Radiation effect on heat transfer of a micropolar fluid through a porous medium

Applied Mathematics and Computation 169 (2005) 500–510 www.elsevier.com/locate/amc

Radiation effect on heat transfer of a micropolar fluid through a porous medium Emad M. Abo-Eldahab *, Ahmed F. Ghonaim Department of Mathematics, Faculty of Science, Helwan University, Ain Helwan, PO Box 11795 Cairo, Egypt

Abstract Radiation effect on heat transfer of a micropolar fluid past unmoving horizontal plate through a porous medium is considered. Numerical solutions for the governing boundary layer equations obtained by applying an efficient numerical technique based on the shooting method. This solution studied for a range of values of permeability, vortex-viscosity and radiation parameters. The effects of these parameters examined on the velocity distribution of fluid, temperature distribution and angular velocity of microstructures as well as the coefficient of heat flux and shearing stress at the plate. Ó 2004 Published by Elsevier Inc.

1. Introduction Eringen [1] first drived the theory of micropolar fluids, which describe the microrotation effects to the microstructures. Since, Navier–Stokes theory does not describe preciously the physical properties of polymer fluids, colloidal solu*

Corresponding author. E-mail address: [email protected] (E.M. Abo-Eldahab).

0096-3003/$ - see front matter Ó 2004 Published by Elsevier Inc. doi:10.1016/j.amc.2004.09.059

E.M. Abo-Eldahab, A.F. Ghonaim / Appl. Math. Comput. 169 (2005) 500–510

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Nomenclature f g G1 r l Pr T k1 S u t x y q H m cp qr U0

Dimensionless velocity functions Dimensionless microrotation angular velocity Microrotation constant Angular velocity Dynamical viscosity Prandtl number Temperature distribution Coupling constant Constant characteristic to the fluid Velocity in x-direction Velocity in y-direction Distance along the surface Distance normal to the surface Density of fluid Dimensionless temperature Kinematics viscosity Specific heat The radiate heat flux Uniform stream velocity

Subscripts w Conditions at the surface 1 Conditions far away from the surface 0 Differentiate with respect to g K Permeability of the porous medium C ForchheimerÕs inertia constant u Porosity

tions, suspension solutions, liquid crystals and fluids containing small additives. Eringen [2] extended his theory to the theory of thermomicropolar fluids, which interest to the effects of microstructures on the fluid flow. Many processes in new engineering areas occur at high temperatures and knowledge of radiate heat transfer becomes very important for the design of the pertinent equipment. Nuclear power plants gas turbines and the various propulsion devices for aircraft, missiles, satellites and space vehicles are examples of such engineering areas. The study of radiation effects on the various types of flows is quite complicated. In the recent years, many authors have studied radiation effects on the boundary layer of radiating fluids past a plate. Radiation effect on heat transfer in an electrically conducting fluid at a stretch-

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ing surface with a uniform free stream has been analyzed by Abo-Eldahab [3]. The radiation effect on convective heat transfer in an electrically conducting fluid at a stretching surface with variable viscosity and uniform free stream was studied by Abo-Eldahab and El Gendy [4]. Raptis [5] studied the flow of a micropolar fluid past a continuously moving plate by the presence of radiation. The heat transfer of a micropolar fluid by the presence of radiation studied by Raptis and C. Perdikis [6]. The problem of micropolar fluids past through a porous media has many applications, such as, porous rocks, foams and foamed solids, aerogels, alloys, polymer blends and microemulsions. The simultaneous effects of a fluid inertia force and boundary viscous resistance upon flow and heat transfer in a constant porosity porous medium were analyzed by Vafai and Tien [7]. A. Raptis [8] studied boundary layer flow of a micropolar fluid through a porous medium. Abo-Eldahab and El Gendy [9] investigated the convective heat transfer past a stretching surface embedded in nondarcian porous medium in the presence of magnetic field. In my opinion, this is the first analysis that contains radiation effects on heat transfer in a micropolar fluid through a porous medium. In the present work we consider the radiation effects on heat transfer in a micropolar fluid through a porous medium. The governing boundary layer equations have been transformed to ordinary differential once, and these have been solved numerically using shooting technique. The effects of the permeability, vortex-viscosity and radiation parameters examined on the velocity of fluid, temperature distribution and angular velocity of microstructures as well as the coefficient of heat flux and shearing stress at the plate.

2. Mathematical formulation Let us consider a steady two-dimensional flow of an incompressible micropolar fluid through a porous medium past a continuously semi-infinite horizontal plate. The origin is located at the spot through which the plate is drawn in the fluid medium, x-axis is chosen along the plate and y-axis is taken normal to it as shown in Fig. 1. The fluid is considered to be a gray, absorbing–emitting radiation but nonscattering medium and the Rosselant approximation is used to describe the radiate heat flux in the energy equation. The radiate heat flux in the x-direction is considered negligible in comparison to y-direction. The governing equations of boundary layer to micropolar fluid through a porous medium by using a generalized DarcyÕs law are [10,11]: ou ov þ ¼ 0; ox oy

ð1Þ

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Boundary layer

y,υ x,u

x

Fig. 1. Coordinate system for the physical model.

u

ou ou o2 u or mu þ v ¼ m 2 þ k1 þ ðU  uÞ þ CuðU 2  u2 Þ; ox oy oy oy K

G1

u

o2 r ou ¼ 0;  2r  oy 2 oy

oT oT k o2 T 1 oqr þv ¼ .  2 ox oy qcp oy qcp oy

ð2Þ

ð3Þ

ð4Þ

The boundary conditions are given by: y ¼ 0 : u ¼ 0; v ¼ 0; r ¼ 0; T ¼ T w ; y ! 1 : u ! U 0 ; r ! 0; T ! T 1 ;

ð5Þ

where k1 = qS, m ¼ lþS : q By using the Rosselant approximation, we have qr ¼ 

4r oT 4 ; 3k oy

ð6Þ

where r* the Stefan–Boltzman constant and k* the mean absorption coefficient. We can consider that the temperature differences within the flux are sufficiently small such that T4 can be written as a linear function of temperature by expanding T4 in a Taylor series about T and neglecting higher order terms, then T 4 ffi 4T 31 T  3T 41 .

ð7Þ

By substitution from equations (6) and (7) in equation (4), so u

oT oT k o2 T 16r T 31 o2 T þv ¼ þ . ox oy qcp oy 2 3qcp k oy 2

ð8Þ

Proceeding with the analysis, we define a stream function W(x, y) such that u¼

oW oW ; v¼ . oy ox

ð9Þ

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Now, let us consider the transformations 1

Wðx; yÞ ¼ ð2mU 0 xÞ2 f ðgÞ;
12 U0 r¼ U 0 gðgÞ; 2mx
12 U0 g¼ y; 2mx T  T1 . Tw  T1

H¼

ð10Þ

Then the system of equation (2), (3), (8) with boundary conditions (5) becomes f 000 þ ff 00 þ Dg0 þ

1 ð1  f 0 Þ þ N ð1  f 02 Þ ¼ 0; M

ð11Þ

Gg00  2ð2g þ f 00 Þ ¼ 0;

ð12Þ

ð3R þ 4ÞH00 þ 3R Pr f H0 ¼ 0:

ð13Þ

With boundary conditions g ¼ 0 : f ¼ 0; f 0 ¼ 0; g ¼ 0; H ¼ 1; g ! 0 : f 0 ! 1; g ! 0; H ! 0.

ð14Þ

In the above equations, Prime denotes differentiation with respect to g only. Where Pr ¼

qmcp ðPrandtl numberÞ k

D¼

k1 ðCoupling constantÞ m

G¼

G1 U 0 ðMicrorotation parameterÞ mx

R¼

k k ðRadiation parameterÞ 4r T 31

M¼

KU 0 ðPermeability parameterÞ 2umx

N ¼ 2uCx ðInertia coefficient parameterÞ

ð15Þ

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The system of equations (11)–(13) with boundary conditions (14) was solved numerically and the velocity, angular velocity and temperature profiles are discussed in Figs. 2–7. Knowing the temperature field, it is interesting to study the effect of mixed convection and radiation on the rate of heat transfer qw. This is given by

 oT 4r oT 4  . ð16Þ qw ¼ k oy y¼0 3k oy y¼0

0.16 0.14 0.12 0.1

M = 0.5

–g 0.08

M = 0.6 M = 0.7

0.06 0.04 0.02 0

0

1

2

3

4

5

6

7

η

Fig. 2. Variation of the dimensionless angular velocity g with the permeability parameter M (Pr = 0.7, D = 0.5, R = 0.1, G = 2, N = 0.1).

1.2 1 0.8

M= 0.5 M = 0.6

f' 0.6

M = 0.7

0.4 0.2 0

0

1

2

3

η

4

5

6

7

Fig. 3. Variation of the dimensionless velocity with the permeability parameter M (Pr = 0.7 D = 0.5, R = 0.1, G = 2, N = 0.1).

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E.M. Abo-Eldahab, A.F. Ghonaim / Appl. Math. Comput. 169 (2005) 500–510 1 0.8 M = 0.5

0.6

M= 3

Θ 0.4 0.2 0

0

1

2

3

4

5

6

7

η Fig. 4. Variation of the dimensionless temperature with the magnetic parameter M (Pr = 0.7, D = 0.5, R = 0.01, G = 2).

1.2 1 ∆= 2 ∆ = 1.1

0.8

∆= 0 f ' 0.6

0.4 0.2 0 0

1

2

3

4

5

6

7

η

Fig. 5. Variation of the dimensionless velocity with the vortex-viscosity parameter (Pr = 0.7, R = 0.1, M = 0.5, G = 2, N = 0.1).

By using equation (7), we can write equation (16) as follow

 16r T 31 oT qw ¼  k þ ; oy y¼0 3k Which is written in dimensionless form as rffiffiffiffiffiffiffi
 U0 4 H0 ð0Þ. qw ¼ kðT w  T 1 Þ 1þ 3R 2mx

ð17Þ

ð18Þ

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507

0.16 0.14 0.12 0.1 −g

0.08 0.06

∆= 2 ∆ = 0.5 ∆= 0

0.04 0.02 0

0

1

2

3

η

4

5

6

7

Fig. 6. Variation of the dimensionless velocity with the vortex-viscosity parameter (Pr = 0.7, R = 0.1, M = 0.5, G = 2, N = 0.1).

1 0.8 R = 0.1

0.6

R = 0.01

Θ

0.4 0.2 0 0

R= 1

1

2

3

η

4

5

6

7

Fig. 7. Variation of the dimensionless temperature with the radiation parameter (Pr = 0.7, R = 0.1, M = 0.5, G = 2, N = 0.1).

The numerical values of H 0 (0) are given in Table 1 for different values of radiation parameter, permeability and vortex-viscosity parameters.

3. Numerical procedure The shooting method for linear equations is based on replacing the boundary value problem by two initial value problems, and the solutions of the boundary value problem is a linear combination between the solutions of the

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Table 1 Variation of f00 , g 0 and H 0 (0) at the plate with M, D and R M

D

R

f00 (0)

g 0 (0)

H 0 (0)

0.5 0.6 0.7 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.5 0.5 0.5 0.5 0 1.1 0.5 0.5 0.5 0.5

0.1 0.1 0.1 0.01 0.1 0.1 1 2 4 10

1.47373 1.36034 1.27359 1.47373 1.519402 1.41719 1.47373 1.47373 1.47373 1.47373

0.534188 0.516889 0.502788 0.534188 0.533808 0.534708 0.534188 0.534188 0.534188 0.534188

0.203581 0.202933 0.202394 0.170621 0.203441 0.203759 0.359834 0.413389 0.453065 0.484069

two initial value problems. The shooting method for the nonlinear boundary value problem is similar to the linear case, except that the solution of the nonlinear problem cannot be simply expressed as a linear combination of the solutions of the two initial value problems. Instead, we need to use a sequence of suitable initial values for the derivatives such that the tolerance at the end point of the range is very small. This sequence of initial values is given by the secant method, and we use the fourth-order Rung–Kutta method to solve the initial value problems. Following [12] and [13], the value of g at infinity is fixed at 6. Equation of (11)–(13) with boundary conditions (14) were solve numerically using Rung– Kutta method algorithm with a systematic guessing f00 (0), g 0 (0) and H 0 (0) by the shooting technique until the boundary conditions at infinity f 0 (g) decay exponentially to one and g(g) and H(g) to zero. The functions f 0 , g and H are shown in Figs. 2–7.

4. Results and discussion In the present analysis, we studied the more general problem considered the radiation effect on heat transfer of a micropolar fluid past on unmoving horizontal plate through a porous medium. After certain transformation, numerical solution has been obtained by using the shooting method. Results are obtained for various values of permeability, vortex-viscosity and radiation parameters. The effects of these parameters examined on the velocity of fluid, temperature distribution and angular velocity of microstructures along Figs. 2–7. First we consider the variation of permeability parameter M, Figs. 2–4. Fig. 2 shows that the velocity of microstructures increased with increase M. Fig. 3 show that the velocity of fluid decreases with increases M because of an

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increasing in permeability means that decreasing in porosity and then the velocity of the fluid decrease. Fig. 4 illustrates that the dimensionless temperature Hincreases with the increase of M. these as we expect from the decreasing of the velocity. Fig. 5 and 6 describe the effects of the vortex-viscosity parameter D on the velocity of the fluid and angular velocity of the microstructures. It is observed that as D increase, the fluid velocity and the angular velocity of the microstructure increase. As D increase, the viscosity decrease and then both of g and f 0 increase. The results incorporating the effect of radiation on the dimensionless temperature H profile are presented in Fig. 7. We noticed that the temperature distribution decrease. The effect of radiation is to increase the rate of energy transport to the gas, thereby making the thermal boundary layer becomes thicker and the fluid becomes warmer. In fact, as the radiation parameter decreases the fluid temperature. This is because as the radiation parameter R increases, the mean Rosselant absorption coefficient k* increases (for some T1 and thermal conductivity k). According to equations (4) and (6), an increase in k* means decrease of the radiation effect. There is no figures for f 0 or g because of there is no buoyancy force applied, and then the system of the ordinary differential equations uncoupled, so, there is no effect for the radiation parameter on both of f 0 and g. The table describes the different effects to the permeability parameter M, vortex-viscosity parameter D and the radiation parameter R on the shear stress f00 , heat flux H 0 and rate of change of angular velocity g 0 at the surface. We observe that the magnitude of both of wall temperature gradient H 0 and rate of change g 0 are increase as the permeability parameter increase while the shear stress decrease. On the other hand, the wall temperature gradient increases as the radiation parameter decrease, while shear stress f00 and rate of change g 0 have no effects. As the vortex-viscosity parameter D increase, the shear stress f00 decrease and both of H 0 and g 0 decrease.

5. Concluding remarks In this paper, we studied the radiation effect on heat transfer of a micropolar fluid past on unmoving horizontal plate through a porous medium. Numerical solutions for the governing boundary layer equations for a range of values of permeability, vortex-viscosity and radiation parameters obtained by applying an efficient numerical technique based on the shooting method. The effects of these parameters examined on the velocity of fluid, temperature distribution and angular velocity of microstructures. The wall values of the velocity, angular velocity and temperature are tabulated. This information would be useful in order to evaluate the friction factor and heat transfer rate.

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Micropolar fluid display drag reduction when compared to Newtonian fluids. We observe that the angular velocity and temperature increase when permeability parameter increases while the velocity decrease. The velocity and angular velocity increase when the vortex-viscosity parameter increases. The temperature decreases with increase radiation parameter.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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