/mn> C 2 .(5)

${\nabla }_{m}\cdot {q}_{m}=0$.(6)

${\rho }_{0}\left[\frac{1}{\epsilon }\frac{\partial {q}_{m}}{\partial t}+\frac{1}{{\epsilon }^{2}}\left({q}_{m}\cdot {\nabla }_{m}\right){q}_{m}\right]=-{\nabla }_{m}{P}_{m}+{\mu }_{m}{\nabla }_{m}^{2}{q}_{m}-\frac{\mu }{K}{q}_{m}$.(7)

$A\frac{\partial {T}_{m}}{\partial t}+\left({q}_{m}\cdot {\nabla }_{m}\right){T}_{m}={\kappa }_{m}{\nabla }_{m}^{2}{T}_{m}$.(8)

$\epsilon \frac{\partial {C}_{m1}}{\partial t}+\left({q}_{m}\cdot {\nabla }_{m}\right){C}_{m1}={\kappa }_{m1}{\nabla }_{m}^{2}{C}_{m1}$.(9)

$\epsilon \frac{\partial {C}_{m2}}{\partial t}+\left({q}_{m}\cdot {\nabla }_{m}\right){C}_{m2}={\kappa }_{m2}{\nabla }_{m}^{2}{C}_{m2}$.(10)

Here $q=\left(u,v,w\right)$.s the velocity vector, ${\rho }_{0}$.s the fluid density, t is the time, $\mu$.s the fluid viscosity, P is the pressure for fluid layer, T is the temperature, $\kappa$.s the thermal diffusivity of the fluid, ${\kappa }_{1}$.nd ${\kappa }_{2}$.s the solute1 and solute2 diffusivity of the fluid in the fluid layer, ${C}_{1}$.s the concentration1 or the salinity field1 for the fluid, ${C}_{2}$.s the concentration2 or the salinity field2 for the fluid in the fluid layer, ${P}_{m}$.s the pressure for porous layer, K is the permeability of

the porous medium, $A=\frac{{\left({\rho }_{0}{C}_{p}\right)}_{m}}{{\left(\rho {C}_{p}\right)}_{f}}$.s the ratio of heat capacities, ${C}_{p}$.s the

specific heat, $\epsilon$.s the porosity, ${\kappa }_{m1}$.nd ${\kappa }_{m2}$.s the solute1 and solute2 diffusivity of the fluid in porous layer, ${C}_{m1},{C}_{m2}$.re the concentration1 and concentration2 for porous layer respectively and the subscripts “m” and “f” refer to the porous medium and the fluid respectively.

The Equations (1) to (10) have a basic steady solution for fluid and porous layer respectively.

$q={q}_{b},P={P}_{b}\left(z\right),T={T}_{b}\left(z\right),{C}_{1}={C}_{1b}\left(z\right),{C}_{2}={C}_{2b}\left(z\right)$.(11)

${q}_{m}={q}_{mb},{P}_{m}={P}_{mb}\left({z}_{m}\right),{T}_{m}={T}_{mb}\left({z}_{m}\right),{C}_{1m}={C}_{1mb}\left({z}_{m}\right),{C}_{2m}={C}_{2mb}\left({z}_{m}\right)$.(12)

$-\frac{\partial {T}_{b}}{\partial z}=\frac{{T}_{0}-{T}_{u}}{d}h\left(z\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{ }0\le z\le d$.(13)

$-\frac{\partial {T}_{mb}}{\partial {z}_{m}}=\frac{{T}_{l}-{T}_{0}}{{d}_{m}}{h}_{m}\left({z}_{m}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{ }-{d}_{m}\le {z}_{m}\le 0$.(14)

${C}_{1b}\left(z\right)={C}_{10}-\frac{\left({C}_{10}-{C}_{1u}\right)z}{d}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{ }0\le z\le d$.(15)

${C}_{1mb}\left({z}_{m}\right)={C}_{10}-\frac{\left({C}_{1l}-{C}_{10}\right){z}_{m}}{{d}_{m}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}-{d}_{m}\le {z}_{m}\le 0$.(16)

${C}_{2b}\left(z\right)={C}_{20}-\frac{\left({C}_{20}-{C}_{2u}\right)z}{d}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{ }0\le z\le d$.(17)

${C}_{2mb}\left({z}_{m}\right)={C}_{20}-\frac{\left({C}_{2l}-{C}_{20}\right){z}_{m}}{{d}_{m}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{ }-{d}_{m}\le {z}_{m}\le 0$.(18)

where ${T}_{0}=\frac{\kappa {d}_{m}{T}_{u}+{\kappa }_{m}d{T}_{l}}{\kappa {d}_{m}+{\kappa }_{m}d}$. ${C}_{10}=\frac{{\kappa }_{1}{d}_{m}{C}_{1u}+{\kappa }_{1m}d{C}_{1l}}{{\kappa }_{1}{d}_{m}+{\kappa }_{1m}d}$. ${C}_{20}=\frac{{\kappa }_{2}{d}_{m}{C}_{2u}+{\kappa }_{2m}d{C}_{2l}}{{\kappa }_{2}{d}_{m}+{\kappa }_{2m}d}$

are the interface temperature and concentrations, $h\left(z\right)$.nd ${h}_{m}\left({z}_{m}\right)$.re temperature gradients in fluid and porous layers respectively and the subscript “b” denotes the basic state.

To examine the stability of the system, we give a small perturbation to the system as

$q={q}_{b}+{q}^{\prime },P={P}_{b}+{P}^{\prime },T={T}_{b}\left(z\right)+\theta ,{C}_{1}={C}_{1b}\left(z\right)+{S}_{1},{C}_{2}={C}_{2b}\left(z\right)+{S}_{2}$.(19)

$\begin{array}{l}{q}_{m}={q}_{mb}+{{q}^{\prime }}_{m},{P}_{m}={P}_{mb}+{{P}^{\prime }}_{m},{T}_{m}={T}_{mb}\left({z}_{m}\right)+{\theta }_{m},\\ {C}_{1m}={C}_{1mb}\left({z}_{m}\right)+{S}_{m1},{C}_{2m}={C}_{2mb}\left({z}_{m}\right)+{S}_{m2}\end{array}$.(20)

where the primed quantities are the dimensionless one. Introducing (19) & (20) are substituted into the (1) to (10), apply curl twice to eliminate the pressure term from (2) to (7) and only the vertical component is retained. The variables

are then nondimensionalised using $\frac{{d}^{2}}{\kappa },\frac{\kappa }{d},{T}_{0}-{T}_{u},{C}_{10}-{C}_{1u},{C}_{20}-{C}_{2u}$.n the fluid layer and $\frac{{d}_{m}^{2}}{{\kappa }_{m}},\frac{{\kappa }_{m}}{{d}_{m}},{T}_{l}-{T}_{0},{C}_{1l}-{C}_{10},{C}_{2l}-{C}_{20}$.s the corresponding characteristic quantities in the porous layer.

To render the equations nondimensional, we choose different scales for the two layers (Chen and Chen  , Nield  ), so that both layers are of unit length such that $\left(x,y,z\right)=d\left({x}^{\prime },{y}^{\prime },{z}^{\prime }\right)$. $\left({x}_{m},{y}_{m},{z}_{m}\right)={d}_{m}\left({{x}^{\prime }}_{m},{{y}^{\prime }}_{m},{{z}^{\prime }}_{m}-1\right)$.

Omitting the primes for simplicity, we get in $0\le z\le 1$.nd $0\le {z}_{m}\le 1$.espectively

$\frac{1}{Pr}\frac{\partial }{\partial t}\left({\nabla }^{2}W\right)={\nabla }^{4}W$.(21)

$\frac{\partial \theta }{\partial t}=Wh\left(z\right)+{\nabla }^{2}\theta$.(22)

$\frac{\partial {S}_{1}}{\partial t}=W+{\tau }_{1}{\nabla }^{2}{S}_{1}$.(23)

$\frac{\partial {S}_{2}}{\partial t}=W+{\tau }_{2}{\nabla }^{2}{S}_{2}$.(24)

$\frac{{\beta }^{2}}{P{r}_{m}}\frac{\partial }{\partial t}\left({\nabla }_{m}^{2}{W}_{m}\right)=\stackrel{^}{\mu }{\beta }^{2}{\nabla }_{m}^{4}{W}_{m}-{\nabla }_{m}^{2}{W}_{m}$.(25)

$A\frac{\partial {\theta }_{m}}{\partial t}={W}_{m}{h}_{m}\left({z}_{m}\right)+{\nabla }_{m}^{2}{\theta }_{m}$.(26)

$\epsilon \frac{\partial {S}_{m1}}{\partial t}={W}_{m}+{\tau }_{m1}{\nabla }_{m}^{2}{S}_{m1}$.(27)

$\epsilon \frac{\partial {S}_{m2}}{\partial t}={W}_{m}+{\tau }_{m2}{\nabla }_{m}^{2}{S}_{m2}$.(28)

Here, for the fluid layer, $Pr=\frac{\nu }{\kappa }$.s the Prandtl number, ${\tau }_{1}=\frac{{\kappa }_{1}}{\kappa }$.s the ratio of salute1 diffusivity to thermal diffusivity fluid in fluid layer, ${\tau }_{2}=\frac{{\kappa }_{2}}{\kappa }$.s the ratio of salute2 diffusivity to thermal diffusivity fluid in fluid layer. For the porous layer, $P{r}_{m}=\frac{\epsilon {\nu }_{m}}{{\kappa }_{m}}$.s the Prandtl number, ${\beta }^{2}=\frac{K}{{d}_{m}^{2}}=Da$.s the Darcy number, $\beta$.s the porous parameter, $\stackrel{^}{\mu }=\frac{{\mu }_{m}}{\mu }$.s the viscosity ratio, ${\tau }_{m1}=\frac{{\kappa }_{m1}}{{\kappa }_{m}}$.s the ratio of salute1 diffusivity to thermal diffusivity of the porous layer, ${\tau }_{m2}=\frac{{\kappa }_{m2}}{{\kappa }_{m}}$.s the ratio of salute2 diffusivity to thermal diffusivity of the porous layer, $h\left(z\right)$.nd ${h}_{m}\left({z}_{m}\right)$.re the non-dimensional temperature gradients with $\underset{0}{\overset{1}{\int }}h\left(z\right)\text{d}z=1$.nd $\underset{0}{\overset{1}{\int }}{h}_{m}\left({z}_{m}\right)\text{d}{z}_{m}=1$. $\theta$.nd ${\theta }_{m}$.re the temperature in fluid and porous layers

respectively, S and ${S}_{m}$.re the concentration in fluid and porous layers respectively and W and ${W}_{m}$.re the dimensionless vertical velocity in fluid and porous layer respectively.

We apply normal mode expansion on dependent variables as follows:

$\left[\begin{array}{c}W\\ \theta \\ {S}_{1}\\ {S}_{2}\end{array}\right]=\left[\begin{array}{c}W\left(z\right)\\ \theta \left(z\right)\\ {S}_{1}\left(z\right)\\ {S}_{2}\left(z\right)\end{array}\right]f\left(x,y\right){\text{e}}^{nt}$.(29)

$\left[\begin{array}{c}{W}_{m}\\ {\theta }_{m}\\ {S}_{m1}\\ {S}_{m2}\end{array}\right]=\left[\begin{array}{c}{W}_{m}\left({z}_{m}\right)\\ {\theta }_{m}\left({z}_{m}\right)\\ {S}_{m1}\left({z}_{m}\right)\\ {S}_{m2}\left({z}_{m}\right)\end{array}\right]{f}_{m}\left({x}_{m},{y}_{m}\right){\text{e}}^{{n}_{m}t}$.(30)

with ${\nabla }_{2}^{2}f+{a}^{2}f=0$.nd ${\nabla }_{2m}^{2}{f}_{m}+{a}_{m}^{2}{f}_{m}=0$. Here a and ${a}_{m}$.re the nondimensional horizontal wave numbers, n and ${n}_{m}$.re the frequencies. Since the dimensional horizontal wave numbers must be the same for the fluid and porous

layers, we must have $\frac{a}{d}=\frac{{a}_{m}}{{d}_{m}}$.nd hence ${a}_{m}=\stackrel{^}{d}a$.

Introducing Equation (29) and Equation (30) into the Equations (21) to (28) and denoting $\frac{\partial }{\partial z}=D$.nd $\frac{\partial }{\partial {z}_{m}}={D}_{m}$.hen we get an eigenvalue problem consisting

of the following ordinary differential equations in $0\le z\le 1$.nd $0\le {z}_{m}\le 1$.espectively

$\left({D}^{2}-{a}^{2}+\frac{n}{Pr}\right)\left({D}^{2}-{a}^{2}\right)W=0$.(31)

$\left({D}^{2}-{a}^{2}+n\right)\theta +Wh\left(z\right)=0$.(32)

$\left({\tau }_{1}\left({D}^{2}-{a}^{2}\right)+n\right){S}_{1}+W=0$.(33)

$\left({\tau }_{2}\left({D}^{2}-{a}^{2}\right)+n\right){S}_{2}+W=0$.(34)

$\left[\left({D}_{m}^{2}-{a}_{m}^{2}\right)\stackrel{^}{\mu }{\beta }^{2}+\frac{{n}_{m}{\beta }^{2}}{P{r}_{m}}-1\right]\left({D}_{m}^{2}-{a}_{m}^{2}\right){W}_{m}=0$.(35)

$\left({D}_{m}^{2}-{a}_{m}^{2}+A{n}_{m}\right){\theta }_{m}+{W}_{m}{h}_{m}\left({z}_{m}\right)=0$.(36)

$\left({\tau }_{m1}\left({D}_{m}^{2}-{a}_{m}^{2}\right)+{n}_{m}\epsilon \right){S}_{m1}+{W}_{m}=0$.(37)

$\left({\tau }_{m2}\left({D}_{m}^{2}-{a}_{m}^{2}\right)+{n}_{m}\epsilon \right){S}_{m2}+{W}_{m}=0$.(38)

It is known that the principle of exchange of instabilities holds for triple diffusive convection in both fluid and porous layers separately for certain choice of parameters. Therefore, we assume that the principle of exchange of instabilities holds even for the composite layers. In other words, it is assumed that the onset of convection is in the form of steady convection and accordingly we take $n={n}_{m}=0$. We get, in $0\le z\le 1$.nd $0\le {z}_{m}\le 1$.espectively

${\left({D}^{2}-{a}^{2}\right)}^{2}W=0$.(39)

$\left({D}^{2}-{a}^{2}\right)\theta +Wh\left(z\right)=0$.(40)

${\tau }_{1}\left({D}^{2}-{a}^{2}\right){S}_{1}+W=0$.(41)

${\tau }_{2}\left({D}^{2}-{a}^{2}\right){S}_{2}+W=0$.(42)

$\left[\left({D}_{m}^{2}-{a}_{m}^{2}\right)\stackrel{^}{\mu }{\beta }^{2}-1\right]\left({D}_{m}^{2}-{a}_{m}^{2}\right){W}_{m}=0$.(43)

$\left({D}_{m}^{2}-{a}_{m}^{2}\right){\theta }_{m}+{W}_{m}{h}_{m}\left({z}_{m}\right)=0$.(44)

${\tau }_{m1}\left({D}_{m}^{2}-{a}_{m}^{2}\right){S}_{m1}+{W}_{m}=0$.(45)

${\tau }_{m2}\left({D}_{m}^{2}-{a}_{m}^{2}\right){S}_{m2}+{W}_{m}=0$.(46)

3. Boundary Conditions

The boundary conditions are nondimensionlised then subjected to normal mode analysis and finally they take the form

${D}^{2}W\left(1\right)+M{a}^{2}\theta \left(1\right)+{M}_{s1}{a}^{2}{S}_{1}\left(1\right)+{M}_{s2}{a}^{2}{S}_{2}\left(1\right)=0,$

$W\left(1\right)=D\theta \left(1\right)=D{S}_{1}\left(1\right)=D{S}_{2}\left(1\right)=0,$

${W}_{m}\left(0\right)=D{W}_{m}\left(0\right)={D}_{m}{\theta }_{m}\left(0\right)={D}_{m}{S}_{m1}\left(0\right)={D}_{m}{S}_{m2}\left(0\right)=0,$

$\stackrel{^}{T}W\left(0\right)={W}_{m}\left(1\right),\stackrel{^}{T}\stackrel{^}{d}DW\left(0\right)={D}_{m}{W}_{m}\left(1\right),$

$\stackrel{^}{T}{\stackrel{^}{d}}^{2}\left({D}^{2}+{a}^{2}\right)W\left(0\right)=\stackrel{^}{\mu }\left({D}_{m}^{2}+{a}_{m}^{2}\right){W}_{m}\left(1\right),$

$\stackrel{^}{T}{\stackrel{^}{d}}^{3}{\beta }^{2}\left({D}^{3}W\left(0\right)-3{a}^{2}DW\left(0\right)\right)={D}_{m}{W}_{m}\left(1\right)+\stackrel{^}{\mu }{\beta }^{2}\left({D}_{m}^{3}{W}_{m}\left(1\right)-3{a}_{m}^{2}{D}_{m}{W}_{m}\left(1\right)\right),$

$\theta \left(0\right)=\stackrel{^}{T}{\theta }_{m}\left(1\right),D\theta \left(0\right)={D}_{m}{\theta }_{m}\left(1\right),{S}_{1}\left(0\right)=\stackrel{^}{{S}_{1}}{S}_{m1}\left(1\right),$

$D{S}_{1}\left(0\right)={D}_{m}{S}_{m1}\left(1\right),{S}_{2}\left(0\right)=\stackrel{^}{{S}_{2}}{S}_{m2}\left(1\right),D{S}_{2}\left(0\right)={D}_{m}{S}_{m2}\left(1\right)$.(47)

where $\stackrel{^}{{S}_{1}}=\frac{{\kappa }_{s1}}{{\kappa }_{s1m}}$. $\stackrel{^}{{S}_{2}}=\frac{{\kappa }_{s2}}{{\kappa }_{s2m}}$.re the ratios of solute1 and solute2 diffusivities of fluid layer to those of porous layer respectively, $\stackrel{^}{d}=\frac{{d}_{m}}{d}$. depth ratio, $\stackrel{^}{T}=\frac{\kappa }{{\kappa }_{m}}$. ratio of thermal diffusivities of fluid, $M=-\frac{\partial {\sigma }_{t}}{\partial T}\frac{\left({T}_{0}-{T}_{u}\right)d}{\mu \kappa }$.s the Thermal Marangoni number, ${M}_{s1}=-\frac{\partial {\sigma }_{t}}{\partial C}\frac{\left({C}_{10}-{C}_{1u}\right)d}{\mu \kappa }$.s the solute1 Marangoni number and ${M}_{s2}=-\frac{\partial {\sigma }_{t}}{\partial C}\frac{\left({C}_{20}-{C}_{2u}\right)d}{\mu \kappa }$.s the solute2 Marangoni number.

4. Method of Solution

From Equation (39) and Equation (43), we get W and ${W}_{m}$.s

$W\left(z\right)={A}_{1}\mathrm{cosh}az+{A}_{2}z\mathrm{cosh}az+{A}_{3}\mathrm{sinh}az+{A}_{4}z\mathrm{sinh}az$.(48)

${W}_{m}\left({z}_{m}\right)={A}_{5}\mathrm{cosh}{a}_{m}{z}_{m}+{A}_{6}\mathrm{sinh}{a}_{m}{z}_{m}+{A}_{7}\mathrm{cosh}{\delta }_{m}{z}_{m}+{A}_{8}\mathrm{sinh}{\delta }_{m}{z}_{m}$.(49)

where ${\delta }_{m}=\sqrt{\frac{{a}_{m}^{2}+1}{\stackrel{^}{\mu }{\beta }^{2}}}$.nd ${{A}^{\prime }}_{i}s\left(i=1,2,\cdots ,8\right)$.re arbitrary constants, $W\left(z\right)$.nd ${W}_{m}\left({z}_{m}\right)$.re suitably written as

$W\left(z\right)={A}_{1}\left[\mathrm{cosh}az+{a}_{1}z\mathrm{cosh}az+{a}_{2}\mathrm{sinh}az+{a}_{3}z\mathrm{sinh}az\right]$.(50)

${W}_{m}\left({z}_{m}\right)={A}_{1}\left[{a}_{4}\mathrm{cosh}{a}_{m}{z}_{m}+{a}_{5}\mathrm{sinh}{a}_{m}{z}_{m}+{a}_{6}\mathrm{cosh}{\delta }_{m}{z}_{m}+{a}_{7}\mathrm{sinh}{\delta }_{m}{z}_{m}\right]$.(51)

where

${a}_{1}=\frac{{a}_{6}{\Delta }_{12}+{a}_{7}{\Delta }_{13}}{\stackrel{^}{T}\stackrel{^}{d}},{a}_{2}=\frac{{a}_{6}{\Delta }_{10}+{a}_{7}{\Delta }_{11}}{{\Delta }_{9}},{a}_{3}=\frac{{a}_{6}{\Delta }_{5}+{a}_{7}{\Delta }_{6}-{\Delta }_{8}}{{\Delta }_{7}},$

${a}_{4}=-{a}_{6},{a}_{5}=\frac{-{a}_{7}{\delta }_{m}}{{a}_{m}},{a}_{6}={\Delta }_{19},{a}_{7}=\frac{{\Delta }_{17}}{{\Delta }_{18}}$

${\Delta }_{1}=\mathrm{cosh}{\delta }_{m}-\mathrm{cosh}{a}_{m},{\Delta }_{2}=\mathrm{sinh}{\delta }_{m}-\frac{{\delta }_{m}\mathrm{sinh}{a}_{m}}{{a}_{m}},$

${\Delta }_{3}={\delta }_{m}\mathrm{sinh}{\delta }_{m}-{a}_{m}\mathrm{sinh}{a}_{m},{\Delta }_{4}={\delta }_{m}\left(\mathrm{cosh}{\delta }_{m}-\mathrm{cosh}{a}_{m}\right),$

${\Delta }_{5}=\left({\delta }_{m}^{2}+{a}_{m}^{2}\right)\mathrm{cosh}{\delta }_{m}-2{a}_{m}^{2}\mathrm{cosh}{a}_{m},$

${\Delta }_{6}=\left({\delta }_{m}^{2}+{a}_{m}^{2}\right)\mathrm{sinh}{\delta }_{m}-2{a}_{m}{\delta }_{m}\mathrm{sinh}{a}_{m},$

${\Delta }_{7}=\frac{2a\stackrel{^}{T}{\stackrel{^}{d}}^{2}}{\stackrel{^}{\mu }},{\Delta }_{8}=\frac{2{a}^{2}\stackrel{^}{T}{\stackrel{^}{d}}^{2}}{\stackrel{^}{\mu }},{\Delta }_{9}=-2{a}^{3}\stackrel{^}{T}{\stackrel{^}{d}}^{3}{\beta }^{2},$

${\Delta }_{10}={a}_{m}\mathrm{sinh}{a}_{m}+2{a}_{m}^{2}\stackrel{^}{\mu }{\beta }^{2}\mathrm{sinh}{a}_{m}-{\delta }_{m}\mathrm{sinh}{\delta }_{m}+{\Delta }_{100},$

${\Delta }_{100}=\stackrel{^}{\mu }{\beta }^{2}\left({\delta }_{m}^{3}\mathrm{sinh}{\delta }_{m}-3{a}_{m}^{2}{\delta }_{m}\mathrm{sinh}{\delta }_{m}\right),$

${\Delta }_{11}={a}_{m}\mathrm{cosh}{a}_{m}+2{a}_{m}^{2}\stackrel{^}{\mu }{\beta }^{2}\mathrm{cosh}{a}_{m}-{\delta }_{m}\mathrm{cosh}{\delta }_{m}+{\Delta }_{110},$

${\Delta }_{110}=\stackrel{^}{\mu }{\beta }^{2}\left({\delta }_{m}^{3}\mathrm{cosh}{\delta }_{m}-3{a}_{m}^{2}{\delta }_{m}\mathrm{cosh}{\delta }_{m}\right),$

${\Delta }_{12}={\Delta }_{3}-a\frac{{\Delta }_{10}}{{\Delta }_{9}},{\Delta }_{13}={\Delta }_{4}-a\frac{{\Delta }_{11}}{{\Delta }_{9}},$

${\Delta }_{14}=\frac{{\Delta }_{12}\mathrm{cosh}a}{\stackrel{^}{T}\stackrel{^}{d}}+\left(\frac{{\Delta }_{10}{\Delta }_{7}+{\Delta }_{5}{\Delta }_{9}}{{\Delta }_{7}{\Delta }_{9}}\right)\mathrm{sinh}a,$

${\Delta }_{15}=\frac{{\Delta }_{13}\mathrm{cosh}a}{\stackrel{^}{T}\stackrel{^}{d}}+\left(\frac{{\Delta }_{11}{\Delta }_{7}+{\Delta }_{6}{\Delta }_{9}}{{\Delta }_{7}{\Delta }_{9}}\right)\mathrm{sinh}a,$

${\Delta }_{16}=\frac{{\Delta }_{8}\mathrm{sinh}a}{{\Delta }_{7}}-\mathrm{cosh}a,{\Delta }_{17}=\stackrel{^}{T}{\Delta }_{14}-{\Delta }_{1}{\Delta }_{16},$

${\Delta }_{18}={\Delta }_{2}{\Delta }_{14}-{\Delta }_{1}{\Delta }_{15},{\Delta }_{19}=\frac{1}{{\Delta }_{1}}\left(\stackrel{^}{T}-\frac{{\Delta }_{2}{\Delta }_{17}}{{\Delta }_{18}}\right).$

We get the species concentration for fluid layer ${S}_{1}$. ${S}_{2}$.rom (41) and (42) also from (45) and (46) species concentration for porous layer ${S}_{m1}$. ${S}_{m2}$.s

${S}_{1}\left(z\right)={A}_{1}\left[{a}_{12}\mathrm{cosh}az+{a}_{13}\mathrm{sinh}az+\frac{f\left(z\right)}{{\tau }_{1}}\right]$.(52)

${S}_{2}\left(z\right)={A}_{1}\left[{a}_{16}\mathrm{cosh}az+{a}_{17}\mathrm{sinh}az+\frac{f\left(z\right)}{{\tau }_{2}}\right]$.(53)

${S}_{m1}\left({z}_{m}\right)={A}_{1}\left[{a}_{14}\mathrm{cosh}{a}_{m}{z}_{m}+{a}_{15}\mathrm{sinh}{a}_{m}{z}_{m}+\frac{{f}_{m}\left({z}_{m}\right)}{{\tau }_{m1}}\right]$.(54)

${S}_{m2}\left({z}_{m}\right)={A}_{1}\left[{a}_{18}\mathrm{cosh}{a}_{m}{z}_{m}+{a}_{19}\mathrm{sinh}{a}_{m}{z}_{m}+\frac{{f}_{m}\left({z}_{m}\right)}{{\tau }_{m2}}\right]$.(55)

where

$f\left(z\right)={R}_{1}-{R}_{2},{f}_{m}\left({z}_{m}\right)=-\left({R}_{3}+{R}_{4}\right),$

${R}_{1}=\frac{z}{4{a}^{2}}\left[\left({a}_{1}-az{a}_{3}\right)\mathrm{cosh}az+\left({a}_{3}-az{a}_{1}\right)\mathrm{sinh}az\right],$

${R}_{2}=\frac{z}{2a}\left(\mathrm{sinh}az+{a}_{2}\mathrm{cosh}az\right),$

${R}_{3}=\frac{1}{{\delta }_{m}^{2}-{a}_{m}^{2}}\left({a}_{6}\mathrm{cosh}{\delta }_{m}{z}_{m}+{a}_{7}\mathrm{sinh}{\delta }_{m}{z}_{m}\right),$

${R}_{4}=\frac{{z}_{m}}{2{a}_{m}}\left({a}_{4}\mathrm{sinh}{a}_{m}{z}_{m}+{a}_{5}\mathrm{cosh}{a}_{m}{z}_{m}\right),$

${a}_{12}=\stackrel{^}{{S}_{1}}\left({a}_{14}\mathrm{cosh}{a}_{m}+{a}_{15}\mathrm{sinh}{a}_{m}\right)-{\Delta }_{27},$

${a}_{13}=\frac{1}{a}\left({a}_{m}{a}_{15}\mathrm{cosh}{a}_{m}+{a}_{14}{a}_{m}\mathrm{sinh}{a}_{m}+{\Delta }_{28}\right),$

${a}_{14}=\frac{{\Delta }_{30}}{{\Delta }_{31}},{a}_{15}=\frac{{\Delta }_{29}}{{a}_{m}},{a}_{16}=\stackrel{^}{{S}_{2}}\left({a}_{18}\mathrm{cosh}{a}_{m}+{a}_{19}\mathrm{sinh}{a}_{m}\right)-{\Delta }_{32},$

${a}_{17}=\frac{1}{a}\left({a}_{18}{a}_{m}\mathrm{sinh}{a}_{m}+{a}_{19}{a}_{m}\mathrm{cosh}{a}_{m}+{\Delta }_{33}\right),$

${a}_{18}=\frac{{\Delta }_{36}}{{\Delta }_{37}},{a}_{19}=\frac{{\Delta }_{35}}{{a}_{m}},{\Delta }_{26}=\frac{1}{{\tau }_{1}}\left[{\Delta }_{22}\right],{\Delta }_{27}=\frac{{\Delta }_{20}\stackrel{^}{{S}_{1}}}{\stackrel{^}{T}{\tau }_{m1}},$

${\Delta }_{28}=\frac{1}{{\tau }_{1}}\left(\frac{2a{a}_{2}-{a}_{1}}{4{a}^{2}}\right)-\frac{1}{{\tau }_{m1}}\left[\frac{1}{2{a}_{m}}\left({a}_{4}\mathrm{sinh}{a}_{m}+{a}_{5}\mathrm{cosh}{a}_{m}\right)+{\Delta }_{280}\right],$

${\Delta }_{280}=\frac{1}{2}\left({a}_{5}\mathrm{sinh}{a}_{m}+{a}_{4}\mathrm{cosh}{a}_{m}\right)+\frac{{\delta }_{m}}{{\delta }_{m}^{2}-{a}_{m}^{2}}\left({a}_{6}\mathrm{sinh}{\delta }_{m}+{a}_{7}\mathrm{cosh}{\delta }_{m}\right),$

${\Delta }_{29}=\frac{1}{{\tau }_{m1}}\left(\frac{{a}_{5}}{2{a}_{m}}+\frac{{a}_{7}{\delta }_{m}}{{\delta }_{m}^{2}-{a}_{m}^{2}}\right),$

${\Delta }_{30}={\Delta }_{26}-{\Delta }_{28}\mathrm{cosh}a+a{\Delta }_{27}\mathrm{sinh}a-{\Delta }_{29}\mathrm{cosh}a\mathrm{cosh}{a}_{m}+{\Delta }_{300},$

${\Delta }_{300}=-\frac{{\Delta }_{29}}{{a}_{m}}\left(\stackrel{^}{{S}_{1}}a\mathrm{sinh}a\mathrm{sinh}{a}_{m}\right),$

${\Delta }_{31}=\stackrel{^}{{S}_{1}}a\mathrm{sinh}a\mathrm{cosh}{a}_{m}+{a}_{m}\mathrm{cosh}a\mathrm{sinh}{a}_{m},{\Delta }_{32}=\frac{{\tau }_{m1}\stackrel{^}{{S}_{2}}{\Delta }_{27}}{{\tau }_{m2}\stackrel{^}{{S}_{1}}},$

${\Delta }_{33}=\frac{1}{{\tau }_{2}}\left(\frac{2a{a}_{2}-{a}_{1}}{4{a}^{2}}\right)-\frac{1}{{\tau }_{m2}}\left[\frac{1}{2{a}_{m}}\left({a}_{4}\mathrm{sinh}{a}_{m}+{a}_{5}\mathrm{cosh}{a}_{m}\right)+{\Delta }_{280}\right],$

${\Delta }_{34}=\frac{1}{{\Delta }_{22}{\tau }_{2}}\left[{\Delta }_{26}{\tau }_{1}\right],{\Delta }_{35}=\frac{1}{{\tau }_{m2}}\left(\frac{{a}_{5}}{2{a}_{m}}+\frac{{a}_{7}{\delta }_{m}}{{\delta }_{m}^{2}-{a}_{m}^{2}}\right),$

${\Delta }_{36}={\Delta }_{34}-{\Delta }_{33}\mathrm{cosh}a+a{\Delta }_{32}\mathrm{sinh}a-{\Delta }_{35}\mathrm{cosh}a\mathrm{cosh}{a}_{m}-{\Delta }_{360},$

${\Delta }_{360}=-\frac{{\Delta }_{35}}{{a}_{m}}\left(\stackrel{^}{{S}_{2}}a\mathrm{sinh}a\mathrm{sinh}{a}_{m}\right),$

${\Delta }_{37}=\stackrel{^}{{S}_{2}}a\mathrm{sinh}a\mathrm{cosh}{a}_{m}+{a}_{m}\mathrm{cosh}a\mathrm{sinh}{a}_{m}$

4.1. Linear Temperature Profile

For this case

$h\left(z\right)={h}_{m}\left({z}_{m}\right)=1$.(56)

Substituting Equation (56) into the heat Equation (40) and Equation (44), we get $\theta$.nd ${\theta }_{m}$.s

$\theta \left(z\right)={A}_{1}\left[{a}_{8}\mathrm{cosh}az+{a}_{9}\mathrm{sinh}az+f\left(z\right)\right]$.(57)

${\theta }_{m}\left({z}_{m}\right)={A}_{1}\left[{a}_{10}\mathrm{cosh}{a}_{m}{z}_{m}+{a}_{11}\mathrm{sinh}{a}_{m}{z}_{m}+{f}_{m}\left({z}_{m}\right)\right]$.(58)

where

${a}_{8}=\stackrel{^}{T}\left({a}_{10}\mathrm{cosh}{a}_{m}+{a}_{11}\mathrm{sinh}{a}_{m}\right)-{\Delta }_{20},$

${a}_{9}=\frac{1}{a}\left({a}_{10}{a}_{m}\mathrm{sinh}{a}_{m}+{a}_{11}{a}_{m}\mathrm{cosh}{a}_{m}-{\Delta }_{21}\right),$

${a}_{10}=\frac{{\Delta }_{24}}{{\Delta }_{25}},{a}_{11}=\frac{{\Delta }_{23}}{{a}_{m}},$

${\Delta }_{20}=\frac{\stackrel{^}{T}}{2{a}_{m}}\left({a}_{4}\mathrm{sinh}{a}_{m}+{a}_{5}\mathrm{cosh}{a}_{m}\right)+\frac{\stackrel{^}{T}}{{\delta }_{m}^{2}-{a}_{m}^{2}}\left({a}_{7}\mathrm{sinh}{\delta }_{m}+{a}_{6}\mathrm{cosh}{\delta }_{m}\right),$

${\Delta }_{21}=-\left(\frac{2a{a}_{2}-{a}_{1}}{4{a}^{2}}\right)+\frac{1}{2{a}_{m}}\left({a}_{4}\mathrm{sinh}{a}_{m}+{a}_{5}\mathrm{cosh}{a}_{m}\right)+{\Delta }_{330},$

${\Delta }_{22}=\frac{1}{4{a}^{2}}\left[\left(\left({a}^{2}-1\right){a}_{3}+2a\right)\mathrm{sinh}a+\left(\left({a}^{2}-1\right){a}_{1}+2a{a}_{2}\right)\mathrm{cosh}a\right]+{\Delta }_{220},$

${\Delta }_{220}=\frac{1}{4a}\left[\left({a}_{1}+2a{a}_{2}\right)\mathrm{sinh}a+\left({a}_{3}+2a\right)\mathrm{cosh}a\right],$

${\Delta }_{23}=\frac{{a}_{5}}{2{a}_{m}}+\frac{{a}_{7}{\delta }_{m}}{{\delta }_{m}^{2}-{a}_{m}^{2}},$

${\Delta }_{24}={\Delta }_{22}+{\Delta }_{21}\mathrm{cosh}a+a{\Delta }_{20}\mathrm{sinh}a+{\Delta }_{23}\mathrm{cosh}a\mathrm{cosh}{a}_{m}-{\Delta }_{240},$

${\Delta }_{240}=\frac{{\Delta }_{23}}{{a}_{m}}\left(\stackrel{^}{T}a\mathrm{sinh}a\mathrm{sinh}{a}_{m}\right),$

${\Delta }_{25}=\stackrel{^}{T}a\mathrm{sinh}a\mathrm{cosh}{a}_{m}+{a}_{m}\mathrm{cosh}a\mathrm{sinh}{a}_{m}.$

The Thermal Marangoni number for this model obtained from (47)1 and is found to be

${M}_{1}=-\frac{{\Lambda }_{1}+{\Lambda }_{2}+{\Lambda }_{3}}{{\Lambda }_{4}}$.(59)

where

${\Lambda }_{1}=\left({a}^{2}+{a}^{2}{a}_{1}+2a{a}_{3}\right)\mathrm{cosh}a+\left({a}^{2}{a}_{3}+{a}^{2}{a}_{2}+2a{a}_{1}\right)\mathrm{sinh}a,$

${\Lambda }_{2}={M}_{s1}{a}^{2}\left[{a}_{12}\mathrm{cosh}a+{a}_{13}\mathrm{sinh}a-\frac{{\Omega }_{1}}{{\tau }_{1}}\right],$

${\Lambda }_{3}={M}_{s2}{a}^{2}\left[{a}_{16}\mathrm{cosh}a+{a}_{17}\mathrm{sinh}a-\frac{{\Omega }_{1}}{{\tau }_{2}}\right],$

${\Lambda }_{4}={a}^{2}\left[{a}_{8}\mathrm{cosh}a+{a}_{9}\mathrm{sinh}a-{\Omega }_{1}\right],$

${\Omega }_{1}=\frac{1}{4{a}^{2}}\left[\left(2a+a{a}_{1}-{a}_{3}\right)\mathrm{sinh}a+\left(2a{a}_{2}-{a}_{1}+a{a}_{3}\right)\mathrm{cosh}a\right]$

4.2. Parabolic Temperature Profile

We consider the profile as following (Sparrow et al.  ):

$h\left(z\right)=2z\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{h}_{m}\left({z}_{m}\right)=2{z}_{m}$.(60)

Substituting Equation (60) into the heat Equation (40) and Equation (44), we get $\theta$.nd ${\theta }_{m}$.s

$\theta \left(z\right)={A}_{1}\left[{a}_{20}\mathrm{cosh}az+{a}_{21}\mathrm{sinh}az+L\left(z\right)\right]$.(61)

${\theta }_{m}\left({z}_{m}\right)={A}_{1}\left[{a}_{22}\mathrm{cosh}{a}_{m}{z}_{m}+{a}_{23}\mathrm{sinh}{a}_{m}{z}_{m}+{L}_{m}\left({z}_{m}\right)\right]$.(62)

where

$L\left(z\right)=-\left({R}_{5}+{R}_{6}\right),{L}_{m}\left({z}_{m}\right)=-\left({R}_{7}+{R}_{8}\right),$

${R}_{5}=\left(\frac{{z}^{2}}{2a}-\frac{{a}_{2}z}{2{a}^{2}}\right)\mathrm{sinh}az+\left(\frac{{a}_{2}{z}^{2}}{2a}-\frac{z}{2{a}^{2}}\right)\mathrm{cosh}az,$

${R}_{6}=\left(\frac{\left(2{a}^{2}{z}^{3}+3z\right){a}_{1}-3a{a}_{3}{z}^{2}}{6{a}^{3}}\right)\mathrm{sinh}az+\left(\frac{\left(2{a}^{2}{z}^{3}+3z\right){a}_{3}-3a{a}_{1}{z}^{2}}{6{a}^{3}}\right)\mathrm{cosh}az,$

${R}_{7}=\frac{{z}_{m}^{2}}{2{a}_{m}}\left({a}_{4}\mathrm{sinh}{a}_{m}{z}_{m}+{a}_{5}\mathrm{cosh}{a}_{m}{z}_{m}\right)-{R}_{70},$

${R}_{70}=\frac{{z}_{m}}{2{a}_{m}^{2}}\left({a}_{5}\mathrm{sinh}{a}_{m}{z}_{m}+{a}_{4}\mathrm{cosh}{a}_{m}{z}_{m}\right),$

${R}_{8}=\frac{2{z}_{m}\left({a}_{7}\mathrm{sinh}{\delta }_{m}{z}_{m}+{a}_{6}\mathrm{cosh}{\delta }_{m}{z}_{m}\right)}{{\delta }_{m}^{2}-{a}_{m}^{2}}-{R}_{80},$

${R}_{80}=\frac{4{\delta }_{m}\left({a}_{6}\mathrm{sinh}{\delta }_{m}{z}_{m}+{a}_{7}\mathrm{cosh}{\delta }_{m}{z}_{m}\right)}{{\left({\delta }_{m}^{2}-{a}_{m}^{2}\right)}^{2}},$

${a}_{20}=\stackrel{^}{T}\left({a}_{22}\mathrm{cosh}{a}_{m}+{a}_{23}\mathrm{sinh}{a}_{m}\right)-{\Delta }_{38},$

${a}_{21}=\frac{{a}_{22}{a}_{m}\mathrm{sinh}{a}_{m}+{a}_{23}{a}_{m}\mathrm{cosh}{a}_{m}-{\Delta }_{39}}{a},{a}_{22}=\frac{{\Delta }_{42}}{{\Delta }_{43}},{a}_{23}=\frac{{\Delta }_{41}}{{a}_{m}},$

${\Delta }_{38}=\stackrel{^}{T}\left[\frac{1}{2{a}_{m}}\left({a}_{4}\mathrm{sinh}{a}_{m}+{a}_{5}\mathrm{cosh}{a}_{m}\right)-{R}_{380}+{\Delta }_{381}\right],$

${\Delta }_{380}=\frac{1}{2{a}_{m}^{2}}\left({a}_{5}\mathrm{sinh}{a}_{m}+{a}_{4}\mathrm{cosh}{a}_{m}\right),$

${\Delta }_{381}=\frac{4{\delta }_{m}}{{\left({\delta }_{m}^{2}-{a}_{m}^{2}\right)}^{2}}\left({a}_{6}\mathrm{sinh}{\delta }_{m}+{a}_{7}\mathrm{cosh}{\delta }_{m}\right)+{\Delta }_{382},$

${\Delta }_{382}=\frac{2}{{\delta }_{m}^{2}-{a}_{m}^{2}}\left({a}_{7}\mathrm{sinh}{\delta }_{m}+{a}_{6}\mathrm{cosh}{\delta }_{m}\right),$

${\Delta }_{39}=-\left(\frac{{a}_{3}-a}{2{a}^{3}}\right)+\frac{2{\delta }_{m}}{{\left({\delta }_{m}^{2}-{a}_{m}^{2}\right)}^{2}}\left({a}_{6}\mathrm{sinh}{\delta }_{m}+{a}_{7}\mathrm{cosh}{\delta }_{m}\right)+{\Delta }_{390}+{\Delta }_{391},$

${\Delta }_{390}=\frac{{a}_{4}\mathrm{sinh}{a}_{m}+{a}_{5}\mathrm{cosh}{a}_{m}}{2{a}_{m}}+\frac{\left({a}_{m}^{2}-1\right)\left({a}_{5}\mathrm{sinh}{a}_{m}+{a}_{4}\mathrm{cosh}{a}_{m}\right)}{2{a}_{m}^{2}},$

${\Delta }_{391}=-\frac{2\left({\delta }_{m}^{2}+{a}_{m}^{2}\right)}{{\left({\delta }_{m}^{2}-{a}_{m}^{2}\right)}^{2}}\left({a}_{7}\mathrm{sinh}{\delta }_{m}+{a}_{6}\mathrm{cosh}{\delta }_{m}\right),$

${\Delta }_{40}=\left(\frac{{a}_{2}{a}^{2}-{a}_{2}}{2{a}^{2}}+\frac{1}{2a}\right)\mathrm{sinh}a+\left(\frac{{a}^{2}-1}{2{a}^{2}}+\frac{{a}_{2}}{2a}\right)\mathrm{cosh}a+{\Delta }_{400},$

${\Delta }_{400}=\left(\frac{3{a}_{1}\left({a}^{2}+1\right)+a{a}_{3}\left(2{a}^{2}-3\right)}{6{a}^{3}}\right)\mathrm{sinh}a+{\Delta }_{401},$

${\Delta }_{401}=\left(\frac{3{a}_{3}\left({a}^{2}+1\right)+a{a}_{1}\left(2{a}^{2}-3\right)}{6{a}^{3}}\right)\mathrm{cosh}a,$

${\Delta }_{41}=-\frac{{a}_{4}}{2{a}_{m}^{2}}-\frac{2{a}_{6}{\left({\delta }_{m}^{2}+{a}_{m}^{2}\right)}^{2}}{{\left({\delta }_{m}^{2}-{a}_{m}^{2}\right)}^{2}},$

${\Delta }_{42}={\Delta }_{40}+{\Delta }_{39}\mathrm{cosh}a+a{\Delta }_{38}\mathrm{sinh}a-{\Delta }_{41}\mathrm{cosh}a\mathrm{cosh}{a}_{m}+{\Delta }_{420},$

${\Delta }_{420}=-\frac{{\Delta }_{41}}{{a}_{m}}\left(\stackrel{^}{{S}_{1}}a\mathrm{sinh}a\mathrm{sinh}{a}_{m}\right),{\Delta }_{43}={\Delta }_{25}.$

The thermal Marangoni number for this model obtained from (47)1 and is found to be

${M}_{2}=-\frac{{\Lambda }_{1}+{\Lambda }_{2}+{\Lambda }_{3}}{{\Lambda }_{5}}$.(63)

where

${\Lambda }_{5}={a}^{2}\left[{a}_{20}\mathrm{cosh}a+{a}_{21}\mathrm{sinh}a-{\Omega }_{3}\right],$

${\Omega }_{3}=\frac{1}{6{a}^{3}}\left[{R}_{9}+{R}_{10}\right],$

${R}_{9}=\left(3a\left(a-{a}_{2}-{a}_{3}\right)+3{a}_{1}+2{a}^{2}{a}_{1}\right)\mathrm{sinh}a,$

${R}_{10}=\left(3a\left(a{a}_{2}-1-{a}_{1}\right)+3{a}_{3}+2{a}^{2}{a}_{3}\right)\mathrm{cosh}a.$

4.3. Inverted Parabolic Temperature Profile

We have

$h\left(z\right)=2\left(1-z\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{h}_{m}\left({z}_{m}\right)=2\left(1-{z}_{m}\right)$.(64)

Substituting Equation (64) into the heat Equation (40) and Equation (44), we get $\theta$.nd ${\theta }_{m}$.s

$\theta \left(z\right)={A}_{1}\left[{a}_{24}\mathrm{cosh}az+{a}_{25}\mathrm{sinh}az+\Psi \left(z\right)\right]$.(65)

${\theta }_{m}\left({z}_{m}\right)={A}_{1}\left[{a}_{26}\mathrm{cosh}{a}_{m}{z}_{m}+{a}_{27}\mathrm{sinh}{a}_{m}{z}_{m}+{\Psi }_{m}\left({z}_{m}\right)\right]$.(66)

where

$\Psi \left(z\right)=-\left({R}_{11}+{R}_{12}+{R}_{13}\right),{\Psi }_{m}\left({z}_{m}\right)=-\left({R}_{14}+{R}_{15}+{R}_{16}+{R}_{17}\right),$

${R}_{11}=\left(\frac{z}{a}-\frac{{z}^{2}}{2a}+\frac{{a}_{2}z}{2{a}^{2}}\right)\mathrm{sinh}az+\left(\frac{{a}_{2}z}{a}-\frac{{a}_{2}{z}^{2}}{2a}+\frac{z}{2{a}^{2}}\right)\mathrm{cosh}az,$

${R}_{12}=\left(\frac{{z}^{2}}{2a}-\frac{{z}^{3}}{3a}-\frac{z}{2{a}^{3}}\right)\left({a}_{1}\mathrm{sinh}az+{a}_{3}\mathrm{cosh}az\right),$

${R}_{13}=\left(\frac{{z}^{2}-z}{2{a}^{2}}\right)\left({a}_{3}\mathrm{sinh}az+{a}_{1}\mathrm{cosh}az\right),$

${R}_{14}=\left(\frac{{z}_{m}}{{a}_{m}}-\frac{{z}_{m}^{2}}{2{a}_{m}}\right)\left({a}_{4}\mathrm{sinh}{a}_{m}{z}_{m}+{a}_{5}\mathrm{cosh}{a}_{m}{z}_{m}\right),$

${R}_{15}=\frac{{z}_{m}}{2{a}_{m}^{2}}\left({a}_{5}\mathrm{sinh}{a}_{m}{z}_{m}+{a}_{4}\mathrm{cosh}{a}_{m}{z}_{m}\right),$

${R}_{16}=\frac{2\left(1-{z}_{m}\right)}{{\delta }_{m}^{2}-{a}_{m}^{2}}\left({a}_{7}\mathrm{sinh}{\delta }_{m}{z}_{m}+{a}_{6}\mathrm{cosh}{\delta }_{m}{z}_{m}\right),$

${R}_{17}=\frac{4{\delta }_{m}}{{\left({\delta }_{m}^{2}-{a}_{m}^{2}\right)}^{2}}\left({a}_{6}\mathrm{sinh}{\delta }_{m}{z}_{m}+{a}_{7}\mathrm{cosh}{\delta }_{m}{z}_{m}\right),$

${a}_{24}=\stackrel{^}{T}\left({a}_{26}\mathrm{cosh}{a}_{m}+{a}_{27}\mathrm{sinh}{a}_{m}\right)-{\Delta }_{44},$

${a}_{25}=\frac{1}{a}\left({a}_{m}{a}_{27}\mathrm{cosh}{a}_{m}+{a}_{m}{a}_{26}\mathrm{sinh}{a}_{m}+{\Delta }_{45}\right),$

${a}_{26}=\frac{{\Delta }_{48}}{{\Delta }_{49}},{a}_{27}=\frac{{\Delta }_{47}}{{a}_{m}},$

${\Delta }_{44}={\Delta }_{440}+\frac{1}{2{a}_{m}^{2}}\left({a}_{5}\mathrm{sinh}{a}_{m}+{a}_{4}\mathrm{cosh}{a}_{m}\right),$

${\Delta }_{440}=\frac{1}{2{a}_{m}}\left({a}_{4}\mathrm{sinh}{a}_{m}+{a}_{5}\mathrm{cosh}{a}_{m}\right)+{\Delta }_{441},$

${\Delta }_{441}=\frac{4{\delta }_{m}}{{\left({\delta }_{m}^{2}-{a}_{m}^{2}\right)}^{2}}\left({a}_{6}\mathrm{sinh}{\delta }_{m}+{a}_{7}\mathrm{cosh}{\delta }_{m}\right),$

${\Delta }_{45}=\left(\frac{2{a}^{2}{a}_{2}+a\left(1-{a}_{1}\right)-{a}_{3}}{2{a}^{3}}\right)-{\Delta }_{450}-{\Delta }_{451},$

${\Delta }_{450}=\frac{1}{2{a}_{m}}\left({a}_{4}\mathrm{sinh}{a}_{m}+{a}_{5}\mathrm{cosh}{a}_{m}\right),$

${\Delta }_{451}=\frac{1+{a}_{m}^{2}}{2{a}_{m}^{2}}\left({a}_{5}\mathrm{sinh}{a}_{m}+{a}_{4}\mathrm{cosh}{a}_{m}\right)+{\Delta }_{452},$

${\Delta }_{452}=\frac{2\left({\delta }_{m}^{2}+{a}_{m}^{2}\right)}{{\left({\delta }_{m}^{2}-{a}_{m}^{2}\right)}^{2}}\left({a}_{7}\mathrm{sinh}{\delta }_{m}+{a}_{6}\mathrm{cosh}{\delta }_{m}\right),$

${\Delta }_{46}=\frac{{a}_{2}+{a}_{2}{a}^{2}+a}{2{a}^{2}}\mathrm{sinh}a+\frac{1+{a}^{2}+a{a}_{2}}{2{a}^{2}}\mathrm{cosh}a-{\Delta }_{460},$

${\Delta }_{460}=\left(\frac{{a}_{1}}{2{a}^{3}}+\frac{{a}_{3}}{6}\right)\mathrm{sinh}a+\left(\frac{{a}_{3}}{2{a}^{3}}+\frac{{a}_{1}}{6}\right)\mathrm{cosh}a,$

${\Delta }_{47}=\frac{{a}_{4}}{2{a}_{m}^{2}}+\frac{{a}_{5}}{{a}_{m}}+2{a}_{6}\frac{{\delta }_{m}^{2}+{a}_{m}^{2}}{{\left({\delta }_{m}^{2}-{a}_{m}^{2}\right)}^{2}}+\frac{2{a}_{7}{\delta }_{m}}{{\delta }_{m}^{2}-{a}_{m}^{2}},$

${\Delta }_{48}={\Delta }_{46}-{\Delta }_{49}\mathrm{cosh}a+a{\Delta }_{44}\mathrm{sinh}a-{\Delta }_{47}\mathrm{cosh}a\mathrm{cosh}{a}_{m}-{\Delta }_{480},$

${\Delta }_{480}=\frac{{\Delta }_{47}}{{a}_{m}}\left(\stackrel{^}{T}a\mathrm{sinh}a\mathrm{sinh}{a}_{m}\right),{\Delta }_{49}={\Delta }_{25}.$

The thermal Marangoni number for this model obtained from (47)1 and is found to be

${M}_{3}=-\frac{{\Lambda }_{1}+{\Lambda }_{2}+{\Lambda }_{3}}{{\Lambda }_{6}}$.(67)

where

${\Lambda }_{6}={a}^{2}\left[{a}_{24}\mathrm{cosh}a+{a}_{25}\mathrm{sinh}a-{\Omega }_{4}\right],$

${\Omega }_{4}=\frac{1}{6{a}^{3}}\left[\left({a}^{2}-3\right)\left({a}_{3}\mathrm{cosh}a+{a}_{1}\mathrm{sinh}a\right)\right]+{R}_{18},$

${R}_{18}=\frac{1}{2{a}^{2}}\left[\left(a+{a}_{2}\right)\mathrm{sinh}a+\left(a{a}_{2}+1\right)\mathrm{cosh}a\right]$

5. Results and Discussion

The Thermal Marangoni numbers ${M}_{1}$.or linear, ${M}_{2}$.or parabolic and ${M}_{3}$.or inverted parabolic temperature profiles are obtained. The constraints are drawn against the depth ratio $\stackrel{^}{d}$. The dimensionless fixed values are $\stackrel{^}{T}=1.0$. $\stackrel{^}{S}=1.0$. $a=1.0$. $\beta =0.03$. ${M}_{s1}=10$. ${M}_{s2}=1$. ${\tau }_{1}={\tau }_{2}={\tau }_{m1}={\tau }_{m2}=\stackrel{^}{{S}_{1}}=\stackrel{^}{{S}_{2}}=0.25$.nd $\stackrel{^}{\mu }=2.5$.

The effects of the parameters $a,\beta ,\stackrel{^}{\mu },{\tau }_{1},{\tau }_{m1},\stackrel{^}{{S}_{1}},{M}_{s1}$.nd ${M}_{s2}$.n all the three thermal Marangoni numbers are depicted in Figures 1 to 8. The main observation that the thermal Marangoni numbers of all the three profiles, the inverted

Figure 1. The effects of horizontal wave number a.

Figure 2. The effects of porous parameter $\beta$.

Figure 3. The effects of viscosity ratio $\stackrel{^}{\mu }$.

Figure 4. The effects of ${\tau }_{1}$.

Figure 5. The effects of ${\tau }_{m1}$.

Figure 6. The effects of $\stackrel{^}{{S}_{1}}$.

parabolic profile is the most stable one and the linear profile is the most unstable one as the thermal Marangoni numbers are highest and lowest respectively, for a given set of fixed values of parameters, specially for porous layer dominant systems. For fluid dominant system, there is no much change in the thermal Marangoni numbers for all the profiles.

The variations of a, horizontal wave number on the thermal Marangoni

Figure 7. The effects of solute1 Marangoni number ${M}_{s1}$.

Figure 8. The effects of solute2 Marangoni number ${M}_{s2}$.

numbers ${M}_{1},{M}_{2}$.nd ${M}_{3}$.re respectively shown in Figures 1(a)-(c) for $a=1.0,1.1$.nd 1.2. We observed that the thermal Marangoni number for the inverted parabolic profile is larger than those for the linear and parabolic profiles. For all the profiles, it is evident from the graph that an increase in the value of a, the thermal Marangoni number increases and its effect is to stabilize the system.

The variations of the porous parameter $\beta$.n the three thermal Marangoni numbers are depicted Figures 2(a)-(c). The curves for $\beta =0.03,0.04,0.05$. Increase in the value of $\beta$. i.e., increasing the permeability, the thermal Marangoni numbers decrease for all the three profiles. Hence the surface tension driven triple diffusive convection occurs earlier on increasing the porous parameter, which is physically reasonable, as there is more way for the fluid to move. So, the system is destabilized.

Figures 3(a)-(c) show the variations of viscosity ratio $\stackrel{^}{\mu }$.or the values $\stackrel{^}{\mu }=2.5,3.0,3.5$. Increase in the value of $\stackrel{^}{\mu }$. the values of the thermal Marangoni numbers ${M}_{1},{M}_{2}$.nd ${M}_{3}$.ncreases. So, the increase in the values of viscosity ratio is to stabilize the system and hence the surface tension driven triple diffusive convection is delayed.

Figures 4(a)-(c) display the effects of ${\tau }_{1}$.s the ratio of salute1 diffusivity to thermal diffusivity fluid in fluid layer for ${M}_{1},{M}_{2}$.nd ${M}_{3}$.espectively for the values ${\tau }_{1}=0.25,0.50,0.75$. For all the three profiles, there is a increase in the values of the thermal Marangoni numbers. Increasing the value of ${\tau }_{1}$.he surface tension driven triple diffusive convection becomes slow and hence the system can be stabilized.

Figures 5(a)-(c) display the variations of the value of ${\tau }_{m1}$.s the ratio of salute1 diffusivity to thermal diffusivity of the porous layer for the values ${\tau }_{m1}=0.25,0.50,0.75$. Increasing this ratio, the thermal Marangoni numbers increase for all the three profiles. So, the surface tension driven triple diffusive convection becomes slow and hence the system can be stabilized.

Figures 6(a)-(c) show the effects of ratio of solute1 diffusivity of the fluid in the fluid layer to that of porous layer $\stackrel{^}{{S}_{1}}=0.25,0.50,0.75$. Increasing this ratio, for all the three profiles, there is a small increase in ${M}_{1},{M}_{2}$.nd ${M}_{3}$.o, the surface tension driven triple diffusive convection becomes slow and hence the system can be stabilized.

Figures 7(a)-(c) show the effects of the ${M}_{s1}$.s the solute1 Maran-goni number for ${M}_{s1}=10,50,100$. By increasing the values of Solute1 Marangoni numbers, the thermal Marangoni numbers increase for all the three temperature profiles. So, the surface tension driven triple diffusive convection can be delayed by increasing solute Marangoni number, hence the system can be stabilized.

Figures 8(a)-(c) illustrate the effects of the ${M}_{s2}$.s the solute2 Marangoni number for ${M}_{s2}=10,25,50$. By increasing the values of Solute2 Marangoni numbers, the thermal Marangoni numbers decrease for all the three temperature profiles. So, the surface tension driven triple diffusive convection can be preponed by increasing solute Marangoni number, hence the system can be destabilized.

6. Conclusions

1) The inverted parabolic temperature profile is the most suitable for the situations demanding the control of Marangoni convection, whereas the linear and parabolic profile is suitable for the situations where the convection is needed.

2) By increasing the values of $a,\stackrel{^}{\mu },{\tau }_{1},{\tau }_{m1},\stackrel{^}{{S}_{1}},{M}_{s1}$.nd by decreasing the values of $\beta$.nd ${M}_{s2}$.he surface tension driven triple diffusive convection in a composite layer under microgravity condition can be delayed and hence the system can be stabilized.

3) In the manufacture of pure crystal growth, our work can be useful. The people who are manufacturing crystals can refer this paper. This can give them an initial insight into the effects of parameters in the multicomponent crystal growth problems.

Acknowledgements

We express our gratitude to Prof. N. Rudraiah and Prof. I. S. Shivakumara, UGC-CAS in Fluid mechanics, Bangalore University, Bengaluru, for their help during the formulation of the problem. The author Manjunatha. N, express his sincere thanks to the management of REVA University, Bengaluru, for their encouragement.

Cite this paper
Manjunatha, N. and Sumithra, R. (2019) Effects of Non-Uniform Temperature Gradients on Triple Diffusive Marangoni Convection in a Composite Layer. Open Journal of Applied Sciences, 9, 640-660. doi: 10.4236/ojapps.2019.98052.
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