The mixed convection along a vertical plate embedded in non-Darcian porous medium with suction and injection

Applied Mathematics and Computation 136 (2003) 139–149 www.elsevier.com/locate/amc

The mixed convection along a vertical plate embedded in non-Darcian porous medium with suction and injection E.M.A. Elbashbeshy Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, Cairo, Egypt

Abstract The effect of surface mass flux on mixed convection along a vertical plate embedded in porous medium is studied. The solutions are obtained for the case of variable surface heat flux in the form qx ðxÞ ¼ Axn . A finite difference scheme was used to solve the system of transformed governing equations. Velocity and temperature profiles increase as the mixed convection parameter increases. As n increases, the velocity decreases and temperature increases. The dimensionless wall shear stress increases and the temperature decreases as the mixed convection parameter increases. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Boundary layer; Mixed convection; Porous medium

1. Introduction Transport processes through porous media play important roles in diverse applications such as petroleum industries, chemical catalytic reactors, and many others. The study of convection heat transfer and fluid flow in porous media has received great attention in recent years. Most of the earlier studies [1–3] were based on Darcy’s law which states that the volume-averaged velocity is proportion to the pressure gradient. Kaviany [4] used the line integral method to study the heat transfer from a semi-infinite flat plate embedded in porous media. Jiin and Chuan [5], studied the mixed convection along a vertical adiabatic surface embedded in porous medium. All of the works mentioned above are conducted for flows over an impermeable surface. The free convections with injection or suction over a vertical 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 2 ) 0 0 0 2 3 - 1

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and horizontal plate in a porous medium were studied by Cheng [6], and Minkowyez et al. [7], respectively. Lai and Kulacki [8,9] investigated the effects of injection and suction on mixed convection over horizontal and inclined surfaces in saturated porous media. Elbashbeshy and Bazid [10], analysed the heat transfer over a continuously moving plate embedded in non-Darcian porous medium. Elbashbeshy and Bazid [11], investigated the effects of suction and injection on mixed convection over horizontal flat plate embedded in nonDarcian porous medium. Even more the mixed convection along a vertical plate embedded in non-Darcian porous medium was studied by Elbashbeshy and Bazid [12]. However, the effect of injection and suction on mixed convection along a vertical plate with variable surface heat flux embedded in nonDarcian porous medium seems not to have been investigated. The object of the present work is to study the problem discussed by Elbashbeshy and Bazid [12], to include a uniform suction and injection.

2. Analysis Assuming that: (a) The magnitude of the free stream velocity is maintained and the flow is steady state and two-dimensional; (b) the boundary layer approximations hold; and (c) the properties are constant, the conservation equations for fluid flow through isotropic and saturated porous medium can be written as [5]. ou ov þ ¼ 0; ox oy
 q ou ou u þv e2 ox oy l o2 u l ¼ q1 gbðT  T1 Þ þ  ðu  u1 Þ  qCðu2  u21 Þ; e oy 2 K u

oT oT o2 T þv ¼ ac 2 ; ox oy oy

ð1Þ

ð2Þ

ð3Þ

where ðu; vÞ are velocity components parallel and perpendicular to the plate, ðx; yÞ are coordinate axes parallel and perpendicular to the plate, q is fluid density, ðe; KÞ are the porosity and permeability of the porous medium, g is the gravitational acceleration, b the coefficients of the thermal expansion, T is the temperature, l is the dynamic viscosity, C is the transport property related to the inertia effect, ac is the effective thermal diffusivity of the saturated porous medium and 1 is the condition at the free stream.

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The boundary conditions for Eqs. (1)–(3) are T ¼ T 1 ; u ¼ u1 ; oT ¼ qx ðxÞ ¼ Axn ; x  0; y ¼ 0; k oy y ! 1; u ¼ u1 ; T ¼ T1 ;

x ¼ 0;

y  0;

u ¼ 0;

v ¼ vx ;

ð4Þ

where k is thermal conductivity and ðA; nÞ are constants.

Fig. 1. Temperature profiles as a function of g for various values of Pr at e ¼ 0:45, c ¼ 2:01, k ¼ 1 and n ¼ 1. (a) fx ¼ 0:2, (b) fx ¼ 0:2.

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By using the scale analysis, we can introduced the following non-dimensional variables rffiffiffiffiffiffi u1 gðx; yÞ ¼ y ; xm xm ; 1ðxÞ ¼ Ku1 ð5Þ wðx; yÞ f ð1; gÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi ; mu1 x rffiffiffiffiffiffiffiffi ðT  T1 Þ u1 x ; hð1; gÞ ¼ xqx ðxÞ m k

where wðx; yÞ is the stream function that satisfies Eq. (1) with u ¼ ow=oy and v ¼ ow=ox. Substituting Eq. (5) into Eqs. (1)–(3), we obtain 1 000 1 f þ 2 ff 00 þ k11=2 h  1ðf 0  1Þ  c1ðf 02  1Þ e 2e 
0 1 0 of 00 of ¼ 2 f ; f e o1 o1

ð6Þ


 1 oh of h00 þ Pr½f h0  ð1 þ 2nÞf 0 h ¼ Pr1 f 0  h0 ; 2 o1 o1

ð7Þ

where the primes denote partial differentiation with respect to g. Pr ¼ m=ac is the Prandtl number, k ¼ Gr=Re2 , is the mixed convection parameter, (Gr is the Grashof number, Re is the Reynolds number) which measures the relative importance of free to force convection. k ¼ 0 corresponds to the case of purely force convection condition. k ! 1 corresponds to the case of purely free convection condition. c ¼ CKu1 =m is the dimensionless inertia parameter expressing the relative importance of the inertia effect. The transformed boundary conditions for Eqs. (6) and (7) are g ¼ 0; g ! 1;

f ð1; 0Þ ¼ fx ; f 0 ð1; 1Þ ¼ 1;

h0 ð1; 0Þ ¼ 1;

f 0 ð1; 0Þ ¼ 0;

ð8Þ

hð1; 1Þ ¼ 0;

Table 1 Nu=Re1=2 for various values of Pr at e ¼ 0:45, f ¼ 5, k ¼ n ¼ 1 and c ¼ 2:01 Pr

pffiffiffiffiffiffi Nu= Re

fx ¼ 0:2 fx ¼ 0:2

0.72

3

5

10

0.8626 0.9479

1.4077 1.7494

1.6379 2.1997

1.9480 3.0537

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Fig. 2. Temperature profiles as a function of g for various values of fx at e ¼ 0:45, c ¼ 2:01, k ¼ n ¼ 1 and Pr ¼ 3.

Fig. 3. Velocity profiles as a function of g for various values of fx at e ¼ 0:45, c ¼ 2:01, k ¼ 1; n ¼ 1 and Pr ¼ 3.

fx ¼ ð2=u1 ÞRe1=2 is the dimensionless injection or suction parameters, is positive for suction ðvx < 0Þ, negative for injection ðvx > 0Þ and fx ¼ 0 for the case of an impermeable surface.

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Table 2 Nu=Re1=2 and f 00 ð5; 0Þ for various values of fx at e ¼ 0:45, k ¼ n ¼ 1, c ¼ 2:01 and Pr ¼ 3 fx

pffiffiffiffiffiffi Nu= Re 00 f (5,0)

)0.2

0.0

0.2

0.4

0.6

1.4077 2.9196

1.5726 3.0180

1.7494 3.1230

1.9374 3.2342

2.1358 3.3513

3. Numerical method In this study, Keller’s box finite-difference method was used. Eqs. (6) and (7) associated with boundary condition (8) were solved by an efficient and accurate implicit finite-difference method similar to that described in Cebeci and Bardshaw [13]. This numerical scheme has several very desirable features that make it appropriate for the solution of parabolic partial differential equations. These features include a second-order accuracy with arbitrary f and g spacings, allowing very rapid f variations and allowing easy programming of the solution of a large number of coupled equations. In the interest of brevity, the details of the solution procedure by this method are not repeated here. The physical quantities of major interest are the wall shear stress sx ¼ lðou=oyÞy¼0 (l is the viscosity) and the local Nusselt number Nu ¼ ðRe1=2 = hðf; 0Þ.

4. Results and discussion Sample of the boundary layer temperature for e ¼ 0:45, c ¼ 2:01, k ¼ n ¼ 1 at f ¼ 5 are present in Fig. 1(a) and (b). The effect of Prandtl number is such that the thermal boundary layer decreases with increasing Pr in the case injection or suction, but the thermal boundary layer thickness in the case suction is greater than the thermal boundary layer thickness in the case injection. The dimensionless heat transfer coefficient Nu=Re1=2 increases with increasing Prandtl number in the case injection or suction, but Nu=Re1=2 in the case injection is greater than in the case suction (see Table 1). The thermal boundary layer thickness increases with increasing suction ðfx < 0Þ and decreases with increasing injection ðfx > 0Þ (see Fig. 2). The velocity decreases with increasing suction ðfx < 0Þ and increases with increasing injection ðfx > 0Þ (see Fig. 3). The dimensionless heat transfer coefficient Nu=Re1=2 increases with increasing injection and decreases with increasing suction. The dimensionless wall shear stress increases with decreasing suction and increases with increasing injection (see Table 2). The thermal boundary layer thickness decreases with increasing exponent of temperature n in the case injection or suction, but thermal boundary layer

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Fig. 4. Temperature profiles as a function of g for various values of n at e ¼ 0:45, c ¼ 2:01, k ¼ 1 and Pr ¼ 3: (a) fx ¼ 0:2, (b) fx ¼ 0:2.

Table 3 Nu=Re1=2 and f 00 ð5; 0Þ for various values of n at e ¼ 0:45, k ¼ 1, c ¼ 2:01 and Pr ¼ 3:0 n

)0.5 0.0 0.5 1.0

fx

pffiffiffiffiffiffi Nu= Re

f 00 ð5; 0Þ

)0.2

0.2

)0.2

0.2

0.6331 0.9869 1.2238 1.4077

1.0017 1.3338 1.5661 1.7494

3.1897 3.0123 2.9519 2.9196

3.2589 3.1787 3.1437 3.1230

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Fig. 5. Velocity profiles as a function of g for various values of c at e ¼ 0:45, k ¼ n ¼ 1 and Pr ¼ 3.

thickness in the case injection is greater than in the case suction (see Fig. 4(a) and (b)). The dimensionless heat transfer coefficient Nu=Re1=2 increases with increasing n for both two cases (suction and injection) whereas the dimensionless wall shear stress decreases with increasing n for both cases. The dimensionless heat

Fig. 6. Temperature profiles as a function of g for various values of c at e ¼ 0:45, k ¼ n ¼ 1 and Pr ¼ 3.

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Fig. 7. Velocity profiles as a function of g for various values of n at c ¼ 2:01, e ¼ 0:45, k ¼ 1 and Pr ¼ 3: (a) fx ¼ 0:2, (b) fx ¼ 0:2.

Table 4 Nu=Re and f 00 ð5; 0Þ for various values of c at e ¼ 0:45, k ¼ n ¼ 1 and Pr ¼ 3:0 c

0.35 0.73 2.01

fx

pffiffiffiffiffiffi Nu= Re

f 00 ð5; 0Þ

)0.2

0.2

)0.2

0.2

1.2543 1.3022 1.4077

1.6034 1.6910 1.7494

1.9446 2.2038 2.9196

2.1234 2.3915 3.1230

transfer coefficient and dimensionless wall shear stress in case injection are greater than in case suction (see Table 3).

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Figs. 5 and 6 show the velocity profiles for various values of c in case suction and injection. The momentum boundary layer thickness of the injection is greater than in the case suction whereas the thermal boundary layer thickness in the case suction is greater than in the case of injection. The velocity distribution increases with increasing c whereas the temperature distribution decreases with increasing c. Fig. 7(a) and (b) show the velocity profiles for values of exponent of temperature n in the case injection or suction. The momentum boundary layer thickness decreases with increasing n. From Table 4 the dimensionless heat transfer coefficient Nu=Re1=2 increases with increasing c for both two cases (suction and injection). Also the dimensionless wall shear stress increases with increasing c. The dimensionless heat transfer coefficient and dimensionless wall shear stress in case injection are greater than in case suction. 5. Conclusions Heat transfer characteristics over a vertical plate with variable surface heat flux embedded in porous medium with suction and injection are discussed. A non-similarity transformation was used to solve the laminar momentum and energy boundary layer equations. It is shown that injection increases the heat transfer whereas, suction causes a decrease in heat transfer for all parameters studied. Finally, it was shown that increasing Prandtl number, exponent temperature n and c enhances the heat transfer coefficient, keeping all other parameters constant.

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[8] F.C. Lai, F.A. Kulacki, The influence of surface mass flux on mixed convection over horizontal plates in saturated porous media, Int. J. Heat Mass Transfer 33 (1990) 576–579. [9] F.C. Lai, F.A. Kulacki, The influence of surface mass flux on mixed convection over inclined plates in saturated porous media, Int. J. Heat Mass Transfer 112 (1990) 515–518. [10] E.M.A. Elbashbeshy, M.A.A. Bazid, Heat transfer over a continuously moving plate embedded in non-Darcian porous medium, Int. J. Heat Mass Transfer 43 (2000) 3087–3092. [11] E.M.A. Elbashbeshy, Laminar mixed convection over horizontal flat plate embedded in nonDarcian porous medium with suction and injection, Appl. Math. Comput. 121 (2001) 123–128. [12] E.M.A. Elbashbeshy, M.A.A. Bazid, The mixed convection along a vertical plate with variable surface heat flux embedded in porous medium. Appl. Math. Comput. 125 (2002) 317–324. [13] T. Cebeci, P. Bradshaw, Finite difference solution of boundary layer equations, in: Physical and Computational Aspect of Convection Heat Transfer, Springer, New York, 1984 (Chapter 13).