Effects of Non-Uniform Temperature Gradients on Triple ...For the fluid layer, Chand [8] has applied...

Open Journal of Applied Sciences, 2019, 9, 640-660 http://www.scirp.org/journal/ojapps

ISSN Online: 2165-3925 ISSN Print: 2165-3917

DOI: 10.4236/ojapps.2019.98052 Aug. 22, 2019 640 Open Journal of Applied Sciences

Effects of Non-Uniform Temperature Gradients on Triple Diffusive Marangoni Convection in a Composite Layer

Narayanappa Manjunatha1, Ramakrishna Sumithra2

1School of Applied Sciences, REVA University, Bengaluru, India 2Department of Mathematics, Government Science College, Bengaluru, India

Abstract The problem of triple diffusive Marangoni convection is investigated in a composite layer comprising an incompressible three component fluid satu-rated, sparsely packed porous layer over which lies a layer of the same fluid. The lower rigid surface of the porous layer and the upper free surface are considered to be insulating to temperature, insulating to both salute concen-tration perturbations. At the upper free surface, the surface tension effects depending on temperature and salinities are considered. At the interface, the normal and tangential components of velocity, heat and heat flux, mass and mass flux are assumed to be continuous. The resulting eigenvalue problem is solved exactly for linear, parabolic and inverted parabolic temperature pro-files and analytical expressions of the thermal Marangoni number are ob-tained. The effects of variation of different physical parameters on the ther-mal Marangoni numbers for the profiles are compared.

Keywords Triple Diffusive, Thermal Marangoni Convection, Composite Layer, Temperature Profiles

1. Introduction

There are many fluid systems containing more than two components occurring in nature and engineering applications. The subject of systems having multi components in porous and fluid layers has attracted many researchers due to its importance in the study of crystal growth, geothermally heated lakes, earth core, solidification of molten alloys, underground water flow, acid rain effects, natural

How to cite this paper: Manjunatha, N. and Sumithra, R. (2019) Effects of Non- Uniform Temperature Gradients on Triple Diffusive Marangoni Convection in a Com- posite Layer. Open Journal of Applied Sci- ences, 9, 640-660. https://doi.org/10.4236/ojapps.2019.98052 Received: July 23, 2019 Accepted: August 19, 2019 Published: August 22, 2019 Copyright © 2019 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/

Open Access

http://www.scirp.org/journal/ojappshttps://doi.org/10.4236/ojapps.2019.98052http://www.scirp.orghttps://doi.org/10.4236/ojapps.2019.98052http://creativecommons.org/licenses/by/4.0/

N. Manjunatha, R. Sumithra

DOI: 10.4236/ojapps.2019.98052 641 Open Journal of Applied Sciences

phenomena such as contaminant transport, warming of stratosphere and mag-mas and their laboratory models and sea water etc. The effect of multicompo-nent convection is studied by Griffiths [1] [2], Rudraiah and Vortmeyer [3], Shivakumara [4], Poulikakos [5], Pearlstein et al. [6] and Lopez et al. [7].

For the fluid layer, Chand [8] has applied the linear stability analysis and a normal mode analysis to study the triple-diffusive convection in a micropolar ferromagnetic fluid layer heated and soluted from below. Suresh Chand [9] has investigated the triple-diffusive convection in a micropolar ferrofluid layer heated and soluted below subjected to a transverse uniform magnetic field in the presence of uniform vertical rotation. Shivakumara and Naveen Kumar [10] have investigated the effect of couple stresses on linear and weakly nonlinear stability of a triply diffusive fluid layer using a modified perturbation technique. Kango et al. [11] have studied the theoretical investigation of the triple-diffusive convection in a micropolar ferrofluid layer heated and soluted below subjected to a transverse uniform magnetic field in the presence of uniform vertical rota-tion. Vivek Kumar and Mukesh Kumar Awasthi [12] have considered the prob-lem of triple-diffusive convection in a horizontal nanofluid layer heated and salted from below using linear stability theory and normal mode technique. A linear stability analysis is carried out for triple diffusive convection in Oldroyd-B liquid and rotating couple stress liquid by Sameena Tarannum and Pranesh [13] [14].

In porous medium, Suresh Chand [15] has obtained closed-form of solution for the rotation in a magnetized ferrofluid with internal angular momentum, heated and soluted from below saturating a porous medium and subjected to a transverse uniform magnetic field. Salvatore Rionero [16] have studied a triple convective-diffusive fluid mixture saturating a porous horizontal layer, heated from below and salted from above and below. Kango et al. [17] have studied the triple-diffusive convection in Walters (Model B’) fluid with varying gravity field is considered in the presence of uniform vertical magnetic field in porous me-dium. Khan et al. [18] investigated the steady triple diffusive boundary layer free convection flow past a horizontal flat plate embedded in a porous medium filled by a water-based nanofluid and two salts. Moli Zhao et al. [19] have investigated the linear stability of triply diffusive convection in a binary Maxwell fluid satu-rated porous layer using modified Darcy-Maxwell model. The triply diffusive convection in a Maxwell viscoelastic fluid is mathematically investigated in the presence of uniform vertical magnetic field through porous medium studied by Pawan Kumar Sharma et al. [20] using linearized stability theory and normal mode analysis. Jyoti Prakash et al. [21] [22] have studied the magnetohydrody-namic triply diffusive convection with one of the components as heat, with dif-fusivity and sparsely distributed porous medium using the Darcy-Brinkman model. Rana et al. [23] have studied the triple-diffusive convection in a horizon-tal layer of nanofluid heated from below and salted from above and below. Goyal et al. [24] have studied the triple diffusive natural convection under Darcy flow

https://doi.org/10.4236/ojapps.2019.98052



over an inclined plate embedded in a porous medium saturated with a binary base fluid containing nanoparticles and two salts using group theory transfor-mations. Patil et al. [25] studied a numerical investigation on steady triple diffu-sive mixed convection boundary layer flow past a vertical plate moving parallel to the free stream in the upward direction. A linear stability analysis is per-formed for the onset of triple-diffusive convection in the presence of internal heat source in a Maxwell fluid saturated porous layer studied by Mukesh Kumar Awasthi et al. [26]. Raghunatha et al. [27] have investigated the weakly nonlinear stability of the triple diffusive convection in a Maxwell fluid saturated porous layer. For the composite layers, Sumithra [28] has studied the triple-diffusive Marangoni convection in a two layer system and obtained the analytical expres-sion for the Thermal Marangoni Number. The combined effects of magnetic field and non uniform basic temperature gradients on two component convec-tion in two layer system is investigated by Manjunatha and Sumithra [29] [30].

This paper investigates the triple diffusive Marangoni convection in a compo-site layer and studies the effects of the linear, parabolic and inverted parabolic temperature gradients on the corresponding thermal Marangoni numbers.

2. Mathematical Formulation

Consider a three different diffusing components with different molecular diffu-sivities, saturating a horizontally isotropic sparsely packed porous layer of thickness md underlying a three component fluid layer of thickness d. The lower surface of the porous layer is considered to be rigid and the upper surface of the fluid layer is free at which the surface tension effects depending on tem-perature and both the species concentrations. Both the boundaries are kept at different constant temperatures and salinities. A Cartesian coordinate system is chosen with the origin at the interface between porous and fluid layers and the z-axis, vertically upwards. The basic equations for fluid and porous layer respec-tively as

0∇⋅ =q (1)

( ) 20 Ptρ µ∂ + ⋅∇ = −∇ + ∇ ∂

q q q (2)

( ) 2T T Tt

κ∂ + ⋅∇ = ∇∂

q (3)

( ) 21 1 1 1C C Ct

κ∂

+ ⋅∇ = ∇∂

q (4)

( ) 22 2 2 2C C Ct

κ∂

+ ⋅∇ = ∇∂

q (5)

0m m∇ ⋅ =q (6)

( ) 20 21 1m

m m m m m m m m mPt Kµρ µ

ε ε∂ + ⋅∇ = −∇ + ∇ − ∂

qq q q q (7)




( ) 2m m m m m m mT

A T Tt

κ∂

+ ⋅∇ = ∇∂

q (8)

( ) 21 1 1 1m m m m m m mC

C Ct

ε κ∂

+ ⋅∇ = ∇∂

q (9)

( ) 22 2 2 2m m m m m m mC

C Ct

ε κ∂

+ ⋅∇ = ∇∂

q (10)

Here ( ), ,u v w=q is the velocity vector, 0ρ is the fluid density, t is the time, µ is the fluid viscosity, P is the pressure for fluid layer, T is the temperature, κ is the thermal diffusivity of the fluid, 1κ and 2κ is the solute1 and solute2 diffusivity of the fluid in the fluid layer, 1C is the concentration1 or the salinity field1 for the fluid, 2C is the concentration2 or the salinity field2 for the fluid in the fluid layer, mP is the pressure for porous layer, K is the permeability of

the porous medium, ( )( )

0 p m

p f

CA

C

ρ

ρ= is the ratio of heat capacities, pC is the

specific heat, ε is the porosity, 1mκ and 2mκ is the solute1 and solute2 diffu-sivity of the fluid in porous layer, 1 2 , m mC C are the concentration1 and concen-tration2 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.

( ) ( ) ( ) ( )1 1 2 2, , , ,b b b b bP P z T T z C C z C C z= = = = =q q (11)

( ) ( ) ( ) ( )1 1 2 2, , , ,m mb m mb m m mb m m mb m m mb mP P z T T z C C z C C z= = = = =q q (12)

( )0 in 0b uT T T h z z dz d

∂ −− = ≤ ≤∂

(13)

( )0 in 0mb l m m m mm m

T T Th z d z

z d∂ −

− = − ≤ ≤∂

(14)

( ) ( )10 11 10 in 0ubC C z

C z C z dd−

= − ≤ ≤ (15)

( ) ( )1 101 10 in 0l mmb m m mm

C C zC z C d z

d−

= − − ≤ ≤ (16)

( ) ( )20 22 20 in 0ubC C z

C z C z dd−

= − ≤ ≤ (17)

( ) ( )2 202 20 in 0l mmb m m mm

C C zC z C d z

d−

= − − ≤ ≤ (18)

where 0m u m l

m m

d T dTT

d dκ κκ κ

+=

+, 1 1 1 110

1 1

m u m l

m m

d C dCC

d dκ κ

κ κ+

=+

, 2 2 2 2202 2

m u m l

m m

d C dCC

d dκ κ

κ κ+

=+

are the interface temperature and concentrations, ( )h z and ( )m mh z are tem-perature 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

( ) ( ) ( )1 1 1 2 2 2, , , ,b b b b bP P P T T z C C z S C C z Sθ′ ′= + = + = + = + = +q q q (19)

( )( ) ( )1 1 1 2 2 2

, , ,

,m mb m m mb m m mb m m

m mb m m m mb m m

P P P T T z

C C z S C C z S

θ′ ′= + = + = +

= + = +

q q q (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 2

0 10 1 20 2, , , ,u u ud T T C C C C

dκ

κ− − − in the fluid

layer and 2

0 1 10 2 20, , , ,m m

l l lm m

dT T C C C C

dκ

κ− − − as the corresponding characteristic

quantities in the porous layer. To render the equations nondimensional, we choose different scales for the

two layers (Chen and Chen [31], Nield [32]), so that both layers are of unit length such that ( ) ( ), , , ,x y z d x y z′ ′ ′= , ( ) ( ), , , , 1m m m m m m mx y z d x y z′ ′ ′= − .

Omitting the primes for simplicity, we get in 0 1z≤ ≤ and 0 1mz≤ ≤ re-spectively

( )2 41 W WPr t∂

∇ = ∇∂

(21)

( ) 2Wh ztθ θ∂ = +∇∂

(22)

211 1

S W St

τ∂

= + ∇∂

(23)

222 2

S W St

τ∂

= + ∇∂

(24)

( )2

2 2 4 2ˆm m m m m mm

W W WPr tβ µβ∂ ∇ = ∇ −∇

∂ (25)

( ) 2m m m m m mA W h ztθ

θ∂

= +∇∂

(26)

211 1

mm m m m

SW S

tε τ∂

= + ∇∂

(27)

222 2

mm m m m

SW S

tε τ∂

= + ∇∂

(28)

Here, for the fluid layer, Pr νκ

= is the Prandtl number, 11κ

τκ

= is the ratio of

salute1 diffusivity to thermal diffusivity fluid in fluid layer, 22κ

τκ

= is the ratio

of salute2 diffusivity to thermal diffusivity fluid in fluid layer. For the porous

layer, mmm

Prενκ

= is the Prandtl number, 2 2m

K Dad

β = = is the Darcy number,

β is the porous parameter, ˆ mµ

µµ

= is the viscosity ratio, 11m

mm

κτ

κ= is the




ratio of salute1 diffusivity to thermal diffusivity of the porous layer, 22m

mm

κτ

κ=

is the ratio of salute2 diffusivity to thermal diffusivity of the porous layer, ( )h z

and ( )m mh z are the non-dimensional temperature gradients with ( )1

0

d 1h z z =∫

and ( )1

0

d 1m m mh z z =∫ , θ and mθ are the temperature in fluid and porous layers

respectively, S and mS are the concentration in fluid and porous layers respec-tively and W and mW are the dimensionless vertical velocity in fluid and porous layer respectively.

We apply normal mode expansion on dependent variables as follows:

( )( )( )( )

( )1 1

2 2

, ent

W W zz

f x yS S zS S z

θ θ =

(29)

( )( )( )( )

( )1 1

2 2

, e m

m m m

m m m n tm m m

m m m

m m m

W W zz

f x yS S zS S z

θ θ =

(30)

with 2 22 0f a f∇ + = and 2 22 0m m m mf a f∇ + = . Here a and ma are the nondi-

mensional horizontal wave numbers, n and mn are the frequencies. Since the dimensional horizontal wave numbers must be the same for the fluid and porous

layers, we must have mm

aad d= and hence ˆma da= .

Introducing Equation (29) and Equation (30) into the Equations (21) to (28) and

denoting Dz∂=

∂ and m

m

Dz∂

=∂

then we get an eigenvalue problem consisting

of the following ordinary differential equations in 0 1z≤ ≤ and 0 1mz≤ ≤ re-spectively

( )2 2 2 2 0nD a D a WPr − + − =

(31)

( ) ( )2 2 0D a n Wh zθ− + + = (32)

( )( )2 21 1 0D a n S Wτ − + + = (33) ( )( )2 22 2 0D a n S Wτ − + + = (34)

( ) ( )2

2 2 2 2 2ˆ 1 0mm m m m mm

nD a D a W

Prβ

µβ

− + − − =

(35)

( ) ( )2 2 0m m m m m m mD a An W h zθ− + + = (36)

( )( )2 21 1 0m m m m m mD a n S Wτ ε− + + = (37)




( )( )2 22 2 0m m m m m mD a n S Wτ ε− + + = (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 para-meters. 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 con-vection is in the form of steady convection and accordingly we take 0mn n= = . We get, in 0 1z≤ ≤ and 0 1mz≤ ≤ respectively

( )22 2 0D a W− = (39) ( ) ( )2 2 0D a Wh zθ− + = (40)

( )2 21 1 0D a S Wτ − + = (41) ( )2 22 2 0D a S Wτ − + = (42)

( ) ( )2 2 2 2 2ˆ 1 0m m m m mD a D a Wµβ − − − = (43) ( ) ( )2 2 0m m m m m mD a W h zθ− + = (44)

( )2 21 1 0m m m m mD a S Wτ − + = (45) ( )2 22 2 0m m m m mD a S Wτ − + = (46)

3. Boundary Conditions

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

( ) ( ) ( ) ( )2 2 2 21 1 2 21 1 1 1 0,s sD W Ma M a S M a Sθ+ + + =

( ) ( ) ( ) ( )1 21 1 1 1 0,W D DS DSθ= = = =

( ) ( ) ( ) ( ) ( )1 20 0 0 0 0 0,m m m m m m m mW DW D D S D Sθ= = = = =

( ) ( ) ( ) ( )ˆˆ ˆ0 1 , 0 1 ,m m mTW W TdDW D W= =

( ) ( ) ( ) ( )2 2 2 2 2ˆˆ ˆ0 1 ,m m mTd D a W D a Wµ+ = + ( ) ( )( ) ( ) ( ) ( )( )3 2 3 2 2 3 2ˆˆ ˆ0 3 0 1 1 3 1 ,m m m m m m mTd D W a DW D W D W a D Wβ µβ− = + − ( ) ( ) ( ) ( ) ( ) ( )1 1 1ˆ0 1 , 0 1 , 0 1 ,m m m mT D D S S Sθ θ θ θ= = =

( ) ( ) ( ) ( ) ( ) ( )1 1 2 2 2 2 20 1 , 0 1 , 0 1m m m m mDS D S S S S DS D S= = = (47)

where 111

s

s m

Sκκ

= , 222

s

s m

Sκκ

= are the ratios of solute1 and solute2 diffusivities

of fluid layer to those of porous layer respectively, ˆ md

dd

= = depth ratio,

ˆm

T κκ

= = ratio of thermal diffusivities of fluid, ( )0 ut T T dM

Tσ

µκ−∂

= −∂

is the

Thermal Marangoni number, ( )10 1

1ut

s

C C dM

Cσ

µκ−∂

= −∂

is the solute1 Maran-




goni number and ( )20 2

2ut

s

C C dM

Cσ

µκ−∂

= −∂

is the solute2 Marangoni number.

4. Method of Solution

From Equation (39) and Equation (43), we get W and mW as

( ) 1 2 3 4cosh cosh sinh sinhW z A az A z az A az A z az= + + + (48)

( ) 5 6 7 8cosh sinh cosh sinhm m m m m m m m m mW z A a z A a z A z A zδ δ= + + + (49)

where 2

2

1ˆm

ma

δµβ+

= and ( )1,2, ,8iA s i′ = are arbitrary constants, ( )W z

and ( )m mW z are suitably written as

( ) [ ]1 1 2 3cosh cosh sinh sinhW z A az a z az a az a z az= + + + (50)

( ) [ ]1 4 5 6 7cosh sinh cosh sinhm m m m m m m m m mW z A a a z a a z a z a zδ δ= + + + (51)

where

6 12 7 13 6 10 7 11 6 5 7 6 81 2 3

9 7

, , ,ˆˆa a a a a a

a a aTd

∆ + ∆ ∆ + ∆ ∆ + ∆ −∆= = =

∆ ∆

7 174 6 5 6 19 7

18

, , ,mm

aa a a a a

aδ− ∆

= − = = ∆ =∆

1 2sinh

cosh cosh , sinh ,m mm m mm

aa

aδ

δ δ∆ = − ∆ = −

( )3 4sinh sinh , cosh cosh ,m m m m m m ma a aδ δ δ δ∆ = − ∆ = −

( )2 2 25 cosh 2 cosh ,m m m m ma a aδ δ∆ = + − ( )2 26 sinh 2 sinh ,m m m m m ma a aδ δ δ∆ = + −

2 2 23 3 2

7 8 9

ˆ ˆˆ ˆ2 2 ˆˆ, , 2 ,ˆ ˆ

aTd a Td a Td βµ µ

∆ = ∆ = ∆ = −

2 210 100ˆsinh 2 sinh sinh ,m m m m m ma a a aµβ δ δ∆ = + − + ∆

( )2 3 2100 ˆ sinh 3 sinh ,m m m m maµβ δ δ δ δ∆ = − 2 2

11 110ˆcosh 2 cosh cosh ,m m m m m ma a a aµβ δ δ∆ = + − + ∆

( )2 3 2110 ˆ cosh 3 cosh ,m m m m maµβ δ δ δ δ∆ = − 10 11

12 3 13 49 9

, ,a a∆ ∆

∆ = ∆ − ∆ = ∆ −∆ ∆

10 7 5 91214

7 9

coshsinh ,ˆˆ

a aTd

∆ ∆ + ∆ ∆∆∆ = + ∆ ∆

13 11 7 6 915

7 9

coshsinh ,ˆˆ

aa

Td ∆ ∆ ∆ + ∆ ∆

∆ = + ∆ ∆

816 17 14 1 16

7

sinh ˆcosh , ,a

a T∆

∆ = − ∆ = ∆ −∆ ∆∆




2 1718 2 14 1 15 19

1 18

1 ˆ, .T ∆ ∆

∆ = ∆ ∆ −∆ ∆ ∆ = − ∆ ∆

We get the species concentration for fluid layer 1S , 2S from (41) and (42) also from (45) and (46) species concentration for porous layer 1mS , 2mS as

( ) ( )1 1 12 131

cosh sinhf z

S z A a az a azτ

= + +

(52)

( ) ( )2 1 16 172

cosh sinhf z

S z A a az a azτ

= + +

(53)

( ) ( )1 1 14 151

cosh sinh m mm m m m m mm

f zS z A a a z a a z

τ

= + +

(54)

( ) ( )2 1 18 192

cosh sinh m mm m m m m mm

f zS z A a a z a a z

τ

= + +

(55)

where

( ) ( ) ( )1 2 3 4, ,m mf z R R f z R R= − = − +

( ) ( )1 1 3 3 12 cosh sinh ,4zR a aza az a aza aza

= − + −

( )2 2sinh cosh ,2zR az a aza

= +

( )3 6 72 21 cosh sinh ,m m m m

m m

R a z a za

δ δδ

= +−

( )4 4 5sinh cosh ,2m

m m m mm

zR a a z a a z

a= +

( )12 1 14 15 27cosh sinh ,m ma S a a a a= + − ∆

( )13 15 14 281 cosh sinh ,m m m ma a a a a a aa

= + + ∆

( )30 2914 15 16 2 18 19 3231

, , cosh sinh ,m mm

a a a S a a a aa

∆ ∆= = = + − ∆∆

( )17 18 19 331 sinh cosh ,m m m ma a a a a a aa

= + + ∆

[ ]

36 35 20 118 19 26 22 27

37 1 1

1, , , ,ˆm m

Sa a

a Tτ τ∆ ∆ ∆

= = ∆ = ∆ ∆ =∆

( )2 128 4 5 28021 1

21 1 1 sinh cosh ,24 m mm m

aa a a a a aaaτ τ

− ∆ = − + + ∆

( ) ( )280 5 4 6 72 21 sinh cosh sinh cosh ,2

mm m m m

m m

a a a a a aa

δδ δ

δ∆ = + + +

−

5 729 2 2

1

1 ,2

m

m m m m

a aa a

δτ δ

∆ = +

−




30 26 28 27 29 300cosh sinh cosh cosh ,ma a a a a∆ = ∆ −∆ + ∆ −∆ + ∆

( )29300 1 sinh sinh ,mm

S a a aa∆

∆ = −

1 2 2731 1 32

2 1

sinh cosh cosh sinh , ,mm m mm

SS a a a a a a

Sττ

∆∆ = + ∆ =

( )2 133 4 5 28022 2

21 1 1 sinh cosh ,24 m mm m

aa a a a a aaaτ τ

− ∆ = − + + ∆

[ ] 5 734 26 1 35 2 222 2 2

1 1, ,2

m

m m m m

a aa a

δτ

τ τ δ

∆ = ∆ ∆ = + ∆ −

36 34 33 32 35 360cosh sinh cosh cosh ,ma a a a a∆ = ∆ −∆ + ∆ −∆ −∆

( )35360 2 sinh sinh ,mm

S a a aa∆

∆ = −

37 2 sinh cosh cosh sinhm m mS a a a a a a∆ = +

4.1. Linear Temperature Profile

For this case

( ) ( ) 1m mh z h z= = (56)

Substituting Equation (56) into the heat Equation (40) and Equation (44), we get θ and mθ as

( ) ( )1 8 9cosh sinhz A a az a az f zθ = + + (57)

( ) ( )1 10 11cosh sinhm m m m m m m mz A a a z a a z f zθ = + + (58)

where

( )8 10 11 20ˆ cosh sinh ,m ma T a a a a= + − ∆

( )9 10 11 211 sinh cosh ,m m m ma a a a a a aa

= + − ∆

232410 11

25

, ,m

a aa∆∆

= =∆

( ) ( )20 4 5 7 62 2ˆ ˆ

sinh cosh sinh cosh ,2 m m m mm m m

T Ta a a a a aa a

δ δδ

∆ = + + +−

( )2 121 4 5 33022 1 sinh cosh ,

24 m mm

aa a a a a aaa

− ∆ = − + + + ∆

( )( ) ( )( )2 222 3 1 2 22021 1 2 sinh 1 2 cosh ,4 a a a a a a aa aa ∆ = − + + − + + ∆

( ) ( )220 1 2 31 2 sinh 2 cosh ,

4a aa a a a a

a ∆ = + + +

5 723 2 2 ,2

m

m m m

a aa a

δδ

∆ = +−




24 22 21 20 23 240cosh sinh cosh cosh ,ma a a a a∆ = ∆ + ∆ + ∆ + ∆ −∆

( )23240 ˆ sinh sinh ,mm

Ta a aa∆

∆ =

25ˆ sinh cosh cosh sinh .m m mTa a a a a a∆ = +

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

1 2 31

4

MΛ +Λ +Λ

= −Λ

(59)

where

( ) ( )2 2 2 21 1 3 3 2 12 cosh 2 sinh ,a a a aa a a a a a aa aΛ = + + + + + 2 1

2 1 12 131

cosh sinh ,sM a a a a a τ Ω

Λ = + −

2 13 2 16 17

2

cosh sinh ,sM a a a a a τ Ω

Λ = + −

[ ]24 8 9 1cosh sinh ,a a a a aΛ = + −Ω

( ) ( )1 1 3 2 1 321 2 sinh 2 cosh

4a aa a a aa a aa a

a Ω = + − + − +

4.2. Parabolic Temperature Profile

We consider the profile as following (Sparrow et al. [33]):

( ) ( )2 and 2m m mh z z h z z= = (60) Substituting Equation (60) into the heat Equation (40) and Equation (44), we get θ and mθ as

( ) ( )1 20 21cosh sinhz A a az a az L zθ = + + (61)

( ) ( )1 22 23cosh sinhm m m m m m m mz A a a z a a z L zθ = + + (62)

where

( ) ( ) ( ) ( )5 6 7 8, ,m mL z R R L z R R= − + = − + 22

2 25 2 2sinh cosh ,2 22 2

a z a zz zR az aza aa a

= − + −

( ) ( )2 3 2 2 3 21 3 3 16 3 3

2 3 3 2 3 3sinh cosh ,

6 6

a z z a aa z a z z a aa zR az az

a a

+ − + − = +

( )2

7 4 5 70sinh cosh ,2m

m m m mm

zR a a z a a z R

a= + −

( )70 5 42 sinh cosh ,2m

m m m mm

zR a a z a a z

a= +

( )7 68 802 2

2 sinh cosh,m m m m m

m m

z a z a zR R

aδ δδ

+= −

−




( )( )

6 780 22 2

4 sinh cosh,m m m m m

m m

a z a zR

a

δ δ δ

δ

+=

−


22 23 39 42 4121 22 23

43

sinh cosh, , ,m m m m

m

a a a a a aa a a

a a+ − ∆ ∆ ∆

= = =∆

( )38 4 5 380 3811ˆ sinh cosh ,

2 m mmT a a a a R

a

∆ = + − + ∆

( )380 5 421 sinh cosh ,

2 m mma a a a

a∆ = +

( )( )381 6 7 38222 2

4sinh cosh ,m m m

m m

a aa

δδ δ

δ∆ = + + ∆

−

( )382 7 62 22 sinh cosh ,m m

m m

a aa

δ δδ

∆ = +−

( )( )339 6 7 390 3913 22 2

2sinh cosh ,

2m

m m

m m

a aa a

a a

δδ δ

δ

− ∆ = − + + + ∆ + ∆ −

( )( )2 5 44 5390 2

1 sinh coshsinh cosh,

2 2m m mm m

m m

a a a a aa a a aa a

− ++∆ = +

( )( )

( )2 2

391 7 622 2

2sinh cosh ,m m m m

m m

aa a

a

δδ δ

δ

+∆ = − +

−

2 22 2 2

40 4002 2

1 1sinh cosh ,2 22 2

a a a aaa aa aa a

− −∆ = + + + + ∆

( ) ( )2 21 3400 4013

3 1 2 3sinh ,

6

a a aa aa

a

+ + − ∆ = + ∆

( ) ( )2 23 1401 3

3 1 2 3cosh ,

6

a a aa aa

a

+ + − ∆ =

( )( )

22 264

41 2 22 2

2,

2m m

m m m

a aaa a

δ

δ

+∆ = − −

−

42 40 39 38 41 420cosh sinh cosh cosh ,ma a a a a∆ = ∆ + ∆ + ∆ −∆ + ∆

( )41420 1 43 25sinh sinh , .mm

S a a aa∆

∆ = − ∆ = ∆

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

1 2 32

5

MΛ +Λ +Λ

= −Λ

(63)




where


[ ]3 9 1031 ,

6R R

aΩ = +

( )( )29 2 3 1 13 3 2 sinh ,R a a a a a a a a= − − + + ( )( )210 2 1 3 33 1 3 2 cosh .R a aa a a a a a= − − + +

4.3. Inverted Parabolic Temperature Profile

We have

( ) ( ) ( ) ( )2 1 and 2 1m m mh z z h z z= − = − (64)

Substituting Equation (64) into the heat Equation (40) and Equation (44), we get θ and mθ as

( ) ( )1 24 25cosh sinhz A a az a az zθ = + +Ψ (65)

( ) ( )1 26 27cosh sinhm m m m m m m mz A a a z a a z zθ = + +Ψ (66)

where

( ) ( ) ( ) ( )11 12 13 14 15 16 17, ,m mz R R R z R R R RΨ = − + + Ψ = − + + + 22

2 2 211 2 2sinh cosh ,2 22 2

a z a z a zz z zR az aza a a aa a

= − + + − +

( )2 3

12 1 33 sinh cosh ,2 3 2z z zR a az a aza a a

= − − +

( )2

13 3 12 sinh cosh ,2z zR a az a az

a −

= +

( )2

14 4 5sinh cosh ,2m m

m m m mm m

z zR a a z a a z

a a

= − +

( )15 5 42 sinh cosh ,2m

m m m mm

zR a a z a a z

a= +

( ) ( )16 7 62 22 1

sinh cosh ,m m m m mm m

zR a z a z

aδ δ

δ−

= +−

( )( )17 6 722 2

4sinh cosh ,m m m m m

m m

R a z a za

δδ δ

δ= +

−


( )25 27 26 451 cosh sinh ,m m m ma a a a a a aa

= + + ∆

48 4726 27

49

, ,m

a aa

∆ ∆= =∆




( )44 440 5 421 sinh cosh ,

2 m mma a a a

a∆ = ∆ + +

( )440 4 5 4411 sinh cosh ,

2 m mma a a a

a∆ = + + ∆

( )( )441 6 722 2

4sinh cosh ,m m m

m m

a aa

δδ δ

δ∆ = +

−

( )2 2 1 345 450 4513

2 1,

2a a a a a

a + − −

∆ = − ∆ −∆

( )450 4 51 sinh cosh ,

2 m mma a a a

a∆ = +

( )2

451 5 4 4522

1sinh cosh ,

2m

m mm

aa a a a

a+

∆ = + + ∆

( )( )

( )2 2

452 7 622 2

2sinh cosh ,m m m m

m m

aa a

a

δδ δ

δ

+∆ = +

−

2 22 2 2

46 4602 2

1sinh cosh ,

2 2a a a a a aaa a

a a+ + + +

∆ = + − ∆

3 31 1460 3 3sinh cosh ,6 62 2

a aa aa aa a

∆ = + + +

( )2 2

5 7447 62 2 2 22 2

22 ,

2m m m

mm m mm m

a a aa aaa aa

δ δδδ

+∆ = + + +

−−

48 46 49 44 47 480cosh sinh cosh cosh ,ma a a a a∆ = ∆ −∆ + ∆ −∆ −∆

( )47480 49 25ˆ sinh sinh , .mm

Ta a aa∆

∆ = ∆ = ∆

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

1 2 33

6

MΛ +Λ +Λ

= −Λ

(67)

where


( )( )24 3 1 1831 3 cosh sinh ,

6a a a a a R

a Ω = − + +

( ) ( )18 2 221 sinh 1 cosh

2R a a a aa a

a= + + +

5. Results and Discussion The Thermal Marangoni numbers 1M for linear, 2M for parabolic and 3M for inverted parabolic temperature profiles are obtained. The constraints are drawn




against the depth ratio d̂ . The dimensionless fixed values are ˆ 1.0T = , ˆ 1.0S = , 1.0a = , 0.03β = , 1 10sM = , 2 1sM = , 1 2 1 2 1 2 0.25m m S Sτ τ τ τ= = = = = =

and ˆ 2.5µ = . The effects of the parameters 1 1 1 1ˆ, , , , , ,m sa S Mβ µ τ τ and 2sM on all the

three thermal Marangoni numbers are depicted in Figures 1 to 8. The main obser-vation 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 β .

Figure 3. The effects of viscosity ratio µ̂ .




Figure 4. The effects of 1τ .

Figure 5. The effects of 1mτ .

Figure 6. The effects of 1S . 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 sys-tems. For fluid dominant system, there is no much change in the thermal Ma-rangoni numbers for all the profiles.

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




Figure 7. The effects of solute1 Marangoni number 1sM .

Figure 8. The effects of solute2 Marangoni number 2sM .

numbers 1 2,M M and 3M are respectively shown in Figures 1(a)-(c) for

1.0,1.1a = and 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 β on the three thermal Marangoni numbers are depicted Figures 2(a)-(c). The curves for 0.03,0.04,0.05β = . In-crease in the value of β , i.e., increasing the permeability, the thermal Maran-goni numbers decrease for all the three profiles. Hence the surface tension dri-ven 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 µ̂ for the values ˆ 2.5,3.0,3.5µ = . Increase in the value of µ̂ , the values of the thermal Marango-

ni numbers 1 2,M M and 3M increases. So, the increase in the values of vis-cosity 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 1τ is the ratio of salute1 diffusivity to thermal diffusivity fluid in fluid layer for 1 2,M M and 3M respectively for the




values 1 0.25,0.50,0.75τ = . For all the three profiles, there is a increase in the values of the thermal Marangoni numbers. Increasing the value of 1τ the sur-face tension driven triple diffusive convection becomes slow and hence the sys-tem can be stabilized.

Figures 5(a)-(c) display the variations of the value of 1mτ is the ratio of sa-lute1 diffusivity to thermal diffusivity of the porous layer for the values

1 0.25,0.50,0.75mτ = . Increasing this ratio, the thermal Marangoni numbers in-crease 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 1 0.25,0.50,0.75S = . Increasing this ratio, for all the three profiles, there is a small increase in 1 2,M M and 3M so, 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 1sM is the solute1 Maran-goni number for 1 10,50,100sM = . 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 2sM is the solute2 Marangoni number for 2 10,25,50sM = . 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 situ-ations 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 1 1 1 1ˆ, , , , ,m sa S Mµ τ τ and by decreasing the val-ues of β and 2sM the surface tension driven triple diffusive convection in a composite layer under microgravity condition can be delayed and hence the sys-tem 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.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this pa-per.

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Effects of Non-Uniform Temperature Gradients on Triple Diffusive Marangoni Convection in a Composite LayerAbstractKeywords1. Introduction2. Mathematical Formulation3. Boundary Conditions4. Method of Solution4.1. Linear Temperature Profile 4.2. Parabolic Temperature Profile 4.3. Inverted Parabolic Temperature Profile

5. Results and Discussion6. ConclusionsAcknowledgementsConflicts of InterestReferences

Effects of Non-Uniform Temperature Gradients on Triple ...For the fluid layer, Chand [8] has applied...

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