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Finite-difference analysis of natural convection flow of a viscous fluid in a porous channel with constant heat sources M.A. Rana a,b,∗ , Rashid Qamar c , A.A. Farooq d , A.M. Siddiqui e a

Quality Assurance Division, PINSTECH, P. O. Nilore, Islamabad 44000, Pakistan

b

Department of Basic Sciences, Riphah International University, Sector I-14, Islamabad 44000, Pakistan

c

Directorate of Management Information System, PAEC HQ, Islamabad 44000, Pakistan

d

Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan

e

Department of Mathematics, Pennsylvania State University, York Campus, York, PA 17403, USA

article

info

Article history: Received 2 August 2010 Received in revised form 3 June 2011 Accepted 8 June 2011 Keywords: Viscous fluid Natural convection Porous channel Numerical techniques

abstract The fully developed natural convection flow of a viscous fluid in a porous channel is modeled and studied numerically. The walls are kept at constant temperatures. The effects of various dimensionless parameters emerging in the model are studied graphically. It has been noted that the velocity and temperature both depend on the heat source and the free convection parameters. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction The natural convection flow between heated vertical plates is a classical problem that occurs in many physical phenomena and engineering applications and has received attention in recent years because of its various applications in the design of nuclear reactors, aircraft cabin insulation, thermal storage systems and cooling of electronic equipments. Ostrach [1,2] has considered the laminar natural convection flow between heated plates when either the temperature changes linearly along the plate or the plates are kept at a constant temperature. Rao [3] extended this to the case of flow in a porous channel. Arora and Agarwal [4] have studied the fully developed free convection flow of a viscous fluid in a porous channel including frictional heating effect in the presence of constant heat sources or sinks distributed uniformly in the fluid. Basant and Abiodun [5] investigated the free convective flow of heat generating/absorbing fluid between vertical parallel porous plates due to periodic heating of the porous plates and performed analysis by considering fully developed flow and steady-periodic regime. Mostafa [6] analyzed the influence of radiation and temperature-dependent viscosity on the unsteady flow and heat transfer of an electrically conducting fluid past an infinite vertical porous plate taking into account the effect of viscous dissipation. The governing equations are converted into a system of nonlinear ordinary differential equations via a local similarity parameter and the resulting coupled nonlinear ordinary differential equations are solved numerically using the fourth order Runge–Kutta integration scheme with the shooting method. The natural convection flow of Powell–Eyring fluids between vertical flat plates has been carried out by Bruce and Na [7]. Rajagopal and Na [8] studied the natural convection flow of a homogeneous incompressible grade three fluid between two infinite parallel vertical plates numerically and investigated the effect of the non-Newtonian nature of fluid on the skin friction and heat transfer.

∗ Corresponding author at: Department of Basic Sciences, Riphah International University, Sector I-14, Islamabad 44000, Pakistan. Tel.: +92 333 512 6773. E-mail address: [email protected] (M.A. Rana). 0893-9659/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2011.06.003

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In the present work, the fully developed free convection flow of a viscous fluid in a porous channel including frictional heating in the presence of constant heat sources or sinks distributed uniformly in the fluid is studied numerically. The usual Boussinesq law is assumed to express the body force term as the buoyancy term. The effects of appropriate dimensionless parameters on velocity, temperature, skin friction and the Nusselt number are discussed in detail. 2. Description of the problem The fully developed steady laminar natural convection flow of a viscous incompressible fluid between two infinite parallel porous plates of equal permeability is considered. The plates are h distance apart. The x-axis is taken along one of the plates and the y-axis perpendicular to it. The flow is oriented in the direction of the body force. It is assumed that both the velocity and temperature depend upon y because the boundaries in the x-direction are of infinite dimensions. Therefore, for the problem under consideration, we seek velocity and temperature fields respectively of the form V = (u(y), v, 0),

(1)

T = T (y).

(2)

3. Mathematical formulation of the problem The equation of continuity Div V = 0

(3)

is identically satisfied by the velocity field (1). The momentum and energy equations take the form

ρv

du dy

=µ

∂p = 0, ∂y dT

ρ CP v

dy

d2 u dy2

+ ρb −

∂p , ∂x

(4)

∂p = 0, ∂z =k

d2 T dy2

(5)

+µ

du

2

dy

+ Φ,

(6)

where ρ denotes the density of the fluid, µ the coefficient of viscosity, b the generated body force per unit mass, p the pressure, CP the specific heat at constant pressure, k the coefficient of thermal conductivity, and Φ the heat added due to constant heat sources. The appropriate boundary conditions to be satisfied are u = 0, u = 0,

v = V0 , v = V0 ,

T = T1 T = T2

at y = 0 at y = h,

(7)

where V0 < 0 represents suction at the plate y = 0 and injection at y = h, whereas V0 > 0 the vice versa. Using the usual Boussinesq assumption for the body force [8], we have

ρ b = −ρ0 [1 − γ (T − TS )]g , (8) where g denotes the gravity, γ the thermal expansion, ρ0 a constant density, and TS reference temperature. Eqs. (4) and (5) imply that ∂ p/∂ x is at most constant. Following Rajagopal and Na [8], we choose this constant to be equal to −ρ0 g, then the Eq. (4) takes the form

ρv

du dy

=µ

d2 u dy2

+ ρ0 γ (T − TS )g .

(9)

Introducing the following dimensionless parameters U =

u VR

,

η=

y h

,

T¯ =

T − TS T0

,

VR =

k

ρ0 λ

gh2

,

T0 =

µV R , ρ0 λgh2

(10)

then the governing Eqs. (6) and (9) with boundary conditions (7) become d2 U

dU

−R + T = 0, dη 2 dU dT + − RPr + Q α = 0, dη 2 dη dη U (0) = 0, U (1) = 0, T (0) = Q , T (1) = mQ , dη2 d2 T

(11)

(12)

where R = V0 h/v is the suction Reynolds number, Pr = µCP /k the Prandtl number, Q = PrGg γ h/CP the dimensionless group, G = g γ h3 (T1 − TS )/v 2 the Grashof number and α = Φ h2 /k(T − T0 ) the heat source parameter.

M.A. Rana et al. / Applied Mathematics Letters 24 (2011) 2087–2092

(a) Velocity profiles.

2089

(b) Temperature profiles.

Fig. 1. Velocity and temperature profiles for various values of the heat source parameter α , fixing R = 2, Q = 3, Pr = 0.75 and m = 1.

4. Numerical procedure Consider a simplest boundary value problem F (u′′ , u′ , u, r ) = 0, u(a) = A and

(13)

u(b) = B.

(14) ′

′′

To solve this boundary value problem, the derivatives u and u involved in the problem are approximated by finite differences of appropriate order. If we employ second order standard central difference formulation, then we can write u′ (r ) = u′′ (r ) =

u(r + h) − u(r − h) 2h

+ O(h2 ),

u(r + h) − 2u(r ) + u(r − h) h2

(15)

+ O(h2 ).

(16)

This converts the given boundary value problem into a linear system of equations involving values of the function u at a, a + h, a + 2h, . . . , b. For higher accuracy, one should choose h small. However, this increases the number of equations in the system which in turn increases the computational time. Depending upon the size of this resulting system of linear equations, it can either be solved by exact methods or approximate methods. In the present problem, the coupled governing differential equations (11) are discretized using second order central finite-difference approximations defined in Eqs. (15) and (16). The resulting system of algebraic equations is solved using successive under relaxation scheme. The difference equations are linearized employing a procedure known as lagging the coefficients [9]. The iterative procedure is repeated until convergence is obtained according to the criterion max |u(n+1) − u(n) | < ε, where the superscript ‘n’ represents the number of iteration and ‘ε ’ is the order of accuracy. In the present case, ε is taken as 10−16 . 5. Results and discussion The fully developed free convection flow of a viscous incompressible fluid between infinite porous plates of equal permeability oriented in the direction of the body force is considered. The walls are kept at constant temperatures. The governing equations of momentum and energy are solved numerically. The effects of the heat source parameter α , the suction Reynolds number R, the dimensionless group Q and the Prandtl number Pr on velocity and temperature are discussed in Figs. 1–4. We varied α = ±3, ± 10, ± 20 fixing R = 2, Q = 3, Pr = 0.75 and m = 1 in Fig. 1. Here, the positive value of α denotes the heat source while the negative value of α corresponds to sink. The velocity, as expected, increases with an increase in the heat source parameter, reaches a maximum almost in the middle. It, however, decreases as −α increases and a complete reverse flow pattern is observed for large sinks (−α ≥ 20). For the case without the heat source (α = 0), the results of Rao [3] are recovered. The effect of α on temperature T is similar to that of velocity U. It is noted that for large sinks, cooling takes place between the gap which is more and more pronounced with an increase in −α .

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(a) Velocity profiles.

(b) Temperature profiles.

Fig. 2. Velocity and temperature profiles for various values of the suction Reynolds number R fixing α = 10, Q = 3, Pr = 0.75 and m = 1.

(a) Velocity profiles.

(b) Temperature profiles.

Fig. 3. Velocity and temperature profiles for various values of the Prandtl number Pr fixing α = 10, Q = 3, R = 2 and m = 1.

(a) Velocity profiles.

(b) Temperature profiles.

Fig. 4. Velocity and temperature profiles for various values of dimensionless group Q fixing α = 10, R = 2, Pr = 0.75 and m = 1.

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Table 1 Skin frictions on walls taking Q = 3, Pr = 0.75, m = 1.

α

−20 −10 −3

dU dη

η=0

dU dη

η=1

R = −2

R = −1

R=1

R=2

−1.4243

−1.2702

−0.7036

−0.4090

0.4090

0.7036

1.2702

1.4243

0.2180 1.3954 2.4241 3.6480 5.4442

0.1952 1.2498 2.1740 3.2775 4.9051

0.2557 0.9459 1.5507 2.2727 3.3374

0.2998 0.8077 1.2512 1.7788 2.5527

−0.2998 −0.8077 −1.2512 −1.7788 −2.5527

−0.2557 −0.9459 −1.5507 −2.2727 −3.3374

−0.1952 −1.2498 −2.1740 −3.2775 −4.9051

−0.2180 −1.3954 −2.4241 −3.6480 −5.4442

3 10 20

R = −2

R = −1

R=1

R=2

Table 2 Nusselt number on the walls.

α

1 Q

dU dη

1 Q

η=0

dU dη

η=1

R = −2

R = −1

R=1

R=2

R = −2

R = −1

R=1

R=2

−20 −10 −3

−11.7985 −5.9669 −1.7171

−10.7450 −5.4417 −1.5662

−8.5042 −4.2951 −1.2422

−7.4261 −3.7386 −1.0858

3 10 20

2.0426 6.5733 13.3348

1.8706 6.0237 12.2471

1.4496 4.6835 9.4929

1.2378 4.0102 8.0938

7.4261 3.7386 1.0858 −1.2378 −4.0102 −8.0938

8.5042 4.2951 1.2422 −1.4496 −4.6835 −9.4929

10.7450 5.4417 1.5662 −1.8706 −6.0237 −12.2471

11.7985 5.9669 1.7171 −2.0426 −6.5733 −13.3348

Fig. 2 exhibits the effect of the suction Reynolds number R on velocity and temperature distribution. The mass injection

(R > 0) reduces the velocity at the plate located at y = 0 and enhances the velocity at the plate located at y = 1. On the other hand, mass suction (R < 0) has an opposite effect on the velocity. The effect of R on T is similar to that of velocity. It can be seen that both the velocity as well as the temperature decrease with an increase in the Prandtl number, Fig. 3 Fig. 4 depicts the effect of the dimensionless group on velocity and temperature profiles. It is noted that velocity as well as temperature increase with an increase in Q .

6. Skin friction and the Nusselt number The very important design parameters skin friction S and the Nusselt number Nu are calculated as follows:

µk dU = , ρ bαβ h3 dη η=0,1 1 dT Nu = , m = 1. Q dν η=0,1 S=

(17)

(18)

The values of skin friction and the Nusselt number for various values of the heat source parameter α and the suction Reynolds number R are tabulated in Tables 1 and 2 respectively. It is observed that the Nusslet number at the upper wall increases as the sink increases. On the other hand, the Nusslet number at the upper wall decreases as the source increases. The effect of the Nusslet number at the lower wall is quite opposite to that of its effect at the upper plate. The behavior of the skin friction with respect to the heat source parameter is quite similar to that of the Nusselt number, Table 1. It is noted from the Table 2 that the Nusslet number at the walls decreases as the value of mass suction increases for a given sink. The Nusslet number, however, has an opposite trend at the walls in the case of mass injection for a sink. For heat source, the behavior of the Nusslet number at the walls due to the suction Reynolds number is completely reversed. The behavior of skin friction with respect to the suction Reynolds number is quite similar to that of the Nusselt number, Table 1.

7. Concluding remarks The effect of several non-dimensional parameters on velocity, temperature, skin friction and the Nusselt number are studied numerically. It is noted that the velocity decreases as the heat sink parameter increases and a complete reverse flow pattern is observed for large sinks. For large sinks, cooling takes place between the gap. The effect of the Nusselt number at the walls is quite opposite for the sink and source. The behavior of skin friction with respect to the suction Reynolds number (heat source parameter) is quite similar to that of the Nusselt number.

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References [1] S. Ostrach, Combined natural and forced-convection laminar flow and heat transfer of fluids with and without heat sources in channels with linearly varying wall temperatures, NACA-TN-3141, 1954. [2] S. Ostrach, Laminar natural-convection flow and heat transfer of fluids with and without heat sources in channels with constant wall temperatures, NACA-TN-2863, 1952. [3] A.K. Rao, Laminar natural convection flow with suction or injection, Appl. Sci. Res. 11 (1) (1962) 1–9. [4] S.S. Arora, R.S. Agarwal, Free convection flow with constant heat sources in a porous channel, Def. Sci. J. 31 (1) (1981) 45–52. [5] Basant K. Jha, Abiodun O. Ajibade, Free convective flow of heat generating/absorbing fluid between vertical porous plates with periodic heat input, Int. Commun. Heat Mass Transfer 36 (6) (2009) 624–631. [6] Mostafa A.A. Mahmoud, Thermal radiation effect on unsteady MHD free convection flow past a vertical plate with temperature-dependent viscosity, Can. J. Chem. Eng. 87 (1) (2009) 47–52. [7] R.W. Bruce, T.Y. Na, Natural convection flow of Powell–Eyring fluids between two vertical plates, in: ASME 67 WA/HT 25, 1967. [8] K.R. Rajagopal, T.Y. Na, Natural convection flow of a non-Newtonian fluid between two vertical plates, Acta Mech. 54 (1985) 239–246. [9] K.A. Hoffmann, S. Chiang, Computational Fluid Dynamics for Engineers, I, Engineering Education SystemTM , USA, 1995.