Soret Effect on MHD Natural Convection Flow with Radiative Heat Transfer Past an Impulsively Moving Plate with Ramped Wall Temperature through Porous Medium
Khem Chand, Nidhi Thakur
Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla171005, India
*Corresponding Author Email: khemthakur99@gmail.com. nidhithakurmaths55@gmail.com
ABSTRACT:
The effect of thermodiffusion (Soret effect) on unsteady hydromagnetic free convective flow of a viscous, incompressible, electrically conducting and optically thick radiating fluid past an infinite vertical plate in a porous medium is analyzed in the present study. An impulsively moving infinitely long vertical plate with ramped temperature distribution on its surface is taken. The exact solutions of governing dimensionless partial differential equations are obtained by using Laplace transform technique. The effects of pertinent flow parameters on velocity, temperature and concentration profile are numerically evaluated, graphically depicted and discussed. In case of skin friction, Nusselt number and Sherwood number, numerical values are presented in tabular form. Results disclosed that fluid velocity and species concentration in the boundary layer region enhanced with Soret effect.
KEYWORDS: Free convection, Ramped temperature, Thermal radiation, Soret effect, Optically thick fluid
INTRODUCTION:
The study of MHD flows with heat and mass transfer have been a subject of great interest for last few decades due to its diverse applications in the field of science and technology. Owing to technological developments in modern metallurgy, such flows in a porous medium have received considerable attention of researchers. The unsteady free convective flow past a vertical plate with heat and mass transfer in a porous medium has been studied by Bejan and Khair [1], Ganesan and Palani [2], Chaudhary and Jain [3], Singh and Kumar [4], Das and Jana [5] and Jain et al. [6]. In all these studies the effects of thermal radiation is not considered, even though flow with thermal radiation have several applications in engineering processes particularly in space technology and high temperature aerodynamics and in plasma physics. On account of their varied importance, several scholars have carried out their research work on it by taking surface conditions uniform or variable. Shankar et al. [7] investigated radiation and mass transfer effects on MHD free convection fluid flow embedded in a porous medium with heat generation/absorption. The problem of unsteady MHD free convection flow and mass transfer near a moving vertical plate in the presence of thermal radiation has been examined by Seethamahalakshmi et al. [8]. Seth et al. [9] studied the influence of radiation on unsteady hydromagnetic natural convection transient flow near an impulsively moving vertical plate with ramped wall temperature. Kishore et al. [10] studied the effects of thermal radiation and viscous dissipation on MHD heat and mass diffusion flow past an oscillating vertical plate embedded in a porous medium with variable surface conditions. Seth et al. [11] investigated the effects of Hall current, radiation and rotation on natural convection heat and mass transfer flow past a moving vertical plate embedded in the saturated porous medium. Balla and Naikoti [12] performed a numerical analysis to study the unsteady magneto hydrodynamic convective flow of a viscous, incompressible, electrically conducting Newtonian fluid along a vertical permeable plate in the presence of a homogeneous first order chemical reaction and taking into account thermal radiation effects. Nandkeolyar and Das [13] considered MHD free convection flow past a flat plate with ramped wall temperature and radiative heat transfer embedded in a porous medium in the presence of an inclined magnetic field.
In nonisothermal systems, a phenomenon of thermodiffusion (Soret effect) is observed. This effect includes the separation of components of fluid mixture, when it is placed in thermal gradient. The separation of components caused by thermal diffusion is generally small, but its effect is quite important which cannot be neglected. Due to its application in hydrocarbon reservoirs, isotope separation and in gaseous mixture, many researchers worked on it. Kostarev and Pshenichnikov [14] examined thermo diffusion separation of a liquid mixture under developed convection conditions. Jha and Singh [15] performed an analytical study on free convection and mass transfer flow past an infinite vertical plate moving impulsively in its own plane by taking Soret effects into account. Singh and Kumar [16] analyzed Soret and Hall current effects on heat and mass transfer in MHD flow of a viscous fluid through porous medium with variable suction. Combined effects of thermophoresis and electrophoresis on particle deposition onto a vertical flat plate from mixed convection flow through a porous medium have been studied by Tsai and Huang [17]. Bakier and Gorla [18] analyzed the effects of thermophoresis and radiation on laminar flow along a semiinfinite vertical plate. Unsteady MHD free convective mass transfer flow past an infinite vertical porous plate with variable suction and Soret effect has been investigated by Ramana Reddy et al. [19]. Pal and Talukdar [20] studied influence of Soret effect on MHD mixed convection oscillatory flow over a vertical surface in a porous medium with chemical reaction and thermal radiation.
The purpose of the present paper is to extend the work of Seth et al. [9] by analyzing the Soret effect on the unsteady free convective radiative heat flow past an impulsively moving infinite vertical plate with ramped wall temperature in a porous medium. The Laplace transform technique is adopted to solve the governing equations to obtain the analytical results for velocity, temperature and concentration profile.
Formulation of the problem:
Consider an unsteady hydromagnetic flow of an electrically conducting, viscous, incompressible and optically thick radiating fluid past an infinite vertical plate embedded in a porous medium by taking Soret effect into account. Coordinate system is chosen in such a way that_{} axis is taken along the plate in the upward direction,_{}axis normal to plane of the plate in the fluid. A uniform magnetic field of strength_{}is applied parallel to_{}axis. Initially, at time _{} both the fluid and the plate are at rest and at uniform temperature_{}. At time_{}_{}the plate starts moving along_{}direction with uniform velocity_{}. The temperature of the plate is raised or lowered to _{}when _{}and thereafter, it is maintained at uniform temperature_{}when_{}. The plate is infinitely long in_{}_{}and_{}directions, therefore all the physical quantities except pressure are functions of _{}and _{}only.
The magnetic Reynolds number for metallic liquids and partially ionized fluids is very small (Cramer and Pai [21]). So we assume that the magnetic Reynolds number for the flow is small and the induced magnetic field can be neglected and the applied magnetic field is considered as _{}In view of the above assumptions and under usual Boussinesq approximation, equations governing the flow are given by
_{} (1)
_{} (2)
_{} (3)
_{} (4)
In above equations _{}and _{} are, respectively, fluid velocity in_{} direction, fluid velocity in _{}direction, time, temperature of the fluid, concentration of the fluid, kinematic viscosity, electrical conductivity, fluid density, acceleration due to gravity, volumetric coefficient of thermal expansion, volumetric coefficient of expansion for species concentration, thermal conductivity, permeability of porous medium, specific heat at constant pressure, radiative flux vector, molecular diffusivity and thermal diffusivity.
The initial and boundary conditions are
_{} for _{}and _{}
_{} at _{}for _{}
_{} at _{} for _{} (5)
_{} at _{} for _{}
_{} as _{}for _{}
By using Rosseland approximation, the radiative flux vector _{} for optically thick fluid becomes
_{}. (6)
where _{}is mean absorption coefficient and_{}is StefanBoltzmann constant. Assuming that the difference between fluid temperature_{}and free stream temperature_{} is small, so expanding_{} by Taylor series about a free stream temperature. After neglecting second and higher order terms in _{}, we get
_{ }(7)
On the use of equations (6) and (7), equation (3) becomes
_{} (8)
Introducing following nondimensional quantities and parameters
_{}
_{} (9)
Equations (2), (4) and (8) in nondimensional form become
_{} (10)
_{} (11)
_{} (12)
where_{}and_{}are magnetic parameter, permeability parameter, Grashof number, solutal Grashof number, radiation parameter, Prandtl number, Schmidt number and Soret number respectively.
In above non dimensionalisation process, the characteristics time _{}can be defined as
_{} (13)
The corresponding initial and boundary conditions are
_{}for _{} and _{}
_{}at _{}for _{}
_{}at_{} for _{} (14)
_{}at _{} for _{}
_{}as _{}for _{}
METHOD OF SOLUTION:
Applying Laplace transform technique, equations (10), (11) and (12) with help of relation (14) reduce to
_{} (15)
_{} (16)
_{} (17)
where _{} and _{} being Laplace transform parameter.
The boundary conditions (14) become
_{} _{} at _{}
_{} as _{} (18)
The solution of equations (15), (16) and (17) subject to the boundary conditions (18) are given by
_{}
_{ }(19)
_{} (20)
_{} (21)
Taking inverse Laplace transform of equations (19), (20) and (21), we get following expressions for velocity, temperature and concentration profile for plate with ramped temperature
_{}_{}
_{} (22)
_{} (23)
_{} (24)
where _{} is the complementary error function and_{} is the unit step function and
_{}
_{}
_{}
_{}
_{}
_{}
_{}
_{}
Solution for the isothermal case:
In order to emphasize the effect of ramped temperature plate on flow field, comparing it with isothermal plate. Keeping in view the assumption made in this paper, we get the following expression for velocity, temperature and concentration profile for fluid flow near an isothermal plate
_{}
_{}
_{}
_{} (25)
_{} (26)
_{}
(27)
Some important characteristics of flow
From the velocity, temperature and concentration field equations (22), (23) and (24), the expressions for skin friction_{}, Nusselt number_{} and Sherwood number_{} at the ramped temperature plate are given by
_{} (28)
_{} (29)
_{} (30)
where_{}
_{}
_{}
From the velocity, temperature and concentration field equations (25), (26) and (27), the expressions for skin friction, Nusselt number and Sherwood number at the isothermal plate are given by
_{}
_{ }_{} (31)
_{} (32)
_{} (33)
All constants used above have been listed in the appendix.
RESULTS AND DISCUSSION:
In order to understand the physical meaning of the problem, the effects of various governing parameters such as magnetic parameter (M), radiation parameter (N), Prandtl number (Pr), permeability parameter (K_{1}), Schmidt number (Sc), Soret number (So), Grashof number (Gr) and solutal Grashof number (Gc) on the nondimensional physical quantities are computed numerically and represented in figures 114.
Fig.1 Velocity profile for various values of magnetic Fig.2 Velocity profile for various values of radiation
parameter (M) when Pr = 0.7, N = 2, Sc = 0.30, So =1, parameter (N) when Pr = 0.7, M = 2, Sc = 0.30, So =1,
Gr =5, Gc =5, K_{1} = 0.2, t = 0.5 Gr =5, Gc =5, K_{1} = 0.2, t = 0.5
Fig.3 Velocity profile for various values of Prandtl Fig.4 Velocity profile for various values of permeability
number (Pr) when M=2, N=2, Sc=0.30, So=1, parameter (K_{1}) when Pr=0.7, N=2, M = 2, Sc = 0.30,
Gr=5, Gc=5, K_{1} = 0.2, t=0.5 So =1, Gr=5, Gc =5, t=0.5
Fig.1 depicts the effect of magnetic field on the fluid velocity for both ramped temperature and isothermal plates. From the figure it is observed that fluid velocity in the boundary layer region decreases with increase in magnetic parameter (M). This happens due to Lorentz force, which is a resistive force and has the tendency to decelerate the fluid flow. The effect of radiation parameter (N) on the fluid velocity for both ramped temperature and isothermal plates is displayed through fig.2. It is found that increase in radiation parameter contributes to increase in fluid velocity by accelerating fluid velocity. From fig.3 it is observed that the velocity for both ramped temperature and isothermal plates decrease with an increase in the Prandtl number (Pr). Increasing value of Prandtl number implies that the fluid is becoming viscous and viscosity has adverse effect on fluid velocity in the boundary layer region. Fig.4 reveals that the fluid velocity increases with an increase in the permeability parameter (K_{1}) for both ramped temperature and isothermal plates as the existence of porous medium decreases the resistance to flow.
Fig.5 Velocity profile for various values of Schmidt Fig.6 Velocity profile for various values of Soret
number (Sc) when M = 2, N = 2, Pr = 0.7, So =1, Gr =5, number (So) when M = 2, N = 2, Pr = 0.7, Sc =0.30,
Gc =5, K_{1} = 0.2, t = 0.5 Gr =5, Gc =5, K_{1} = 0.2, t = 0.5
Fig.7 Velocity profile for various values of Grashof Fig. 8 Velocity profile for various values of solutal
number (Gr) when M = 2, N = 2, Pr = 0.7, Sc =0.30, Grashof number (Gc) when M = 2, N = 2, Pr = 0.7,
So =1, Gc =5, K_{1} = 0.2, t = 0.5 Sc =0.30, So =1, Gr =5, K_{1} = 0.2, t = 0.5
It is noticed from fig.5 that the fluid velocity for both ramped temperature and isothermal plates decreases with increase in the Schmidt number (Sc). Increase in the value of Schmidt number corresponds to decrease in molecular diffusivity which depresses fluid velocity. Fig.6 demonstrate the effect of Soret number (So) on the fluid velocity for both ramped temperature and isothermal plates. Fluid velocity in the boundary layer region increases with increase in the Soret number. This happens due to greater thermal diffusion. Fig.7 and fig.8 display that fluid velocity for both ramped temperature and isothermal plates increases with increase in Grashof number (Gr) and solutal Grashof number (Gc) respectively. This is due to the fact that thermal and concentration bouyancy forces tends to accelerate the fluid flow in the boundary layer region. Fig.9 presents the effect of Schmidt number (Sc) on species concentration. It is noticed that with increase in Schmidt number, species concentration in the boundary layer region for both ramped temperature and isothermal plates decreases. Mass diffusion tends to enhance species concentration. Higher value of Schmidt number implies lower mass diffusivity which further implies low species concentration. It is evident from fig.10 that concentration distribution in the boundary layer region increases with increase in Soret number (So) for both ramped temperature and isothermal plates.
Fig.9 Concentration profile for various values of Fig.10 Concentration profile for various values of
Schmidt number (Sc) when M = 2, N = 2, Pr = 0.7, So =1, Soret number (So) when M = 2, N = 2, Pr = 0.7,
Gr =5, Gc =5, K_{1} = 0.2, t = 0.5 Sc =0.30, Gr =5, Gc =5, K_{1} = 0.2, t = 0.5
Fig.11 Temperature profile for various values of Fig.12 Temperature profile for various values of
Prandtl number (Pr) when M = 2, N = 2, Sc =0.30, radiation parameter (N) when M = 2,Pr = 0.7,
So =1 Gr =5, Gc =5, K_{1} = 0.2, t = 0.5 Sc =0.30,So =1 Gr =5, Gc =5, K_{1} = 0.2, t = 0.5
Fig.13 Temperature profile for various values of Fig.14 Velocity profile for various values of radiation
time (t) when M = 2, N = 2, Pr = 0.7, Sc = 0.30, parameter (N) when Pr = 0.7, M = 2, Gr =5,
So =1 Gr =5, Gc =5, K_{1} = 0.2 K_{1} = 0.2, Sc =0, So =0, Gc =0, t = 0.5
Fig. 1113 illustrate the effects of Prandtl number (Pr), radiation parameter (N) and time (t) on temperature profile in the boundary layer region for both ramped temperature and isothermal plates. Fig.11 shows that fluid temperature decreases on increasing Prandtl number since large Pr has low thermal diffusivity. It is noticed from fig.12 and fig.13 that fluid temperature increases with increase in radiation parameter and time. Thermal radiation provide additional means to diifuse energy which rises fluid temperature in the boundary layer region. In order to verify the correctness of the present approach, we have made comparisons with available study by Seth et al. [9]. The results are found in good agreement and one of the comparisons is shown in fig.14. for different values of radiation parameter (N).
Table 1 Numerical values of skin friction (–τ)
Pr M N Sc So Gr Gc K_{1} t At ramped At isothermal
temperature plate plate
0.7 2 2 0.30 1.0 5 5 0.2 0.5 0.72911 0.76798
7.0 2 2 0.30 1.0 5 5 0.2 0.5 0.87193 80.394
0.7 4 2 0.30 1.0 5 5 0.2 0.5 3.0389 2.5925
0.7 2 4 0.30 1.0 5 5 0.2 0.5 0.70534 0.53937
0.7 2 2 0.60 1.0 5 5 0.2 0.5 0.73986 0.84111
0.7 2 2 0.30 3.0 5 5 0.2 0.5 0.69571 0.5465
0.7 2 2 0.30 1.0 10 5 0.2 0.5 0.13889 0.3530
0.7 2 2 0.30 1.0 5 10 0.2 0.5 0.94347 0.34153
0.7 2 2 0.30 1.0 5 5 0.5 0.5 0.23671 0.5583
0.7 2 2 0.30 1.0 5 5 0.2 1.2 0.17308 0.04591
Table 2 Numerical values of Nusselt number (Nu)
Pr N t At ramped At isothermal
temperature plate plate
0.7 2 0.4 0.34473 0.43091
1.0 2 0.4 0.41203 0.51503
7.0 2 0.4 1.0901 1.3626
0.7 4 0.4 0.26702 0.33378
0.7 6 0.4 0.22568 0.28209
0.7 2 0.7 0.45603 0.32574 0.7 2 1.2 0.35332 0.24878
Table 3 Numerical values of Sherwood number (Sh)
Sc So At ramped At isothermal
temperature plate plate
0.30 1.0 0.3006 0.38283
0.35 1.0 0.4022 0.4114
0.60 1.0 0.4912 0.52922
0.30 3.0 0.0277 0.27447
0.30 5.0 0.0148 0.16610
Table 1, table 2 and table 3 present, respectively, the numerical values of skin friction (–τ), Nusselt number (Nu) and Sherwood number (Sh) for different values of governing parameters. It is concluded from the table 1 that skin friction decreases on increasing radiation parameter (N), Soret number (So), Grashof number (Gr), solutal Grashof number (Gc), permeability parameter (K_{1})_{ }and time (t), while increases on increasing magnetic parameter (M), Prandtl number (Pr) and Schmidt number (Sc). Table 2 expresses that the rate of heat transfer at both ramped temperature and isothermal plates increases with increase in Prandtl number but decreases with increase in radiation parameter (N). It is also noticed that with progress of time, Nusselt number decreases for isothermal plate but increases for ramped temperature plate. It is evident for table 3 that rate of mass transfer increases on increasing Schmidt number, but decreases on increasing Soret number at both ramped temperature and isothermal plates.
CONCLUSIONS:
After analyzing graphs and tables, following conclusions are drawn:
· Thermal and concentration buoyancy forces, porosity, thermal radiation promote the flow throughout the boundary layer region by accelerating fluid velocity whereas applied magnetic field and mass diffusion retards the flow.
· Species concentration in the boundary layer region increases with mass diffusion.
· Thermal radiation increases fluid temperature in the boundary layer region.
· Skin friction reduces with thermal and concentration buoyany forces, porosity, thermal radiation and Soret number but it enhances with applied magnetic field, thermal and mass diffusion.
· The rate of heat transfer at the plate increases with thermal diffusion and the rate of mass transfer at the plate increases with mass diffusion.
CONFLICT OF INTEREST:
We declare no conflict of interest.
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Appendix:
_{}
Received on 08.07.2017 Accepted on 11.01.2018 ©A&V Publications all right reserved Research J. Engineering and Tech. 2018;9(2): 125134. DOI: 10.5958/2321581X.2018.00018.1 
