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sical M Phy ath of Kumar et al., J Phys Math 2016, 7:1 em al Journal of Physical Mathematics Journ atics http://dx.doi.org/10.4172/2090-0902.1000156 ISSN: 2090-0902 Research Research Article Article Open OpenAccess Access Thermal Diffusive Free Convective Radiating Flow Over an Impulsively Started V...

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Journ al

atics em

sical M ath Phy of

Journal of Physical Mathematics

Kumar et al., J Phys Math 2016, 7:1 http://dx.doi.org/10.4172/2090-0902.1000156

ISSN: 2090-0902

Research Article Research Article

Open OpenAccess Access

Thermal Diffusive Free Convective Radiating Flow Over an Impulsively Started Vertical Porous Plate in Conducting Field Kumar VR1, Raju MC1*, Raju GSS2 and Varma SVK3 1

Department of Mathematics, Annamacharya Institute of Technology and Sciences, India Department of Mathematics, JNTUA College of Engineering, India 3 Department of Mathematics, S.V. University, India 2

Abstract In this manuscript we have studied the laminar convective heat and mass transfer low of an incompressible, viscous, electrically conducting luid over a luid over an impulsively started vertical plate with conduction-radiation embedded in a porous medium in the occurrence of transverse magnetic ield. 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.

Keywords: MHD; Porous medium; hermal difusion; hermal

βc: Coeicient of volume expansion for mass transfer

radiation; Shear stress; Nusselt number and Sherwood number

κ: hermal conductivity

Nomenclature

ρ : Density

/

C : Species concentration luid

τ: Shearing stress

Cp : Speciic heat at constant pressure

w : Condition on the wall ∞ : Free stream condition

/ w

C : Concentration of the luid for away from the plate

Cw/ : Concentration level near the plate/wall D: Chemical molecular difusivity g: Acceleration due to gravity qr: Radiative heat lux Gr: hermal Grashof number Gm: Modiied Grashof number Kr : Permeability parameter M: Hartmann number Nu: Nusselt number Pr: Prandtl number S0: Soret number Sh: Sherwood number

Tw/ : Fluid temperature at the surface u: Dimensional velocity components

Introduction Several transport processes exist in industries and technology where the transfer of heat and mass occurs simultaneously as an outcome of thermal difusion and difusion of chemical species. Natural convection induced by the simultaneous achievement of buoyancy forces resulting from thermal and mass difusion is of considered interest in nature and in many industrial applications such as cosmic luid dynamics, meteorology, chemical industry, cooling of nuclear reactors, magneto hydrodynamics power generators and the earth’s core. Bharat et al.[1] investigated the efects of mass transfer on MHD free convective radiation low over an impulsively started vertical plate embedded in a porous medium. Ahmed et al. [2] discussed convective laminar radiating low over an accelerated vertical plate embedded in a porous medium with an external magnetic ield. Chamka et al. [3] studied thermal radiation and buoyancy efects on hydro magnetic low over an accelerating porous surface with heat source or sink. Ahmed et al. [4] examined Non-linear magneto hydrodynamic low more an impulsively started vertical plate in a saturated porous regime Laplace and Numerical approach. Ravi Kumar et al. [5] examined MHD

Sc: Schmidt number

T / : Temperature u0: Plate velocity β : Coeicient of volume expansion for heat transfer θ: Dimensional luid n: Kinematic viscosity σ : Electrical conductivity C: Dimensionless species concentration J Phys Math ISSN: 2090-0902 JPM, an open access journal

*Corresponding author: Raju MC, Department of Mathematics, Annamacharya Institute of Technology and Sciences, India, Tel: 009457759877; E-mail: [email protected] Received January 11, 2016; Accepted January 31, 2016; Published 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.

Volume 7 • Issue 1 • 1000156

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

Page 2 of 8 double difusive and chemically reactive low through porous medium bounded by two vertical plates. Palani et al. [6] studied free convection MHD low with thermal radiation from an impulsively-started vertical plate. Ravi Kumar et al. [7] discussed heat and mass transfer efects on MHD low of viscous luid through non-homogeneous porous medium in occurrence of temperature dependent heat source. Chen et al. [8] discussed heat and mass transfer in MHD low by ordinary convection from a permeable, inclined surface with variable wall temperature and concentration. Ravi Kumar et al. [9] discussed combined efects of heat absorption and MHD on convective Rivlin-Ericksen low past a semi-ininite 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 efects on MHD low past a vertical oscillating plate among variable temperature efects variable mass difusion. Hossain et al. [12] studied radiation efect 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 low through porous media with time dependent temperature and concentration. Das et al. [15] discussed mass transfer efects on MHD low and heat transfer past a vertical porous plate throughout a porous medium below oscillatory suction and heat source [16]. Seth et al. [17] studied efects of thermal radiation and rotation on unsteady hydro magnetic free convection low past an impulsively moving vertical plate with ramped temperature in a porous medium. Das et al. [18] discussed heat and mass transfer efects on unsteady MHD free convection low near a moving vertical plate in porous medium. Kumar et al. [19] examined magnetic ield efect on transient free of charge convection low through porous medium past an impulsively started vertical plate with luctuating temperature and mass difusion. Mamtha et al. [20] discussed thermal difusion efect on MHD mixed convection unsteady low of a micro polar liquid past a semi-ininite vertical porous plate with radiation and mass transfer. Redddy et al. [21] examined unsteady MHD free convection low of a Kuvshinski luid 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 low of a non-Newtonian liquid low past a permeable moving perpendicular plate in the presence of thermal difusion and heat sink. Reddy at al. [23] discussed mass transfer and heat generation efects on magnetic hydro magnetic free convection low past an incline vertical surface in a porous medium. Senapati et al. [24] examined magnetic efects on mass and heat transfer of hydrodynamics low past an oscillating vertical plate in presence of chemical reaction. Raju et al. [25] investigated MHD convective low through porous medium in a vertical channel with insulated and impermeable base wall in the presence of viscous dissipation and joule heating. he Efects of mass transfer on MHD free convective radiation low 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 difusion efect. hough it is an extension to the previous work, it will difer in several aspects like governing equations, non-dimensional parameters, igures etc. he novelty of this study is the investigation of various physical parameter on the low quantities in the presence of thermal difusion. J Phys Math ISSN: 2090-0902 JPM, an open access journal

Mathematical Formulations he laminar convective heat as well as mass transfer low of an incompressible, viscous, electrically conducting luid over an impulsively started vertical plate among conduction-radiations embedded in a porous medium in presence of transverse magnetic / ield has been studied. he x axis is taken the length of plate in the vertical upward direction and the y / axis is taken normal to the plate. A transverse magnetic ield of identical strength B0 is assumed to be applied normal to the plate. It is also implicit that the thermal radiation along the plate and viscous dissipation is implicit to be negligible. he induced magnetic ield and viscous dissipation is understood to be negligible. Initially it is assumed that the plate and luid are at same temperature T∞/ in the stationary situation with concentration level C∞/ at all the points. At time, t / > 0 the plate is speciied an impulsive motion in its own plane with velocity u0. he temperature of the plate and the concentration stage are also raised to Tw/ and Cw/ . hey are / maintained at the similar level for all time t > 0 . hen under the above assumption the unsteady low with usual Boussinesq’s estimate is governed by the following equations.

∂u / ∂ 2u /  σ B 2 v  g β (T / − T∞/ ) + g β c ( C / − C∞/ ) + v / 2 −  0 + /  u / = / K  ∂t ∂y  ρ

∂T / ∂t

ρ C= κ p /

∂ 2T / ∂y

/2

−

∂qr ∂y /

(1)

(2)

∂C / ∂ 2C / ∂ 2T / = D / 2 + D1 / 2 / ∂t ∂y ∂y

(3)

he initial and boundary conditions are

t / ≤ 0; u / = 0, T / = T∞/ , C / = C∞/ for ever y t / > 0; u / = u0 , T / = Tw/ , C / = Cw/

at y = 0

t > 0; u → 0, T → T , C → C at y → 0 /

/

/

/ w

/

(4)

/ w

he radiation heat lux term is simpliied by making use of the Rosseland approximation [16] as

qr =

−4 σ / ∂T / 3 a / ∂y /

4

Where

(5)

σ / and a / are the Stefan-Boltzmann steady and the mean

absorption coeicient respectively. It should be noted that by using the Roseland approximation, we limit our investigation to optically thick luids. It temperature diferences within the low are suiciently small, 4 such that T / may be expressed as a linear function of the temperature, 4 then the Taylors series for T / and T∞/ , ater neglecting higher order terms is given by

T / ≅ 4T /T∞/ − 3T∞/ 4

3

4

(6)

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

ρC p

∂T /  16 σ / /  ∂ 2T / T∞  = k + 2 ∂t /  3 a /  ∂y /

(7)

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

Volume 7 • Issue 1 • 1000156

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

Page 3 of 8 = y = θ

ρ vC p u0 y / u/ v u02t / u02 K / = ,u = , Pr = , Sc = ,t ,= , Kr v u0 k D v v2

y L   2 t 

σ B02v T / − T∞/ C / − C∞/ ka / = = = , , , C M N 3 , a ρ u02 Tw/ − T∞/ Cw/ − C∞/ 4σ /T∞/

/ / vg β (Tw/ − T∞/ ) vg β c ( Cw/ − C∞/ ) D1 (Tw − T∞ ) = = , Gm , S0 Gr = 3 3 / / u0 u0 v ( Cw − C∞ ) .

(8)

∂θ 3Pr N a= ∂t

( 3N a + 4 )

(9)

∂ 2θ ∂y 2

(10)

∂C 1 ∂ 2C ∂ 2θ = + S 0 ∂t Sc ∂y 2 ∂y 2

(11)

he transformed initial and boundary conditions are t ≤ 0 : u = 0, θ = 0, C = 0 for every y t > 0 : u = 1, θ = 1, C = 1 at y= 0 t > 0 : u → 0,θ → 0, C → 0 as y→∞

(12)

Method of Solution he equations (9) to (11) are nonlinear, coupled partial diferential equations, so we want to solve them by using Laplace transform technique. Taking Laplace transform, the equations (9), (10) and (11) reduce to ∂2 u = ( s + M + K r−1 ) u − Gr θ − Gm C ∂y 2

(13)

∂ 2θ = L1sθ ∂y 2

(14)

∂2 C ∂ 2θ ScsC S Sc − = − 0 ∂y 2 ∂y 2

(15)

Where‘s’ is the Laplace transform parameter. he boundary condition (12) reduces to the following form ater applying Laplace transform. 1 1 1 = = ,θ ,C s s s = θ 0,= u 0,= C 0

= u

y 0 when =

(16)

when y → ∞

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

θ = e− y

1 Ls s 1 L C = e − y Scs − 2 e − y s s

u=

L12 − y e s

s + L5

L8 − y e s − L7

sL

− −

(17) Scs

+

L8 − y e s − L7 L11 − y e s

sSc

L2 − y e s s + L5

+

−

(18)

Ls

L11 − y e s − L10

L11 − y e s − L10

s + L5

−

sSc

Inverting the equations (17), (18) and (19) we get J Phys Math ISSN: 2090-0902 JPM, an open access journal

L8 − y e s

sL

(20)

 y Sc   y Sc  y L C= erfc   − L2erfc   + L2erfc    2 t   2 t  2 t 

(21)

+

(19)

( L12 ) e− y

 e L7 t   y  erfc  + L5t   − ( L8 )   2 t   2   − y L7 + L5  y  y L +L  y  erfc  − ( L7 + L5 ) t  + e 7 5 erfc  + ( L7 + L5 ) t   − e 2 t  2 t    e L10 t    y   y  y L +L − ( L10 + L5 ) t  + e 10 5 erfc  + ( L10 + L5 ) t   − ( L11 )   e− y L10 + L5 erfc  2 t  2 t   2 

u =

Hence the non-dimensional form of (1), (2) and (3) are ∂u ∂ 2u = Grθ + GmC + 2 − Mu − K r−1u ∂t ∂y

θ = erfc 

 2 

L5

 y  y erfc  − L5t  + e 2 t 

y L  e L7 t   − y L ( L8 ) erfc   + ( L8 )   e  2    2 t   y Sc   e L10 t   − y ( L11 ) erfc   + ( L11 )  2  e     2 t 

L7

L5

y L  − L7t  + e y erfc   2 t 

Sc L10

L L7

 y Sc  − L10t  + e y erfc   2 t 

y L  + L7t   − erfc   2 t   Sc L10

(22)

 y Sc  + L10t   erfc   2 t  

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

 ∂u   ∂y  y =

1  ∂u   2 t  ∂y  y 0=

−  = − τ= 

(

(

)) −

 e L7 t   e − L5 t  − ( L8 )   π   2   e L10 t  4 − ( L7 + L5 ) t   e (23)  L7 + L5 −2erfc ( L7 + L5 ) t −  − ( L11 )  2  π      L  4 − ( L10 + L5 ) t   e  +  L10 + L5 −2erfc ( L10 + L5 ) t −  − ( L8 )  π    πt   Sc   e L7 t   4  ( L8 )    LL7 −2erfc L7t − e− ( LL7 )t  − ( L11 )   + π   2   πt   e L10 t   4  ( L11 )    ScL10 −2erfc L10t − e−( ScL10 )t  2 π    

 1  = ( L12 )    L5 −2erfc  2 

(

(

(

(

0

L5t

)

4

)

)

)

he Nusselt number and Sherwood number at the plate are respectively  ∂θ  L − Nu =  = πt  ∂y  y = 0

(24)

 ∂C  Sc Sc L and S h = − − L2 + L2  = ∂ y π π π t t t   y =0

(25)

Result and Discussion 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. he efects of various physical parameters on these low quantities such as Hartmann number M, Prandtl number Pr, Soret number S0, Schmidt number Sc, Permeability parameter Kr, Grashof number Gr, modiied Grashof number Gm and Radiation Parameter Na are studied though graphs. he concentration proiles are plotted in Figure 1 for various values of Schmidt number Sc. From this igure, it is noticed that the concentration decreases with an increase in the values of the Schmidt number Sc. A comparison of curves in the igure 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 difusivity and therefore decreases in concentration boundary layer. he efects of increasing the Soret number S0 on the species concentration proiles have been shown in Figure 2. From this igure, it is noticed that an increase in Soret number S0 results an increase in the concentration

Volume 7 • Issue 1 • 1000156

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

Page 4 of 8 a decrease in the temperature proiles. he efect 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 proile for diferent values of modiied Grashof number. From this igure it is observed that an increase in the values of modiied Grashof number Gm results in increase in the velocity proiles. Figure 7 shows the velocity proiles for diferent values of radiation parameter Na. From this igure it is notice that velocity decreases with increase in Na. Figure 8 revels the efect of Prandtl number Pr on the velocity

1 0.9

t=1,Pr=0.71,Na=1,,S0=0.1, Gr=5,Gm=5,Kr=0.5,M=2;

Sc=0.22,0.60,0.78,0.96

0.8 0.7

c

0.6 0.5 0.4 0.3 0.2 0.1 0

1 0

1

2

3

4

5

6

7

8

9

10

y

0.9

Figure 1: Effect of Schmidt number on Concentration.

Pr=0.71,t=1,S0=0.1,Na=1,Sc=0.22, Na=1,Gm=5,Kr=0.5,M=2;

Na=1,2,3,4

0.8 0.7 0.6

T

1

t=1,Pr=0.71,Na=1,,S0=0.1, Gr=5,Gm=5,Kr=0.5,M=2;

c

0.9

0.5 0.4

0.8

0.3

0.7

0.2

0.6

0.1

0.5

0

0

1

2

3

4

5

6

7

8

9

10

9

10

y

0.4 So=0.1,0.3,0.5,0.7,0.9

0.3

Figure 4: Effect of Radiation parameter on Temperature.

0.2 0.1 1.8

0

0

1

2

3

4

5

6

7

8

9

10

Pr=0.71,t=1,S0=0.1,Na=1,Sc=0. 22, Na=1,Gm=5,Kr=0.5,M=2;