Effect of nonlinear thermal radiation on unsteady

Global Journal of Pure and Applied Mathematics (GJPAM) ISSN 0973-1768 Volume 12,Number 1 (2016) © Research India Publications : http://www.ripublication.com

Effect of nonlinear thermal radiation on unsteady MHD flow between parallel plates M.Sathish Kumar, N.Sandeep* and B. Rushi Kumar Department of Mathematics School of Advanced Sciences VIT University, Vellore- 632014 *Email: [email protected] Abstract: An unsteady two-dimensional MHD flow between the infinite parallel plates has been investigated. The effects of thermal radiation and magnetic field have been included in this model. The governing partial differential equations are transformed into the system of ordinary differential equations and then solved numerically by using Runge-Kutta based shooting technique. The impact of innumerable non-dimensional governing constraints on the velocity and temperature profiles along with the local Nusselt number, Squeeze number, Eckert number and the Prandtl number are analyzed in detail. The results indicate that the absolute skin friction coefficient increase with the increase in Squeeze number. It is also found that the Nusselt number is an increasing function of Squeeze number.

resultant velocity decreases with an increase in the Hartmann number. Israel-Cookey et al. [8] studied the influence of viscous dissipation and radiation on unsteady MHD freeconvection flow past an infinite heated vertical plate in a porous medium in the presence of time-dependent suction and concluded that an increase in the radiation and magnetic field parameters lead to a decrease in the temperature profile on cooling. The viscous dissipation and Joule heating effects on MHD-free convection flow from a vertical plate with powerlaw variation in the presence of hall and ion-slip currents was discussed by Abo-Eldahab and Aziz [9] and found that the dimensionless wall shear stress in the x-direction and local Nusselt number are depressed while the dimensionless wall shear stress in the z-direction is increased with the increasing values of magnetic field parameter. Mbeledogu and Ogulu [10] examined the heat and mass transfer of an unsteady MHD natural convection flow of a rotating fluid past a vertical porous flat plate in the presence of radiative heat transfer and concluded that an increase in the Schmidt number results a decrease in the concentration profiles, while an increase in the rate constant results a negligible change in the concentration profiles. MHD flow of a couple stress fluid past a parallel plate channel in presence of the inclined magnetic field has been investigated by Syamala Sarojini et al [11]. Sandeep et al. [12] examined the radiation and chemical reaction effects on MHD flow over a flat plate. Aligned magnetic field, radiation, and rotation effects of an unsteady hydro magnetic free convection flow past an impulsively moving vertical plate in a porous medium was examined by Sandeep et al. [13]. In this paper they found that an increase in the radiation parameter decreases the fluid flow in the isothermal plate, whereas increases in the ramped temperature plate. An unsteady MHD flow of a Casson fluid in a parallel plate with chemical reaction was analyzed by Kirubhashankar and Ganesh [14] and concluded that an increase in the heat source and Prandtl number decreases the temperature profiles. Shekholeslami [15] was discussed a steady nano fluid flow between parallel plates by considering thermophoresis and Brownian effects. In this paper they found that the skin friction increases with an increase in the viscosity and Magnetic field parameters. Radiation and Soret effect of MHD nano fluid flow over a moving vertical plate in a porous medium has been examined by Raju et al. [16] and found that the magnetic field and porosity parameters effectively

Keywords— MHD; Squeeze number; Nusselt number; Eckert number; Thermal radiation; Parallel plates

I. INTRODUCTION The MHD flow in a parallel plate channel is an exciting area in the study of fluid mechanics because of its significance to various engineering applications. Some more application can be seen in the geophysics, thermal insulation engineering, geothermal reservoirs, gas fields, water movements in geothermal reservoirs and exploration of Petroleum, etc. An unsteady boundary layer flow over a stationary semi-infinite flat plate in the presence of magnetic field has been studied by Das [1]. Ibrahim and Hady [2] examined the mixed convection flow over a horizontal plate with vectored mass transfer in a transverse magnetic field. Takhar et al. [3] examined the radiation effects on the MHD free convection flow of a gas past a semi-infinite vertical plate. An unsteady flow and heat transfer of a semi-infinite flat plate with an aligned magnetic field was investigated by Takhar et al. [4]. Youn J Kim [5] analyzed an unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction. They concluded that an increase in the magnetic parameter decreases the velocity field, whereas the permeability parameter and Grashof number respectively increases the velocity profiles. The effects of radiation and variable viscosity on unsteady free convection flow past a semi-infinite flat plate with aligned magnetic field was studied by Seddek [6]. Bargava et al. [7] examined the numerical solution of free convection MHD micro polar fluid flow between two parallel porous vertical plates and found the

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improves the mass transfer rate and reduces the friction factor. Jayachandra Babu et al. [17] investigated the nonlinear thermal radiation and induced magnetic field effects on stagnation-point flow of ferro fluids and found that the magnetic field parameter has a tendency to enhance the heat transfer rate in the presence of the induced magnetic field. The effects of aligned magnetic field and radiation on the flow of ferrofluid over a flat plate with non-uniform heat source/sink have been examined by Raju et al. [18]. Mohyud-Din et al. [19] analyzed the heat and mass transfer in a nano fluid between rotating parallel plates. Raju and Sandeep [20] have studied the dual solution for unsteady heat and mass transfer in bio-convection flow towards a rotating cone/plate in a rotating fluid. Recently, the reachers [21-25] have discussed and produced important results on MHD flows by considering different channels.

 ∂2T ∂2T  ∂T ∂T ∂T µ +u + v =α  2 + 2  + ∂t ∂x ∂y  ∂x ∂y  ( ρcp )

f

  ∂u 2  TH      ∂y     (4)

16σ ∂T  3 ∂T  T , 3k(ρcp ) f ∂z  ∂z  *

+

The corresponding boundary conditions are

dh  , T = T H at y = h ( t ),  dt   ∂u ∂T  v= = = 0 at y = 0 ∂y ∂y 

C = 0, v = v w =

(5)

To convert the governing equations into a set of nonlinear ordinary differential equations, we introduce the following similarity transformations:

y

η=

II. MATHEMATICAL FORMULATION

,u =

γx f ′(η ), 2 (1 − γ t )

l (1 − γ t )    γl v=− f (η ), 0.5 2 (1 − γ t )

Consider an unsteady two-dimensional MHD flow between infinite parallel plates as shown in Fig.1. The two plates are 1

located at l (1 − γ t ) 2 = h (t ) .When γ < 0 the two plates are separated and for γ > 0 , the two plates are squeezed. The generation of heat because of friction induced by shear in the flow and the viscous dissipation effect are retained. It is also assumed that a magnetic field of strength B0 is applied to the flow. The Nonlinear thermal radiation effect is taken into account.

θ=

0.5

(6)

T (or)T = TH (1 + (θ w − 1)θ (η ) , TH

Substituting (6) into (2) and (3) and then eradicating the pressure gradient from the resulting equations, we get the following nonlinear ordinary differential equation:

f IV − S (η f '''+ 2 f ''+ f ' f ''− ff ''') − Ha f '' = 0,

(7)

Substituting (6) into (4), we get the following nonlinear ordinary differential equations:

R[1 + (θ w − 1)θ ]3 + 3R[1 + (θ w − 1)θ ]2 (θ w − 1)θ '2 = 0, (8) The transformed boundary conditions are

f (0) = 0, f ′′ (0) = 0, θ ′ (0 ) = 0,   f (1) = 1, f ′ (1) = 0, θ (1) = 1, 

Fig.1 Flow Geometry

Here S , Pr, Ec, Sc, Ha,

The governing equations for unsteady two-dimensional flow are given by

∂u ∂v + = 0, ∂x ∂y  ∂u ∂u ∂u  ∂P  ∂2u ∂2u  ρf  +u + v  =− + µ  2 + 2  −σ B02u,  ∂t ∂x ∂y  ∂x  ∂x ∂y   ∂ 2 v ∂ 2v   ∂v ∂v ∂v  ∂p ρ f  + u + v  = − + µ  2 + 2 , ∂x ∂y  ∂y  ∂t  ∂x ∂y 

(1)

σ Ha = lB (1− γ t), µ Nusselt number Nu is given by

(2)

(3)

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are defined as: 2

γl µ 1  γx  , Pr = , Ec =   , cp  2(1− γ t)  2µ ρf α 2

S=

(9)

(10)


 ∂T  − lk    ∂y  y = h (t ) Nu = TH

(11)

By using (6), (11) becomes

Nu* =

1 − γ t N u = − θ ′ (1)

(12)

III. RESULT AND DISSCUSSION The impacts of various natural parameters on the velocity, temperature, skin friction coefficient and the local Nusselt number are discussed in this section. The self-similar boundary value problem is solved numerically using RungeKutta based Shooting technique Figs. 2 and 3 show the velocity and temperature profiles for different values of the Squeeze number (S). It is observed from the figure that the velocity and temperature profiles of the flow are reduced by increasing values of the Squeezing number (S). The effect of Hartmann number on the velocity and temperature profiles is displayed in Figs. 4 and 5. It is observed that the increasing the values of Hartmann number decreases both velocity and temperature profiles. The influence of the Eckert number on temperature profiles is shown in Fig.6. It is clear that increasing the values of Eckert number enhances the temperature profiles. The effect of thermal radiation (R) on the temperature profiles is shown in Fig. 7. It is observed that increasing values of R increases the temperature profiles of the flow. Fig.8 illustrates the influence of θ w on the temperature profile. It is observed that an

Fig.2. Effects of S on velocity profile

increase in θ w enhances the temperature profiles. Fig. 9 represents the influence of thermal radiation on the Nusselt number. It is clear that an increase in the value of R declines the Nusselt number. Figs. 10 and 11 describe the influence of Hartmann number on the Skin friction coefficient and local Nusselt number. We observed a mixed response in the friction factor as well as heat transfer rate. It has been observed a similar type of effects with an increase in the squeeze number, which is displayed in Figs. 13 and 14. But Eckert number have tendency to reduce the heat transfer rate, which is displayed in Fig.12.

Fig.3. Effects of S on temperature profile

Fig.4. Effects of Ha on velocity profile

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Fig.8. Effects of

Fig.5. Effects of Ha on temperature profile

θw

on temperature profile

Fig.6. Effects of Ec on temperature profile Fig.9. Effects of R on Nusselt number

Fig.7. Effects of R on temperature profile Fig.10. Effects of Ha on Skin friction

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Fig. 14. Effects of S on Nusselt number

Fig.11. Effects of Ha on Nusselt number

IV. CONCLUSION The conclusions of the present study are as follows: • Increasing the Hartmann number enhances the Nusselt number and improves the Skin friction coefficient. • Squeeze number have tendency to decrease both the velocity and temperature profiles. • Rise in the Hartmann number depreciate both velocity and temperature profiles. • An increase in the thermal radiation parameter enhances the temperature profiles. • Magnetic field parameter helps to control the momentum boundary layer. V. REFERENCES Fig.12. Effects of Ec on Nusselt number

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Fig.13. Effects of S on Skin friction

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