Reach Us
+32-10-28-02-25

**Kumar VR ^{1}, Raju MC^{1*}, Raju GSS^{2} and Varma SVK^{3}**

^{1}Department of Mathematics, Annamacharya Institute of Technology and Sciences, India

^{2}Department of Mathematics, JNTUA College of Engineering, India

^{3}Department of Mathematics, S.V. University, India

- *Corresponding Author:
- Raju MC

Department of Mathematics, Annamacharya

Institute of Technology and Sciences, India

**Tel:**009457759877

**E-mail:**[email protected]

**Received Date:** January 11, 2016; **Accepted Date:** January 31, 2016; **Published Date:** Feruary 04, 2016

**Citation:** Kumar VR, Raju MC, Raju GSS, Varma SVK (2016) Thermal Diffusive Free Convective Radiating Flow Over an Impulsively Started Vertical Porous Plate in Conducting Field. J Phys Math 7:156. doi:10.4172/2090-0902.1000156

**Copyright:** © 2016 Kumar VR, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Physical Mathematics

In this manuscript we have studied the laminar convective heat and mass transfer flow of an incompressible, viscous, electrically conducting fluid over a fluid over an impulsively started vertical plate with conduction-radiation embedded in a porous medium in the occurrence of transverse magnetic field. An exact solution is derived by solving the dimensionless main coupled partial differential equations using Laplace transform technique. The properties of important physical parameters on the velocity, temperature, concentration, Skin friction, Sherwood number and Nusselt number have been studied through graphs.

MHD; Porous medium; Thermal diffusion; Thermal
radiation; **Shear stress**; Nusselt number and Sherwood number

C^{/} : Species concentration fluid

Cp : Specific heat at constant pressure

: Concentration of the fluid for away from the plate

: Concentration level near the plate/wall

*D*: Chemical molecular diffusivity

g: Acceleration due to gravity

*q _{r}*: Radiative heat flux

*Gr*: Thermal Grashof number

*Gm*: modified Grashof number

*K _{r}* : Permeability parameter

*M*: Hartmann number

*Nu*: Nusselt number

*P _{r}*: Prandtl number

*S _{0}*: Soret number

*S _{h}*

: Fluid temperature at the surface

u: Dimensional velocity components

S_{c}: Schmidt number

*T ^{/}*: Temperature

*u _{0}*: Plate velocity

*β* : coefficient of volume expansion for heat transfer

θ: Dimensional fluid

n: Kinematic viscosity

σ : Electrical conductivity

C: Dimensionless species concentration

*β _{c}*: Coefficient of volume expansion for mass transfer

κ: Thermal conductivity

ρ : Density

τ: Shearing stress

*w*: Condition on the wall

∞ : Free stream condition

Several transport processes exist in industries and technology where the transfer of heat and mass occurs simultaneously as an outcome of thermal diffusion and diffusion of chemical species. Natural convection induced by the simultaneous achievement of buoyancy forces resulting from thermal and mass diffusion is of considered interest in nature and in many industrial applications such as cosmic fluid dynamics, meteorology, chemical industry, cooling of nuclear reactors, magneto hydrodynamics power generators and the earth’s core. Bharat et al.[1] investigated the effects of mass transfer on MHD free convective radiation flow over an impulsively started vertical plate embedded in a porous medium. Ahmed et al. [2] discussed convective laminar radiating flow over an accelerated vertical plate embedded in a porous medium with an external magnetic field. Chamka et al. [3] studied thermal radiation and buoyancy effects on hydro magnetic flow over an accelerating porous surface with heat source or sink. Ahmed et al. [4] examined Non-linear magneto hydrodynamic flow more an impulsively started vertical plate in a saturated porous regime Laplace and Numerical approach. Ravi Kumar et al. [5] examined MHD double diffusive and chemically reactive flow through porous medium bounded by two vertical plates. Palani et al. [6] studied free convection MHD flow with thermal radiation from an impulsively-started vertical plate. Ravi Kumar et al. [7] discussed heat and mass transfer effects on MHD flow of viscous fluid through non-homogeneous porous medium in occurrence of temperature dependent heat source. Chen et al. [8] discussed heat and mass transfer in MHD flow by ordinary convection from a permeable, inclined surface with variable wall temperature and concentration. Ravi Kumar et al. [9] discussed combined effects of heat absorption and MHD on convective Rivlin-Ericksen flow past a semi-infinite perpendicular porous plate with variable temperature and suction. Ahmed et al. [10] examined Numerical/Laplace transform investigation for MHD radiating heat/mass transport in a Darcian porous regime bounded by an oscillating vertical surface. Kumar et al. [11] discussed thermal radiation and mass transfer effects on MHD flow past a vertical oscillating plate among variable temperature effects variable mass diffusion. Hossain et al. [12] studied radiation effect on mixed convection along a perpendicular plate with uniform surface temperature. Ibrahim et al. [13] examined similarity solution of heat and mass transfer for normal convection over a moving vertical plate with internal heat generation and a convective boundary state in the presence of thermal radiation, viscous dissipation, and chemical reaction. Pradyumna kumar et al. [14] examined analytical solution of magnetic hydro magnetic free convective flow through porous media with time dependent temperature and concentration. Das et al. [15] discussed mass transfer effects on MHD flow and heat transfer past a vertical porous plate throughout a porous medium below oscillatory suction and heat source [16]. Seth et al. [17] studied effects of thermal radiation and rotation on unsteady hydro magnetic free convection flow past an impulsively moving vertical plate with ramped temperature in a porous medium. Das et al. [18] discussed heat and mass transfer effects on unsteady MHD free convection flow near a moving vertical plate in porous medium. Kumar et al. [19] examined magnetic field effect on transient free of charge convection flow through porous medium past an impulsively started vertical plate with fluctuating temperature and mass diffusion. Mamtha et al. [20] discussed thermal diffusion effect on MHD mixed convection unsteady flow of a micro polar liquid past a semi-infinite vertical porous plate with radiation and mass transfer. Redddy et al. [21] examined unsteady MHD free convection flow of a Kuvshinski fluid past a vertical porous plate in the presence of chemical reaction and heat source/sink. Kumar et al. [22] investigated theoretical investigation of an unsteady magnetic hydro magnetic free convection heat and mass transfer flow of a non-Newtonian liquid flow past a permeable moving perpendicular plate in the presence of thermal diffusion and heat sink. Reddy at al. [23] discussed mass transfer and heat generation effects on magnetic hydro magnetic free convection flow past an incline vertical surface in a porous medium. Senapati et al. [24] examined magnetic effects on mass and heat transfer of hydrodynamics flow past an oscillating vertical plate in presence of chemical reaction. Raju et al. [25] investigated MHD convective flow through porous medium in a vertical channel with insulated and impermeable base wall in the presence of viscous dissipation and joule heating. The Effects of mass transfer on MHD free convective radiation flow over an impulsively started vertical plate embedded in a porous medium was studied by Bharat and Nityananda [1]. We have extended this work by including the thermal diffusion effect. Though it is an extension to the previous work, it will differ in several aspects like governing equations, non-dimensional parameters, figures etc. The novelty of this study is the investigation of various physical parameter on the flow quantities in the presence of thermal diffusion.

The **laminar convective **heat as well as mass transfer flow of
an incompressible, viscous, electrically conducting fluid over an
impulsively started vertical plate among conduction-radiations
embedded in a porous medium in presence of transverse magnetic
field has been studied. The *x ^{/}* axis is taken the length of plate in the
vertical upward direction and the

(1)

(2)

(3)

The initial and boundary conditions are

(4)

The radiation heat flux term is simplified by making use of the Rosseland approximation [16] as

(5)

Where *σ ^{/}* and

(6)

Substitute (5) and (6) in (2) we have

(7)

Let us introduce the following non-dimensional terms in (1), (7) and (3)

(8)

Hence the non-dimensional form of (1), (2) and (3) are

(9)

(10)

(11)

The transformed initial and boundary conditions are

(12)

The equations (9) to (11) are nonlinear, coupled partial **differential
equations**, so we want to solve them by using **Laplace transform **technique. Taking Laplace transform, the equations (9), (10) and (11)
reduce to

(13)

(14)

(15)

Where‘s’ is the Laplace transform parameter. The boundary condition (12) reduces to the following form after applying Laplace transform.

(16)

Solving (13), (14) and (15) with boundary condition (16) we get

(17)

(18)

(19)

Inverting the equations (17), (18) and (19) we get

(20)

(21)

(22)

The Skin friction at the surface of the plate is given by

(23)

The Nusselt number and Sherwood number at the plate are respectively

(24)

and (24)

To discuss the physical implication of various parameters involved
in the results (20) - (25), the numerical calculation has been carried
out for the distributions of velocity, temperature, concentration,
Skin friction, Nusselt number and **Sherwood number. **The effects
of various physical parameters on these flow quantities such as
Hartmann number M, Prandtl number Pr, Soret number S_{0}, Schmidt
number Sc, Permeability parameter K_{r}, Grashof number Gr, modified
Grashof number Gm and Radiation Parameter Na are studied though
graphs. The concentration profiles are plotted in **Figure 1** for various
values of Schmidt number Sc. From this figure, it is noticed that the
concentration decreases with an increase in the values of the Schmidt
number Sc. A comparison of curves in the figure shows a decrease in
concentration with an increase in Schmidt number Sc. Actually it is
true, since the increase of Sc means decrease of molecular diffusivity
and therefore decreases in concentration boundary layer. The effects
of increasing the Soret number S_{0} on the species concentration profiles
have been shown in **Figure 2**. From this figure, it is noticed that an
increase in Soret number S_{0} results an increase in the concentration profiles. **Figure 3** revels the temperature profiles for different values of
Prandtl number Pr. It is observed that the temperature decrease as an
increase in the values of Prandtl number Pr. The reason is that smaller
values of Prandtl number are equivalent to increase in the thermal
conductivity of the fluid and therefore heat is able to diffuse away from
the heated surface extra rapidly for higher values of Pr (Appendix).
Hence, in the case of larger Prandtl number the thermal boundary
layer is thinner and the rate of heat transfer is reduced. **Figure 4** shows
the temperature profile for different values of Radiation Parameter Na.
From this figure it is noticed that an increase in the values of Na results a decrease in the temperature profiles. The effect of Grashof number
Gr on velocity is presented in **Figure 5**. It is observed that an increase
in Gr leads to a rise in the velocity boundary layer. **Figure 6** shows
the velocity profile for different values of modified Grashof number.
From this figure it is observed that an increase in the values of modified
Grashof number Gm results in increase in the velocity profiles. **Figure
7** shows the velocity profiles for different values of radiation parameter
Na. From this figure it is notice that velocity decreases with increase
in Na. **Figure 8** revels the effect of Prandtl number Pr on the velocity profile. It is evident from the figure that the velocity decreases with
an increase in Pr. **Figure 9** illustrates the velocity profiles for different
values of Schmidt number Sc. It has been observed that the velocity
decreases with increase in Sc. **Figure 10** shows the velocity profiles for
different values of Soret number S_{0}. It was found that an increase in the
value of S_{0} leads to an increase in the velocity distribution across the
boundary layer. This is true, as the S_{0} increases, small light molecules
and large heavy molecules get separated under a temperature gradient, which intern increases the velocity of a fluid. **Figure 11** illustrates the
velocity profiles for different values of Hartmann number M. From this
figure it is notice that velocity decrease with an increase in Hartmann
number M. **Figure 12** reveals the effect of time t on the transient velocity
profiles. It is evident from the figure that the velocity decreases with
increase in t. The velocity profiles are plotted in **Figure 13** for various
values of permeability parameter *K _{r}* From this figure, it is noticed that
the velocity increases with the increase in the values of the permeability parameter

In this paper a theoretical examination has been carried out to
study **thermal diffusion **on MHD free convective radiating flow more
an impulsively started vertical plate embedded in a porous medium.
Solutions for the model has been derived by using Laplace transform
technique. Some conclusions of the study are as follow:

• Concentration distributed is observed to decrease with increase in Schmidt number and it increases with increase in Soret number.

• Temperature decreases with increase in Pr and Na.

• Velocity increases with increase in Gr, Gm, S_{0} and Kr while it
decreases with increase in Na, Pr, Sc, M and t.

• Sherwood number increase with increase in Sc and decrease
with increase S_{0}.

• Nusselt number increases with increase in Pr and Na.

• Skin-friction increases with an increase in Gr, Gm, Kr, Sc and Pr and decreases with increase in M.

- Bharat KS, NityanandaS (2015) The Effects of mass transfer on MHD free convective radiation flow over an impulsively started vertical plate embedded in a porous medium.Journal of Applied analysis and Computation 5: 18-27.
- Ahmed S, BatinA (2013) Convective laminar radiating flow over an accelerated vertical plate embedded in a porous medium with an external magnetic field. IJET 3: 66-72.
- Chamka AJ (2013) Thermal radiation and buoyancy effects on hydro magnetic flow over an accelerating permeable surface with heat source or sink. IJHMT 38: 1699-1712.
- Ahmed A, Kalith K, Zueco J (2014) Non-linear magneto hydrodynamic flow over an impulsively started vertical plate in a saturated porous regime Laplace and Numerical approach. Jof Engg Physics and Thermo physics 87: 1169-1182.
- Ravikumar V, Raju MC,Raju GSS Chamkha AJ (2013)MHD double diffusive and chemically reactive flow through porous medium bounded by two vertical plates. International Journal of Energy & Technology 5:01-08.
- Palani G, Abbas IA (2009) FreeConvection MHD flow with thermal radiation from an impulsively-started vertical plate.Nonlinear Analysis: Modelling and Control 14: 73-84.
- kumarRV, Raju MC ,Raju GSS (2012) Heat and mass transfere on MHD flow of viscous fluid through non-homogeneous porous medium in presence of temperature dependent heat source. International Journal of Contemporary Mathematical sciences 7: 1597-1604.
- Chen CH (2004) Heat and mass transfer in MHD flow by natural convection from a permeable, inclined surface with variable wall temperature and concentration. Act Mechanica 172: 219-235.
- Ravikumar V, Raju MC, Raju GSS (2014) Combined effects of heat absorption and MHD on convective Rivlin-Ericksen flow past a semi-infinite vertical porous plate with variable temperature and suction.Ain Shams Engineering Journal 5: 867-875
- Ahamed S, Abdul B, Chamkha AJ (2015) Numerical/Laplace transform analysis for MHD radiating heat/mass transport in a Darcian porous regime bounded by an oscillating vertical surface. Alexandria Engineering Journal 54: 45-54.
- Kumar AGV, Varma SVK (2011)Thermal radiation and mass transfer effects on MHD flow past a vertical oscillating plate with variable temperature effects variable mass diffusion. Int J Eng 3: 493-499.
- Hossain MA, Takhar HS (1996)Radiation effect on mixed convection along a vertical plate with uniform surface temperature. Heat Mass Transfer 31: 243-248.
- Ibrahim SM, Reddy NB(2013) Similarity solution of heat and mass transfer for natural convection over a moving vertical plate with internal heat generation and a convective boundary condition in the presence of thermal radiation, viscous dissipation, and chemical reaction. ISRN Thermodyn 5: 01-10.
- Kumar PP, Trilochan B(2015)Analytical solution of MHD free convective flow through porous media with time dependent temperature and concentration.Walailak J Sci& Tech 12: 749-762.
- Das SS, Satapathy A, Das JK,Panda JP (2009) Mass transfer effects on MHD flow and heat transfer past a vertical porous plate through a porous medium under oscillatory suction and heat source.Int J Heat Mass Tran52: 5962-5969.
- Siegel R, Howell JR (2002) Thermal radiation heat transfer. Taylor and Francis Group.
- SethG, Nandkeolyar S,Ansari MS (2013)Effects of thermal radiation and rotation on unsteady hydro magnetic free convection flow past an impulsively moving vertical plate with ramped temperature in a porous medium.Journal of Applied Fluid Mechanics 6: 27-38.
- Das K, Jana S(2010) Heat and mass transfer effects on unsteady MHD free convection flow near a moving vertical plate in porous medium. Bull Soc Math Banja Luka 17: 15-32.
- Ravikumar V, Raju MC, Raju GSS, Varma SVK (2013)Magnetic field effect on transient free convection flow through porous medium past an impulsively started vertical plate with fluctuating temperature and mass diffusion. International Journal of Mathematical Archive 4: 198-206.
- Mamtha B, Raju MC, Varma SVK(2015)Thermal diffusion effect on MHD mixed convection unsteady flow of a micro polar fluid past a semi-infinite vertical porous plate with radiation and mass transfer. International Journal of Engineering research in Africa 13: 21-37.
- Harinath Reddy S, Raju MC, Reddy K (2015) Unsteady MHD free convection flow of a kuvshinski fluid past a vertical porous plate in the presence of chemical reaction and heat source/sink. International Journal of Engineering Research in Africa 14: 13-27.
- RavikumarV, Raju MC, RajuGSS (2015)Theoretical investigation of an unsteady MHD free convection heat and mass transfer flow of a non-Newtonian fluid flow past a permeable moving vertical plate in the presence of thermal diffusion and heat sink.International Journal of Engineering Research in Africa 16: 90-109.
- Reddy GM, Reddy BN (2011) Mass transfer and heat generation effects on MHD free convection flow past an incline vertical surface in a porous medium.J of A FM 4: 07-11.
- Senapati N, Dhal RK (2011) Magnetic effects on mass and heat transfer of hydrodynamics flow past an oscillating vertical plate in presence of chemical reactionAMSE B-2: 60-66.
- Raju KVS, Reddy TS, Raju MC, SatyaNarayana PV ,VenkataramanaS (2013) MHD convective flow through porous medium in a horizontal channel with insulated and impermeable bottom wall in the presence of viscous dissipation and joule heatingAin Shams Eng J 5: 543-551.

Select your language of interest to view the total content in your interested language

- Algebraic Geometry
- Analytical Geometry
- Axioms
- Behaviometrics
- Big Data Analytics
- Binary and Non-normal Continuous Data
- Binomial Regression
- Biometrics
- Biostatistics methods
- Clinical Trail
- Complex Analysis
- Cross-Covariance and Cross-Correlation
- Differential Equations
- Fourier Analysis
- Genetic Linkage
- Hamilton Mechanics
- Hypothesis Testing
- Integration
- Large-scale Survey Data
- Matrix
- Microarray Studies
- Multivariate-Normal Model
- Noether's theorem
- Non rigid Image Registration
- Physical Mathematics
- Quantum Mechanics
- Quantum electrodynamics
- Regressions
- Relativity
- Riemannian Geometry
- Robust Method
- Soft biometrics
- Spatial Gaussian Markov Random Fields
- Statistical Methods
- Theoretical Physics
- Theory of Mathematical Modeling
- Topology
- mirror symmetry
- vector bundle

- Total views:
**8912** - [From(publication date):

March-2016 - Nov 15, 2019] - Breakdown by view type
- HTML page views :
**8715** - PDF downloads :
**197**

**Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals**

International Conferences 2019-20