoyancy parameters(G0c,G0t)→+∞. This limiting case corresponds to the increase
in the buoyancy force due to an increase in the density difference caused by
temperature and chemical concentration differences. Solving the eigenrelations
(4.1) and (4.9), we obtain, in the limit (G0c,G0t)→+∞ with D0∼O(1), (5.1)α0=(γ2λ1λ2−γ1)G0t+(μ2λ2λ1−μ1)G0c+⋯,c0=γ2λ2G0t+μ2λ2G0c+⋯. These results agree with those
obtained by Motsa et al. [7] for the case G0c=0 and with those of Shateyi et al. [6] when G0t=0 and help to quantify the effects of the
combined buoyancy on the normal modes. The results show that, depending on
whether the buoyancy terms reinforce or cancel out (e.g., in the case of an
endothermic reaction), the normal modes may grow without limit thus hastening
the transition to turbulence. A viable transition delay mechanism would be to
ensure that the buoyancy terms act contrary to each other so as to reduce the
growth of the disturbances.Solving the eigenrelations (4.1) and (4.9) in the
limit D0→+∞ with G0t,G0c∼O(1), we get (5.2)α0=[(γ2G0t+μ2G0c)2λ2πD0ln|D0|]1/3,c0=[(γ2G0t+μ2G0c)2λ2λ13πD0ln|D0|]1/3. The results correspond to the
case when the chemical time scale is much more pronounced than the flow time
scale. The asymptotic limit (α0,c0)∼O(D0ln|D0|)1/3 predicts that the disturbances would grow
without limit with increasing Damkohler numbers.Solving the eigenrelations (4.1) and (4.9) in the
limit, G0c→∞ with G0t,D0∼O(1), we get (5.3)α0=(μ2λ2λ1−μ1)G0c,c0=μ2λ2G0c. The limit G0c tends to infinity and  corresponds to the increase in the buoyancy force through, for example, an increase in the density difference between the reacted fluid and the unreacted fluid.In Figures2–4, we show the response of the
two-dimensional disturbances to changes in parametric values. Figure 2(a) gives
a comparison between the predictions from the full eigenrelation and the
limiting case for (G0c,G0t)→∞ when D0=0.1.
For small buoyancy, the flow is stabilized by the presence of a solute.
However, as the buoyancy increases, the stabilizing effects of the solute are
overcome by destabilizing effects of large buoyancy values. When G0t,G0c≥2,
the asymptotic results match with the response curve for the full eigenrelation.Comparison of (a) the growth
of the wave number, and (b) wave speed with increasing buoyancy for the
limiting case(⋆⋆⋆) and the full eigenrelation (—) when D0=0.1.
(a)(b)Full eigenrelation prediction
of (a) the linear neutral wave number, and (b) the linear neutral wave speed(—), with (G0t,G0c)=0.1,0.5, and 1.
(a)(b)Linear neutral (a) wave number and (b) wave speed against the buoyancy parameters(G0t,G0c) when D0=0.01,0.02,0.03 and 0.04.
(a)(b)Figure3(a) shows the response of the wave number α0 to increasing Damkohler numbers for different
buoyancy parameter values. For large Damkohler numbers, these results are in
line with the predictions of (5.2) and show that both α0 and c0 increase without limit. In this instance, the
flow is highly destabilized and increasing buoyancy serves to compound this
destabilization. For small buoyancy (≤0.1),
the neutral wave number and phase speeds initially decreased before steadily increasing. However, for
large buoyancy (≥0.5), the neutral wave number rises steeply to a
maximum value before it steadily grows to infinity.The
introduction of the second buoyancy parameterG0t results in a further destabilization of the
flow giving larger wave numbers and wave speeds at any fixed Damkohler number.
For fixed buoyancy values, the Damkohler number on its own has weakly
destabilizing effects on the fluid flow. Similar trends are observed in Figure 3(b) for the buoyancy effects on the phase speed.Figure4 shows
the response of the wave number and speed to increasing buoyancy for small
Damkohler numbers. In Figure 4(a), three distinct branches are observed, a
branch for negative buoyancy and two branches for positive buoyancy, one of these branches is a slow traveling wave with small wave numbers. For small positive buoyancy, the instability wave
starting at the origin is stabilized by increasing buoyancy with wave numbers
rapidly reducing to zero when (G0t,G0c)∼2.5.
The second instability forms an open loop whose turning point is at
progressively smaller wave numbers for increasing Damkohler numbers. In the
case of moderate negative buoyancy values, reducing the Damkohler numbers has a
stabilizing effect on the boundary layer flow.Figure4(b) shows the response of the wave speed to
increasing buoyancy when the chemical reaction is slow in comparison to the
other fluid dynamical processes. The instability has three branches when D0≤0.02,
one branch when buoyancy is negative and two branches when the buoyancy is
positive. By raising the Damkohler number to 0.03,
only two branches associated with positive buoyancy remain. Large negative
buoyancy impacts positively on the boundary layer flow by stabilizing the flow.
Increasing buoyancy values in the positive sense leads to a convergence of the
instability curves obtained when D0∼1 to a single curve such as those shown in
Figure 3(b). The value D0∼1 gives the greatest interaction between the
chemical kinematics and the fluid dynamics. The increase in buoyancy is
confirmed to be destabilizing for all Damkohler numbers. For a fixed buoyancy
parameter value, the wave speed increases with increases in the Damkohler
number in the case of the branch of the instability with larger wave numbers.
However, the wave speed decreases with increasing Damkohler numbers for the
part of the graph with lower wave speed. Only one branch of the instability
remains if the Damkohler number is raised to values greater or equal to one.
The boundary layer flow is relatively stabilized by small Damkohler numbers,
that is, (D0≤0.03).Figure5 shows  the destabilizing nature of buoyancy.
Both the neutral wave number and phase speed increase with increasing positive
buoyancy.Variation of the linear neutral
(a) wave number and (b) wave speed against the Damkohler number for increasing
buoyancy parameters.
(a)(b)


## 6. Conclusion

In this paper,
we have considered the hydrodynamic stability of two-dimensional flow. We considered
two factors that affect the onset of transition to turbulence. We investigated
the effects of fluid buoyancy and slow chemical reaction kinematics (as
measured by the Damkohler number) on the linear stability of disturbance wave
modes in a two-dimensional boundary layer flow using self-consistent asymptotic
methods. Physically, since the Damkohler number is the ratio of the flow time
scale of the fluid to that of the chemical reaction time scale, the limit of
large Damkohler numbers means that chemical kinematics proceed at much faster
rate compared to fluid hydrodynamics. If the Damkohler number is close to zero,
as is the case in this study, the chemical reactions are slow compared to the
motion of the fluid. In such a phenomenon, the system is said to be
kinematically controlled. Examples of such processes are low-temperature
reactors and bioreactors.The effect of increasing fluid buoyancy was shown to
be destabilizing confirming the earlier findings by, for example, Motsa et al.
[7] and Shateyi et al.
[6]. By extending the
work of Shateyi et al. [6], we have shown that the presence of an extra heat
source term G0t further destabilizes the boundary layer flow.
With this extra source term, the instability has three branches when D0≤0.02 instead of the one branch found in Shateyi et
al. [6]. However,
increasing the reaction kinematics so that the Damkohler number is equal to one
collapses the branches to just one. We have further shown that the boundary
layer flow can be stabilized by slowing down the reaction kinematics so that
the Damkohler number is small (<0.03).
It is understood that if the onset of transition is delayed, then, among other
factors, drag will be reduced.



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*Source: 835380-2008-10-16.xml*