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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 Manjunatha

, Ramakrishna Sumithra

School of Applied Sciences, REVA University, Bengaluru, India

Department 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 e

igenvalue 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

, 640-660.

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

Received:

July 23, 2019

Accepted:

August 19, 2019

Published:

August 22, 2019

9 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

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

N. Manjunatha, R. Sumithra

DOI:

10.4236/ojapps.2019.98052 642 Open Journal of Applied Sciences

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

underlying a three component fluid layer of thickness

. 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

-axis, vertically upwards. The basic equations for fluid and porous layer respec-

tively as

0∇⋅ =q

(1)

( )

ρµ

∂



+ ⋅∇ = −∇ + ∇



∂



(2)

( )

∂

+ ⋅∇ = ∇

∂

(3)

( )

111

∂

+ ⋅∇ = ∇

∂

(4)

( )

222

∂

+ ⋅∇ = ∇

∂

(5)

∇⋅ =q

(6)

( )

m m m mm m mm m

ρµ

∂



+ ⋅∇ =−∇+∇−



∂



q q qq

(7)

N. Manjunatha, R. Sumithra

DOI:

10.4236/ojapps.2019.98052 643 Open Journal of Applied Sciences

(

)

m m m m mm

A TT

∂

+ ⋅∇ = ∇

∂

(8)

( )

11 1

m m m m mm

εκ

∂

+ ⋅∇ = ∇

∂

(9)

( )

22 2

m m m m mm

εκ

∂

+ ⋅∇ = ∇

∂

(10)

Here

( )

,,uvw=q

is the velocity vector,

is the fluid density,

is the time,

is the fluid viscosity,

is the pressure for fluid layer,

is the temperature,

is the thermal diffusivity of the fluid,

and

is the solute1 and solute2

diffusivity of the fluid in the fluid layer,

is the concentration1 or the salinity

field1 for the fluid,

is the concentration2 or the salinity field2 for the fluid

in the fluid layer,

is the pressure for porous layer,

is the permeability of

the porous medium,

( )

0 p

is the ratio of heat capacities,

is the

specific heat,

is the porosity,

and

is the solute1 and solute2 diffu-

sivity of the fluid in porous layer,

are the concentration1 and concen-

tration2 for porous layer respectively and the subscripts “

” and “

” 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.

( ) ( ) ( ) ( )

11 2 2

,,, ,

bb b b b

P PzT TzC C zC C z= = = = =qq

(11)

( ) ( ) ( ) ( )

11 2 2

,,, ,

m mb m mb m m mb m m mb m m mb m

PPzTTzC C zC C z= = = = =qq

(12)

( )

in 0

T TT

hz z d

∂−

− = ≤≤

∂

(13)

( )

in 0

mb l

mm m m

T TT

hz d z

∂−

− = −≤≤

∂

(14)

( )

10 1

1 10

in 0

C Cz

Cz C zd

−

= − ≤≤

(15)

( )

1 10

in 0

mb m m m

C Cz

Cz C dz

−

= − −≤≤

(16)

( )

20 2

2 20

in 0

C Cz

C z C zd

−

= − ≤≤

(17)

( )

2 20

in 0

mb m m m

C Cz

Cz C dz

−

= − −≤≤

(18)

where

mu m l

d T dT

κκ

11 11

mu m l

d C dC

κκ

22 22

mu m l

d C dC

κκ

are the interface temperature and concentrations,

( )

and

( )

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

N. Manjunatha, R. Sumithra

DOI:

10.4236/ojapps.2019.98052 644 Open Journal of Applied Sciences

system as

( ) ( ) ( )

1 1 12 2 2

,, , ,

bbb b b

PPPTTz CCzSCCzS

′′

=+=+= += + = +qq q

(19)

( )

( ) ( )

1 1 12 2 2

,, ,

m mb m m mb m m mb m m

m mb m m m mb m m

P P PT T z

CCzSCCzS

′′

=+=+= +

= += +

qq 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

0 10 1 20 2

,, , ,

uu u

T TC C C C

−− −

in the fluid

layer and

0 1 10 2 20

,, , ,

ll l

T TC C C C

−− −

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

( ) (

)

,, , ,xyz d x y z

′′′

( ) ( )

,, ,, 1

mmm m m mm

xyz d xyz

′′′

= −

Omitting the primes for simplicity, we get in

01z

≤≤

and

z≤≤

re-

spectively

( )

Pr t

∂

∇=∇

∂

(21)

( )

Wh z

∂

= +∇

∂

(22)

∂

= +∇

∂

(23)

∂

= +∇

∂

(24)

( )

2 24 2

mm mm mm

W WW

Pr t

µβ

∂

∇ = ∇ −∇

∂

(25)

( )

mm m mm

A Wh z

∂

= +∇

∂

(26)

m m mm

ετ

∂

= +∇

∂

(27)

m m mm

ετ

∂

= +∇

∂

(28)

Here, for the fluid layer,

is the Prandtl number,

is the ratio of

salute1 diffusivity to thermal diffusivity fluid in fluid layer,

is the ratio

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

layer,

εν

is the Prandtl number,

= =

is the Darcy number,

is the porous parameter,

is the viscosity ratio,

is the

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