Meccanica DOI 10.1007/s11012-015-0285-y

Transient MHD free convective flow past an infinite vertical plate embedded in a porous medium with viscous dissipation Siva Reddy Sheri . R. Srinivasa Raju

Received: 28 August 2013 / Accepted: 10 September 2015 Ó Springer Science+Business Media Dordrecht 2015

Abstract A numerical investigation of transient magnetohydrodynamic free convection flow past an infinite vertical plate embedded in a porous medium with viscous dissipation. The governing differential equations are transformed into a set of non-linear coupled partial differential equations and are solved numerically using an efficient finite element method. Numerical results for the velocity, temperature and concentration profiles within the boundary layer are separately discussed. Finally the skin-friction, rate of heat and mass transfer coefficients are obtained and reported in tabular forms for different values of physical parameters of the problem. The numerical results are compared and found to be in good agreement with previously published results under special cases. Keywords Transient  Free convection  MHD  Viscous dissipation  Finite element method List of symbols Tw0 Wall dimensional temperature 0 Free stream dimensional temperature T1

S. R. Sheri  R. Srinivasa Raju (&) Department of Mathematics, GITAM University, Hyderabad, Medak, Telangana State 502329, India e-mail: [email protected]

0 U1 0 t t u g K0 ðu0 ; v0 Þ ðx0 ; y0 Þ Cp M Pr Gr Gc Sc Ec D Cw0 0 C1 U0 Bo Qo Q u0p n0

U1 K Nu Sh

Dimensional free stream velocity Dimensional time Time Non-dimensional velocity Acceleration due to gravity Dimensional porosity parameter Dimensional velocity components Dimensional Cartesian coordinates Specific heat capacity Magnetic parameter Prandtl number Thermal Grashof number Solutal Grashof number Schmidt number Eckert number Molecular diffusivity Wall dimensional concentration Free stream dimensional concentration Mean velocity Magnetic field Non dimensional heat absorption parameter Heat absorption parameter Plate velocity Dimensional free stream frequency of oscillation Free stream velocity Permeability parameter Nusselt number Sherwood number

S. R. Sheri e-mail: [email protected]

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Meccanica

Greek symbols b Coefficient of volume expansion b Volumetric coefficient of expansion with concentration v Kinematic viscosity r Electrical conductivity q Fluid density k Thermal conductivity h Non-dimensional temperature / Non-dimensional concentration s Skin-friction coefficient Subscripts x Wall condition ? Free stream condition Superscripts Differentiation w.r.t g

0

1 Introduction Free convection flow is a significant factor in several practical applications that include, for example, cooling of electronic components, in designs related to thermal insulation, material processing, and geothermal systems etc. Transient natural convection is of fundamental interest in many industrial and environmental situations such as air conditioning systems, atmospheric flows, motors, thermal regulation process, cooling of electronic devices, and security of energy systems. Buoyancy is also of importance in an environment where difference between land and air temperatures can give rise to complicated flow patterns. Magnetohydrodynamic has attracted the attention of a large number of scholars due to its diverse applications. In astrophysics and geophysics, it is applied to study the stellar and solar structures, interstellar matter, radio propagation through the ionosphere etc. In engineering it finds its application in MHD pumps, MHD bearings etc. Convection in porous media has applications in geothermal energy recovery, oil extraction, thermal energy storage and flow through filtering devices. Convective heat transfer in porous media has received considerable attention in recent years owing to its importance in various technological applications such as fibers and granular insulation, electronic system cooling, cool combustors, and porous material regenerative heat exchangers. Gebhart [1] has shown that the viscous dissipation

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effect plays an important role in natural convection in various devices which are subjected to large deceleration or which operate at high rotative speeds and also in strong gravitational field processes on large scales (on large planets) and in geological processes. The effect of thermal dispersion on the non-Darcy natural convection over a vertical flat plate in a fluid saturated porous medium were studied by Abbas et al. [2]. The non-Darcy natural convection over a vertical flat plate in a fluid-saturated porous medium was studied numerically by El-Amin et al. [3]. In this paper, authors considered forchheimer extension in the flow equations and the non-dimensional governing equations were solved by the finite element method. Rashidi et al. [4] was examined free convective heat and mass transfer in a steady two-dimensional magnetohydrodynamic fluid flow over a stretching vertical surface in porous medium. In this study thermal radiation and non-uniform magnetic field are taken into consideration. The effect of viscoelasticity on the natural free convective unsteady laminar heat transfer fluid flowing past an impulsively started vertical plate with variable surface temperature and mass concentration discussed by Kumar et al. [5]. The magnetohydrodynamic flow of an electrically conducting, viscous incompressible fluid past a semi-infinite vertical plate with variable surface temperature under the action of transversely applied magnetic field was investigated by Abbas and Palani [6]. Mohamed et al. [7] studied the flow, chemical reaction and mass transfer of a steady laminar boundary layer of an electrically conducting and heat generating fluid driven by a continuously moving porous surface embedded in a non-Darcian porous medium in the presence of a transfer magnetic field. The combined effects of magnetohydrodynamics and radiation on free convection flow past an impulsively started isothermal vertical plate with Rosseland diffusion approximation was studied by Palani and Abbas [8]. Rashidi and Keimanesh [9] was studied an approximate the stream function and temperature distribution of the MHD flow in a laminar liquid film from a horizontal stretching surface. The effects of magnetic interaction number, slip factor and relative temperature difference on velocity and temperature profiles as well as entropy generation in magnetohydrodynamic (MHD) flow of a fluid with variable properties over a rotating disk were investigated by Rashidi et al. [10] using numerical methods. Chamkha [11] studied

Meccanica

unsteady, two-dimensional, laminar, boundary layer flow of a viscous, incompressible, electrically conducting and heat-absorbing fluid along a semi-infinite vertical permeable moving plate in the presence of a uniform transverse magnetic field and thermal and concentration buoyancy effects. The phenomena of mass transfer are also very common in theory of stellar structure and observable effects are detectable, at least on the solar surface. The study of effects of magnetic field on free convection flow is important in liquid-metals, electrolytes and ionized gases. The thermal physics of hydromagnetic problems with mass transfer is of interest in power engineering and metallurgy. Thermal radiation in fluid dynamics has become a significant branch of the engineering sciences and is an essential aspect of various scenarios in mechanical, aerospace, chemical, environmental, solar power, and hazards engineering. Viscous mechanical dissipation effects are important in geophysical flows and also in certain industrial operations and are usually characterized by the Eckert number. El-Amin et al. [12] studied the problem of viscous dissipation effects on non-Darcy natural convection over a vertical flat plate embedded in a fluid-saturated porous medium. In this paper Forchheimer extension is considered in the flow equations and its contribution to viscous dissipation is considered in the energy equation and the non-dimensional governing equations are solved, numerically, by the finite element method (FEM). Abbas et al. [13] were considered viscous dissipation by studying the case of semi-infinite flat plate embedded in saturated porous medium and is kept at constant, higher temperature compared with the surrounding fluid. The thermaldiffusion and diffusion-thermo effects on combined heat and mass transfer of a steady magnetohydrodynamic (MHD) convective and slip flow due to a rotating disk with viscous dissipation and Ohmic heating was studied by Rashidi and Erfani [14]. Sivaiah and Srinivasa Raju [15] studied effect of heat and mass transfer flow with Hall current, heat source and viscous dissipation. Siva Reddy Sheri and Srinivasa Raju [16] studied the effect of Soret on an unsteady magnetohydrodynamics free convective flow past a semi-infinite vertical plate in the presence viscous dissipation. Zaib and Shafie [17] was investigated the effects of thermaldiffusion and diffusion-thermo on an unsteady MHD free convection boundary layer flow with heat and mass transfer in the presence of Hall current, thermal

stratification, chemical reaction, heat generation, thermal radiation, Joule heating and viscous dissipation. Gnaneswara Reddy [18] studied the effects of thermal radiation, viscous dissipation, and Hall current effects on the hydromagnetic convection flow of an electrically conducting, viscous, incompressible fluid past over a stretching vertical flat plate. The effect of variable viscosity on laminar mixed convection flow and heat transfer along a semi-infinite unsteady stretching sheet taking into account the effect of viscous dissipation was studied by El-Aziz [19]. The combined effects of viscous dissipation and Joule heating on the momentum and thermal transport for the magnetohydrodynamic flow past an inclined plate in both aiding and opposing buoyancy situations have been carried out by Das et al. [20]. The objective of the present paper is to study the transient magnetohydrodynamic free convection flow past an infinite vertical plate embedded in a porous medium with viscous dissipation. The problem is governed by the system of coupled non-linear partial differential equations whose exact solutions are difficult to obtain, if possible. So, finite element method has been adopted for its solution, which is more economical from computational point of view. The behaviour of velocity, temperature, concentration, coefficient of skin-friction, Nusselt number and Sherwood number has been discussed for variations in the governing parameters. The results obtained are good agreement with the results of and the results obtained are good agreement with the results of Chamkha [11]. The rest of the paper is arranged as follows. The rest of the paper is structured as follows. The problem formulation is given in Sect. 2. Section 3 enclosed the numerical solutions by finite element method. Our results are presented and discussed in Sect. 4. Finally, Conclusions are presented in Sect. 5.

2 Mathematical formulation Consider the an unsteady free convective boundary layer flow of a viscous, incompressible, electrically conducting fluid past an infinite vertical porous plate in the presence of viscous dissipation is considered. The x0 -axis is taken in the upward direction along the plate and y0 axis normal to it. The physical model and coordinate system are shown in Fig. 1. A uniform magnetic field is applied in the direction perpendicular to the plate.

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x′

t0 \0 : u0 ¼ 0;T 0 ¼ 0;C 0 ¼ 0 for all y0 t0 0 : ( 0 0 0 0 0 0 0 0 0 þeðTW0 T1 Þen t ;C 0 ¼ C1 þeðCW C1 Þen t at y0 ¼ 0 u0 ¼ u0p ; T 0 ¼ T1

Concentration b. l. Thermal b. l. Momentum b. l.

0 0

0 0 0 ¼ UO ð1þeAen t Þ;T 0 ! T1 ;C 0 ! C1 as y0 ! 1 u0 ¼ U 1

T∞′ C ∞′ Bo

g

O

Tw′

Cw′



Fig. 1 Physical model and coordinate system

Due to infinite length in x0 -direction, the flow variables are functions of y0 and t0 only. Under the usual Boussinesq approximation, governing equations (Chamkha [11]) for this unsteady problem are given by Continuity equation ð1Þ

Momentum equation 0 ou0 1 op0 o2 u0 0 ou 0 þ v ¼  þ v þ gbðT 0  T1 Þ q ox0 ot0 oy0 oy02 2 rB v 0 Þ  O u0  0 u0 þ gb ðC 0  C1 K q

Energy equation

ð3Þ

In order to write the governing equations and the boundary conditions in dimensionless form, the following non dimensional quantities are introduced. u0 v0 Vo y 0 t0 V 2 ; t¼ o; ; v¼ ; g¼ Uo Vo v v 0 0 qCp v v T  T1 ; Sc ¼ ; h ¼ 0 Pr ¼ ; 0 k D Tw  T1

u¼

Q¼

vQo ; qCp Vo2

/¼

0 C0  C1 ; 0 Cw0  C1

EC ¼

n¼

vn0 ; Vo2

0 vgb ðCw0  C1 Þ ; 2 Uo Vo



U1 ¼

K 0 Vo2 ; v2

K¼

M¼

0 U1 ; Uo

Gr ¼

Up ¼

u0p Uo

0 gbmðTw0  T1 Þ ; 2 U o Vo

rB2o v ; qVo2

Uo2  0  T1

ð8Þ

Cp Tw0

ou ou dU1  ð1 þ eAent Þ ¼ þ Grh þ Gc/ ot og dt þ

ð4Þ

Under these assumptions, the appropriate boundary conditions for the velocity, temperature and concentration fields are

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ð7Þ

In view of Eqs. (5), (6), (7) and (8), Eqs. (2), (3) and (4) reduce to the following dimensionless form: Momentum equation

Concentration equation oC 0 oC 0 o2 C 0 þ v0 0 ¼ D 02 0 ot oy oy

0 1 op0 dU1 v 0 r 0 ¼ þ 0 U1 þ B2o U1 0 0 q ox K q dt

Gc ¼ ð2Þ

0 oT 0 1 o2 T 0 Qo 0 oT 0 þ v ¼ k  ðT 0  T1 Þ ot0 oy0 qCP oy02 qCP  0 2 v ou þ Cp oy0

From Eq. (1) it is clear that the suction velocity at the plate is either a constant or function of time only. Hence the suction velocity normal to the plate is assumed in the form   0 0 v0 ¼ Vo 1 þ eAen t ð6Þ where A is a real positive constant, e and eA are small less than unity. Here Vo is mean suction velocity, which has a non-zero positive constant and the minus sign indicates that the suction is towards the plate. Outside the boundary layer, Eq. (2) gives

y′

ov0 ¼0 oy0

ð5Þ

o2 u þ NðU1  uÞ og2

ð9Þ

Energy equation  2 oh oh 1 o2 h ou  ð1 þ eAent Þ ¼  Qh þ Ec ot og Pr og2 og

ð10Þ

Meccanica

3 Method of solution

Concentration equation o/ o/ 1 o2 /  ð1 þ eAent Þ ¼ ot og Sc og2

ð11Þ

where N ¼ M þ K1 . The corresponding dimensionless boundary conditions are: t  0 : u ¼ 0; h ¼ 0; / ¼ 0 for all g  u ¼ Up ; h ¼ 1 þ Aeent ; / ¼ 1 þ Aeent at g ¼ 0 t[0 : u ! U1 ¼ 1 þ eent ; h ! 0; / ! 0 as g ! 1

ð12Þ 2.1 Skin-friction Knowing the velocity field, the skin-friction at the plate can be obtained, which in non-dimensional form is given by

 s0w ou Cf ¼ ¼ ð13Þ og g¼0 qUo Vo 2.2 Nusselt number Knowing the temperature field, the heat transfer coefficient can be obtained which in the non-dimensional form, in terms of the Nusselt number, is given by

   x0 oT 0 Nu ¼  ) Nu Re1 x 0 0 0 ðT  T1 Þ oy y0 ¼0

w oh ¼ ð14Þ og g¼0 2.3 Sherwood number Knowing the concentration filed, the rate of mass transfer coefficient can be obtained, which in the nondimensional form, in terms of the Sherwood number, is given by

   x0 oC0 Sh ¼  ) Sh Re1 x 0 0 0 ðC  C1 Þ oy y0 ¼0

w o/ ¼ ð15Þ og g¼0 where Rex ¼  Vmo x is the Reynold’s number.

3.1 Finite element method The finite element method (FEM) is a numerical and computer based technique of solving a variety of practical engineering problems that arise in different fields such as, in heat transfer, fluid mechanics [21], chemical processing [22], rigid body dynamics [23], solid mechanics [24], and many other fields. It is recognized by developers and users as one of the most powerful numerical analysis tools ever devised to analyze complex problems of engineering. The sophistication of the method, its accuracy, simplicity, and computability all make it a widely used tool in the engineering modeling and design process. It has been applied to a number of physical problems, where the governing differential equations are solved by transforming them into a matrix equation. The primary feature of FEM ([25] and [26]) is its ability to describe the geometry or the media of the problem being analyzed with great flexibility. This is because the discretization of the domain of the problem is performed using highly flexible uniform or non uniform patches or elements that can easily describe complex shapes. The method essentially consists in assuming the piecewise continuous function for the solution and obtaining the parameters of the functions in a manner that reduces the error in the solution. The steps involved in the finite element analysis areas follows. Step 1: Discretization of the domain The basic concept of the FEM is to divide the domain or region of the problem into small connected patches, called finite elements. The collection of elements is called the finite element mesh. These finite elements are connected in a non overlapping manner, such that they completely cover the entire space of the problem. Step 2: Generation of the element equations 1.

2.

A typical element is isolated from the mesh and the variational formulation of the given problem is constructed over the typical element. Over an element, an approximate solution of the variational problem is supposed, and by substituting this in the system, the element equations are generated.

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3.

The element matrix, which is also known as stiffness matrix, is constructed by using the element interpolation functions.

Step 3: Assembly of the element equations The algebraic equations so obtained are assembled by imposing the inter element continuity conditions. This yields a large number of algebraic equations known as the global finite element model, which governs the whole domain. Step 4: Imposition of the boundary conditions The physical boundary conditions defined in (12) are imposed on the assembled equations. Step 5: Solution of assembled equations The assembled equations so obtained can be solved by any of the numerical techniques, namely, Gauss elimination method, LU decomposition method, and the final matrix equation can be solved by a direct or indirect (iterative) method. For computational purposes, the coordinate g is varied from 0 to gmax = 5 where gmax represents infinity i.e., external to the momentum,

Zgeþ1

2 4 ðw1 Þ

ge



Zgeþ1

   2 

  ou ou ou dU1 w1 B   ot og og2 dt

ge

þ Nu  NU1  Grh  Gc/dg ¼ 0 "

Zgeþ1 w2

    2 #   oh oh 1 o2 h ou B   Qh  Ec ot og Pr og2 og

ge

dg ¼ 0

ð17Þ



Zgeþ1 w3

     o/ o/ 1 o2 / dg ¼ 0 B  ot og Sc og2

ge

ð18Þ where B ¼ 1 þ eAent and w1, w2, w3 are arbitrary test functions and may be viewed as the variation in u, h and / respectively. After reducing the order of integration and non-linearity, we arrive at the following system of equations

3        

 geþ1 ou ou ow1 ou dU1 ou ðw1 Þ þ N ðw1 Þu 5  Bðw1 Þ þ  ¼0 dy  ðw1 Þ ot og og og dt og ge N ðw1 ÞU1  ðGr Þðw1 Þh  ðGcÞðw1 Þ/

          Zgeþ1

oh oh 1 ow2 oh o u ou ðw2 Þ  Bðw2 Þ þ  Qðw2 Þh  Ecðw2 Þ dg ot og Pr og og og og ge

  geþ1 w2 oh  ¼0 Pr og ge

energy and concentration boundary layers. The whole domain is divided into a set of 50 line segments of equal width 0.1, each element being two-noded. 3.2 Variational formulation The variational formulation associated with Eqs. (9)–   (11) over a typical two-noded linear element ge ; geþ1 is given by

123

ð16Þ

Zgeþ1

ge



ð19Þ

ð20Þ

       o/ o/ 1 ow3 o/ ðw3 Þ  Bðw3 Þ  dg ot og Sc og og



  w3  o/ geþ1 ¼0 Sc og ge

ð21Þ

Meccanica

3.3 Finite element formulation The finite element model may be obtained from Eqs. (19)–(21) by substituting finite element approximations of the form: u¼

2 X

uej wej ;

h¼

j¼1

2 X

hej wej ;

/¼

j¼1

2 X

ði ¼ 1; 2Þ; where and with w1 ¼ w2 ¼ w3 ¼ e /j are the velocity, temperature and concentration respectively at the jth node of typical eth element   ge ; geþ1 and wei are the shape functions for this   element ge ; geþ1 and are taken as:

The finite element model of the equations for eth element thus formed is given by ½K 11  ½K 12 

½K 13 

32

fue g

3

f/e g ½K 31  ½K 32  ½K 33  2 11 3 2 0e 3 fu g ½M  ½M 12  ½M 13  6 7 6 21 7 þ 4 ½M  ½M 22  ½M 23  5 4 fh0e g 5 31

32

33

½M  ½M  ½M  2 3 b1e 6 2e 7 7 ¼6 4 b 5 3e b

Zgeþ1



wei

   owej dg og

ge

þ

Zgeþ1  ge

owei og

ge

 e  owj dg og

Zgeþ1

Mij11 ¼



wei

 e  wj dg;

ge

Mij12 ¼ Mij13 ¼ 0;Kij21 ¼ ðEcÞ

Zyeþ1



wei

  e   o u owj dy; og og

ye

Mij21 ¼ Mij23 ¼ 0; Mij22 ¼

Zgeþ1



wei



 wej dg;

ge

Kij31 ¼ 0; Kij23 ¼ Q

Mij31 ¼ Mij32 ¼ 0;

Zgeþ1 h



wei



wej

i dg;

Mij33 ¼

Zgeþ1



wei



 wej dg;

ge

ð24Þ

Kij22 ¼ B

0e

f/ g

where f½K mn ; ½M mn g andffue g; fhe g; f/e g; fu0e g; fh0e g; f/0e g and fbme gg ðm; n ¼ 1; 2; 3Þ are the set 0 of matrices of order 2  2 and 2  1 respectively and d (dash) indicates dg . These matrices are defined as follows: Kij11 ¼ B

 e  e  wi wj dg;

ge

7 76 ½K 23  5 4 fhe g 5

6 21 4 ½K  ½K 22 

Kij13 ¼ ½Gr þGc

Zgeþ1

ge  g  geþ1 ð23Þ

2

ge

  Zgeþ1   e dU 1 wi dg; Kij12 ¼  þNU1 dt

uej ;hej

geþ1  g g  ge and we2 ¼ ; geþ1  ge geþ1  ge

 e  i wj dg;

ge

ð22Þ

we1 ¼

wei

/ej wej

j¼1

wej

þN

Zgeþ1 h 

Zgeþ1



wei



 owej og

 dg

ge

1 þ Pr

Zgeþ1 

owei og



owej og

 dg;

ge

Kij32 ¼ 0;

Kij33 ¼ B

Zgeþ1



wei

   owej dg; og

ge

   e  ou geþ1 1e ; bi ¼ wi og ge

 e  geþ1

 e  geþ1 wi oh wi o/ 3e ¼ ; b ¼ b2e i i og ge og ge Pr Sc In one-dimensional space, linear element, quadratic element, or element of higher order can be taken. The entire flow domain is divided into 10,000 quadratic elements of equal size. Each element is three-noded, and therefore the whole domain contains 20,001 nodes. At

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Meccanica Table 1 Effects of Gr on Skin-friction, Nusselt number and Sherwood number when Ec = 0.0

Table 2 Effects of Q on Skin-friction, Nusselt number and Sherwood number when Ec = 0.0

Table 3 Effects of Sc on Skin-friction, Nusselt number and Sherwood number when Ec = 0.0

Gr

Present results s

Previous results of Chamka [11] Nu

s

Nu

Sh

0.0

2.720001

-1.716701

-0.809801

2.7200

-1.7167

-0.8098

1.0

3.277202

-1.716702

-0.809802

3.2772

-1.7167

-0.8098

2.0 3.0

3.834301 4.391501

-1.716701 -1.716701

-0.809801 -0.809801

3.8343 4.3915

-1.7167 -1.7167

-0.8098 -0.8098

4.0

4.948701

-1.716701

-0.809801

4.9487

-1.7167

-0.8098

Q

Present results s

Previous results of Chamka [11] Nu

Sh

s

Nu

Sh

0.0

3.459501

-1.069901

-0.809802

3.4595

-1.0699

-0.8098

1.0

3.277201

-1.716701

-0.809801

3.2772

-1.7167

-0.8098

2.0 3.0

3.193301 3.137802

-2.119301 -2.438802

-0.809801 -0.809801

3.1933 3.1378

-2.1193 -2.4388

-0.8098 -0.8098

Sc

Present results s

Previous results of Chamka [11] Nu

Sh

s

Nu

Sh

0.16

3.432801

-1.716701

-0.223101

3.4328

-1.7167

-0.2231

0.6

3.277201

-1.716701

-0.809801

3.2772

-1.7167

-0.8098

1.0 2.0

3.184702 3.048101

-1.716702 -1.716701

-1.342502 -2.674101

3.1847 3.0481

-1.7167 -1.7167

-1.3425 -2.6741

each node, four functions are to be evaluated; hence, after assembly of the element equations, we obtain a system of 80,004 equations which are nonlinear. Therefore, an iterative scheme must be utilized in the solution. After imposing the boundary conditions, a system of equations has been obtained which is solved by the Gauss elimination method while maintaining an accuracy of 0.00001. A convergence criterion based on the relative difference between the current and previous iterations is employed. When these differences satisfy the desired accuracy, the solution is assumed to have been converged and iterative process is terminated. The Gaussian quadrature is implemented for solving the integrations. The code of the algorithm has been executed in MATLAB. Excellent convergence was achieved for all the results.

4 Results and discussions The formulation of the problem that accounts for the transient magnetohydrodynamic free convection flow

123

Sh

past a vertical porous plate in presence of viscous dissipation is performed in the preceding sections. The governing equations of the flow field are solved numerically by using a finite element method. The above presented equations enable us to carry out numerical computations. The following parameter values are adopted for computations unless otherwise indicated in the figures and tables: Gr = 1.0, Gc = 1.0, M = 5.0, K = 1.0, Pr = 0.71, Q = 1.0, Ec = 0.001, Sc = 0.22, Up = 0.5, A = 0.5, e = 0.2, n = 0.1 and t = 1.0. The boundary conditions for g ? 1are replaced by those at gmax where the value of gmax is sufficiently large, so that the velocity at g = gmax is equal to the relevant free stream velocity. We choose gmax = 5. To assess the accuracy of the present method, comparisons between the present results and previously published data Chamkha [11], Table 1 shows the comparison between values of skinfriction coefficient g. Table 2 shows the comparison between values of Nusselt number of Nu, also Table 3 shows the comparison between values of Sherwood number Sh. In fact, this results show a close

Meccanica

agreement, hence an encouragement for further study of the effects of other varies of parameters on the continuous moving surface. Figures 2 and 3 exhibit the effect of thermal Grashof number and solutal Grashof numbers on the velocity profile with other parameters are fixed. The Grashof number signifies the relative effect of the thermal buoyancy force to the viscous hydrodynamic force in the boundary layer. As expected, it is observed that there is a rise in the velocity due to the enhancement of thermal buoyancy force. Also, as Gr increases, the peak values of the velocity increases rapidly near the porous plate and then decays smoothly to the free stream velocity. The solutal Grashof number defines the ratio of the species buoyancy force to the viscous hydrodynamic force. As expected, the fluid velocity increases and the peak value is more distinctive due to increase in the species buoyancy force. The velocity distribution attains a distinctive maximum value in the vicinity of the plate and then decreases properly to approach the free stream value. It is noticed that the velocity increases with increasing values of the solutal Grashof number. The effect of the Hartmann number M is shown in Fig. 4. It is observed that the velocity of the fluid decreases with the increase of the magnetic field number values. The decrease in the velocity as the Hartmann number M increases is because the presence of a magnetic field in an electrically conducting fluid introduces a force called the Lorentz force, which acts against the flow if the magnetic field is applied in the normal direction, as in the present study. This resistive force slows down

Gr = 0.0, 1.0, 2.0, 3.0

1.5

u 1

0.5

0

1

2

η

3

Fig. 2 Effect of Gr on velocity profiles

4

5

Gc = 0.0, 1.0, 2.0, 3.0 1.5

u 1

0.5

0

1

2

η

3

4

5

Fig. 3 Effect of Gc on velocity profiles

M = 0.0, 1.0, 2.0, 3.0 1.5

u 1

0.5

0

1

2

η

3

4

5

Fig. 4 Effect of M on velocity profiles

the fluid velocity component as shown in Fig. 4. The velocity variation due to the increase of permeability of the porous medium K is shown in Fig. 5. The velocity goes on increasing as the permeability increases. Physically, this means that with the increasing permeability of the porous medium the resistance posed by the porous matrix goes on decreasing which consequently leads to the gain in the velocity. This is in agreement with the fact that the velocity is more in the ordinary medium than in porous medium. The effect of the viscous dissipation parameter i.e., the Eckert number Ec on the velocity and temperature are shown in Figs. 6 and 7 respectively. The Eckert number Ec expresses the relationship between the kinetic energy in the flow and the enthalpy. It

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Meccanica

1

K = 1.0, 2.0, 3.0, 4.0

1.5

u

θ 1

0.5

Ec = 0.001, 0.01, 0.1, 1.0

0.5

0

1

2

η

3

4

0

5

Fig. 5 Effect of K on velocity profiles

0

1

2

η

3

4

5

Fig. 7 Effect of Ec on temperature profiles

Ec = 0.001, 0.01, 0.1, 1.0 1.5

Q = 0.0, 1.0, 2.0, 3.0

1.5

u

u 1

0.5

1

0

1

2

η

3

4

5

0.5

0

1

2

η

3

4

5

Fig. 6 Effect of Ec on velocity profiles Fig. 8 Effect of Q on velocity profiles

embodies the conversion of kinetic energy into internal energy by work done against the viscous fluid stresses. Greater viscous dissipative heat causes a rise in the temperature, as well as the velocity and cross flow velocity. This behaviour is evident from Figs. 6 and 7. Figures 8 and 9 illustrate the influence of heat absorption parameter on the velocity and temperature at t = 1.0 respectively. Physically speaking, the presence of heat absorption (thermal sink) effects has the tendency to reduce the fluid temperature. This causes the thermal buoyancy effects to decrease resulting in a net reduction in the fluid velocity. These behaviors are clearly obvious from Figs. 8 and 9 in which both the velocity and temperature distributions decrease as Q increases. It is also observed that the

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both the hydrodynamic (velocity) and the thermal (temperature) boundary layers decrease as the heat absorption effects increase. For different values of the Schmidt number Sc, the velocity and concentration are plotted in Figs. 10 and 11. The Schmidt number Sc embodies the ratio of momentum diffusivity to the mass (species) diffusivity. It physically relates the relative thickness of the hydrodynamic boundary layer and mass transfer (concentration) boundary layer. As the Schmidt number increases the concentration decreases. This causes the concentration buoyancy effects to decrease, yielding a reduction in the fluid velocity. The reduction in the velocity and concentration profiles is

Meccanica

1

1

Sc = 0.16, 0.60, 1.0, 2.0

ϕ

θ 0.5

0

0.5

Q = 0.0, 1.0, 2.0, 3.0

0

0

1

2

η

3

4

5

0

1

2

η

3

4

5

Fig. 11 Effect of Sc on concentration profiles Fig. 9 Effect of Q on temperature profiles

reductions in the skin-friction coefficient and the Sherwood number while the Nusselt number remains constant.

Sc = 0.16, 0.60, 1.0, 2.0

1.5

u 5 Conclusions 1

0.5

0

1

2

η

3

4

5

Fig. 10 Effect of Sc on velocity profiles

This paper considered transient magnetohydrodynamic free convection flow past an infinite vertical plate embedded in a porous medium with viscous dissipation. The non-dimensional governing equations are solved with the help of finite element method, which have wide application in different fields of engineering. The conclusions of the study are as follows: 1.

accompanied by simultaneous reductions in the velocity and concentration boundary layers, which is evident from Figs. 10 and 11. Tables 1, 2 and 3 depict the effects of the solutal Grashof number Gc, the heat absorption coefficient Q and the Schmidt number Sc on the skin-friction coefficient s, Nusselt number Nu and the Sherwood number Sh, respectively. It is observed from these tables that as Gc increases, the skin-friction coefficient increases whereas the Nusselt and Sherwood numbers remain unchanged. However, as the heat absorption effects increase, both the skinfriction coefficient and the Nusselt number decrease whereas the Sherwood number remains unaffected. Also, increases in the Schmidt number cause

2. 3. 4. 5. 6.

The velocity increases with the increase in thermal Grashof number and solutal Grashof number. The velocity decreases with an increase in the magnetic parameter. The velocity increases with an increase in the permeability of the porous medium parameter. An increase in the Eckert number increases the velocity and temperature. Increasing the heat absorption parameter reduces both velocity and temperature. The velocity as well as concentration decreases with an increase in the Schmidt number.

Acknowledgments The authors are thankful to the University Grant Commission, New Delhi, India for providing financial

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Meccanica assistance to carry out this research work under UGC-Major Research Project [F. No. 42-22/2013 (SR)]. 14.

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