Heat transfer in a hydromagnetic flow of a micropolar fluid over a stretching surface with variable heat flux and generation.

Abdel-Rahman, Gamal M. ; sudais, Noura S. Al-

Introduction

The dynamic fluid over a stretching surface is important in many practical applications such as extrusion of plastic sheets, paper production, glass blowing, metal spinning and drawing plastic films. The theory is expected to provide a mathematical model for non-Newtonian fluid behavior, which can be used to analyze the behavior of exotic lubricants, the flow of colloidal suspensions or polymeric fluids, liquid crystals, additive suspensions, animal blood and turbulent shear flows. The quality of the final product depends on the rate of heat transfer on the stretching surface.

One of the important non-Newtonian fluids is the micropolar fluid in which the theory was first introduced by Eringin [1, 2], Lukaszewicz [3] and Eringin [4]. Elbashbeshy [5] studied the heat transfer over a stretching surface immersed in an incompressible Newtonian fluid when the surface is subjected to a variable heat flux. The purpose of the present study is extending the work of Elbashbeshy [5] to micropolar fluids, which display the effects of local rotary inertia and couple stresses, many researchers have considered various problems in micropolar fluids, see for example Seddeek[6] and Ishak et al. [7,8].

Very recently, Ishak et al. [9] studied the above mention heat transfer over a stretching surface with variable heat flux in micropolar fluids in the absence of the magnetic field (H = 0), the temperature buoyancy parameter (G = 0), and the local heat generation parameter (Q = 0). Hence, the objective of the present paper is to study the heat transfer in a hydromagnetic flow of a micropolar fluid over a stretching surface with variable heat flux, in presence of heat generation. Numerical results were presented for velocity, temperature and microrotation profiles within the boundary layer for different parameters of the problem as magnetic parameter, material parameter, heat generation parameter, Prandtl parameter, velocity exponent parameter, and heat flux exponent parameter and others. Also, the effects of the pertinent parameters on the skin friction and local Nusselt number are also discussed.

Mathematical Formulation

Consider a steady two-dimensional laminar flow of an incompressible micropolar fluid on a continuous, stretching surface with velocity [U.sub.w](x) = a[x.sup.m] and variable surface heat flux [q.sub.w] (x) = b[x.sup.n] (where a, b, m and n are constants) by moving the surface through electrically conducting fluid of electric conductivity [sigma]. The magnetic field B0 is applied perpendicular to the stretching sheet and the effect of the induced magnetic field is neglected since the magnetic Reynolds number is assumed to be small and have constant physical properties. The flow is assumed to be in the x--axis running along the continuous surface in the direction of the motion and the y--axis is taken to be normal to it. In our analysis, the motion is steady and under usual boundary layer approximations, the governing equations of continuity, momentum, angular momentum, conservation of micro-inertia and energy may be described by Eqs. (1)- (5) and boundary conditions (6).

[partial derivative]u/[partial derivative]x + [partial derivative]v [partial derivative]y = 0, (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[rho]j(u [partial derivative]N/[partial derivative]x + v [partial derivative]N/[partial derivative]y) = [partial derivative]/ [partial derivative]y ([gamma] [partial derivative]N/[partial derivative]y) -[kappa](2N+[partial derivative]u/[partial derivative]y), (3)

u [partial derivative]j/[partial derivative]x + v [partial derivative]j/[partial derivative]y = 0 (4)

u [partial derivative]T/[partial derivative]x + v [partial derivative]T/[partial derivative]y= [alpha] [[partial derivative].sup.2]T/[partial derivative][y.sup.2] + [Q.sub.0]/[rho][c.sub.p](T-[T.sub.[infinity]]). (5)

with the appropriate boundary condition;

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

In the above equations u , v and N are the fluid velocity component along and perpendicular to the surface of the flat plate and the components of microrotation or angular velocity whose a rotation is in the x - y plane, respectively.

We assume that j and [gamma] are functions of the coordinates x and y, and not constants as in many other papers.

Following Ahmadi [10], Kline [11] or Gorla [12], we assume that the spin gradient viscosity [gamma] is given by:

[gamma] = ([mu] + [kappa]/2) j = [mu](1 + K/2) j, (7)

Where K = [kappa] / [mu] denotes the dimensionless viscosity ratio and is called the material parameter. This assumption is invoked to allow the field of equations predicting the correct behavior in the limiting case when the microstructure effects become negligible and the total spin N reduces to the angular velocity (see Ahmadi [10]).

The continuity equation (1) is satisfied by introducing a stream function [psi] such that u = [partial derivative][psi]/[partial derivative]y and v = -[partial derivative][psi]/[partial derivative]x Further, we introduce the similarity transformation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Substituting expression (8) into the equations (2)-(5) we get the following ordinary differential equations:

(1 + K) [f.sup.///] + f [f.sup.//] - 2m/m+1 [f.sup./2] + [Kh.sup./] - [Hf.sup./] + G[theta] = 0 (9)

(1 + K/2) (i[h.sup./].sup./) + i(f[h.sup./] - 3m-1/m+1 [f.sup./]h) - K(2h + [f.sup.//] = 0 (10)

2(1-m)i [f.sup./] - (m+1)f [i.sup./] = 0 (11)

1/[P.sub.r] [[tetha].sup./] + f [[theta].sup./] - 2n-m+1/m+1 [f.sup./] [theta] + Q[theta] = 0 (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where primes denote partial differentiation with respect to the variable n and the dimensionless parameters are defined as:

H = 2[sigma][B.sup.2.sub.0]x/[rho][U.sub.w](m+1) (Magnetic field parameter),

[G.sub.r] = 2g[beta][q.sub.w][x.sup.2]/[kappa][U.sup.2.sub.w] [square root of (2/m+1)] (Local temperature Grashof number),

[R.sub.e] = [U.sub.w]x/v (Local Reynolds number),

G = [G.sub.r]/[summation over ([R.sub.e]) (Temperature buoyancy parameter),

[P.sub.r] = (m + 1)v/[alpha] (prandtl number),

And

Q = 2[Q.sub.0]x/[rho][c.sub.p](m+1)[U.sub.w] (Local heat generation parameter),

If we integrate Eq. (11) with Eq. (13), we get:

i = [Af.sup.2(1-m)/(m+1) (14)

Where A is a non-dimensional constant of integration.

We notice that when K = 0 (Newtonian fluids), H = 0, G = 0 and Q = 0 the problem is reduced to those considered by Elbashbeshy [5], for an impermeable surface. The solution of Eqs (9)-(12) is subjected to the boundary conditions (13) in the absence of all the magnetic field (H = 0), the temperature buoyancy parameter (G = 0) and the local heat generation parameter (Q = 0) can be found in Pop et al. [9] through a stationary fluid.

Skin-friction coefficient and Nusselt number

The parameters of engineering interest for the present problem are the local skin-friction coefficient and the local Nusselt number which indicate physically wall shear stress and rate of heat transfer respectively. The wall skin-friction is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Hence the skin-friction coefficient can be written as:

[C.sub.f] = 2[[tau].sub.w] / [rho][U.sup.2.sub.w] = 2/[square root of ([R.sub.e])] [square root of (m+1/2](1+k/2)[f.sup.// (0))], (16)

Now the local surface heat flux ([q.sub.w]) through the plate is given by:

[q.sub.w](x) = -k ([partial derivative]T/[partial derivative]y).sub.y=0,

Hence the Nusselt number ([N.sub.u]) is:

[N.sub.u] = x[q.sub.w]/k[square root of (2[upsilon]x/(m+1)[U.sub.w])] = -[square root of [R.sub.e]] [square root of (m+1/2 [[theta].sup./])] (17)

Numerical Results and discussion

The system of non-linear ordinary differential equations (9)-(12) together with the boundary condition (13) are locally similar and solved numerically using Shooting method. In order to get an insight into the physical situation of the problem, we have computed numerical values of the velocity, temperature and microrotation. The velocity, temperature and microrotation are found for the different values of various parameters occurring in the problem (K, m, H, G, [P.sub.r], n and Q). With the above-mentioned flow parameters, the results are displayed in Figs.1-7, for the velocity, temperature and microrotation profiles.

We observe that both velocity and microrotation profiles increase with the increase of the material parameter(K), the velocity exponent parameter (m) and the temperature buoyancy parameter (G), while the temperature decreases with the increase of (K), (m) and (G) as shown in Figs. 1, 2 and 4, respectively.

The effects of H on the velocity and microrotation profiles are shown in Fig. 3(a) and 3(c), respectively. We see that both velocity and microrotation profiles decrease with the increase of magnetic field parameter H. While the effects of H on the temperature profile are displayed in Figs. 3(b). We observe that the temperature increases with the increase of H.

We also find that the temperature increases with the increase of Prandtl number [P.sub.r], the heat generation parameter Q respectively, and with the decrease of the heat flux exponent parameter n as shown in Figs. 5, 6 and 7, respectively.

The numerical values of the Skin-friction and the Nusselt number are given in Table (1). It may be noted that with an increase in K, H and G the Skin-friction increases, while we observe that the Skin-friction coefficient and the Nusselt number decrease as the velocity exponent parameter m decreases.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

Conclusions

In this paper we have studied numerically the heat transfer in a hydromagnetic flow of a micropolar fluid over a stretching surface with variable heat flux, in presence of heat generation. The governing partial differential equations are transformed into a system of ordinary differential equations using similarity variability and then solved numerically by using shooting method. We discussed the effects of the magnetic parameter H, the material parameter K, the heat generation parameter Q, the Prandtl parameter [P.sub.r], the velocity exponent parameter m and heat flux exponent parameter n on the velocity, temperature and microrotation profiles. While the values of the skin friction coefficient and the local Nusselt number were presented in tables, for various values of the pertinent parameters. From the present study we have found that:

* Both the velocity and microrotation profiles decrease whereas the temperature profile increases with the increase of magnetic field.

* The velocity and microrotation profiles increase whereas the temperature profile decreases with the increase of the temperature buoyancy parameter.

* Both the skin-friction coefficient [C.sub.f] and the Nusselt number [N.sub.u] decrease with the decrease of the velocity exponent parameter.

* The skin-friction coefficient [C.sub.f] increases with the increase of the magnetic parameter, whereas the skin-friction coefficient [C.sub.f] decreases with the increase of the temperature buoyancy parameter.

Symbols used

T Fluid temperature

g Acceleration due to gravity

j Micro-inertia density

[C.sub.p] Specific heat at constant pressure

[Q.sub.0] Heat generation constant

[T.sub.[infinity]] Fluid temperature in the free stream

Greek symbols

[beta] Coefficient of volume expansion

[gamma] Spin-gradient viscosity

[rho] Density

[mu] Dynamic viscosity

[kappa] Gyro-viscosity (or vortex viscosity)

v Kinematic viscosity

[alpha] Thermal diffusivity

References

[1] A.C. Eringen, J. Math. Mech. , 16 (1969)1.

[2] A.C. Eringen, J. Math. Anal. Appl., 38 (1972) 480.

[3] G. Lukaszewicz, Micropolar fluids: theory and application. Basel: Birkhauser; 1973.

[4] A.C. Eringen, Microcontinuum field theories. II: fluent media. New York: Springer; 2001.

[5] E.M.A. Elbashbeshy, J. Phys. D: Appl. Phys., 31(1998)1951.

[6] M.A. Seddeek, Phys. Lett. A, 306(2003)255.

[7] A. Ishak, R. Nazar, I. Pop, Int. J. Eng. Sci.,44(2006)1225.

[8] A. Ishak, R. Nazar, I. Pop, Fluid Dyn. Res., 38(2006)489.

[9] A. Ishak, R. Nazar, I. Pop, Phys. Lett. A, 372(2008)559.

[10] G. Ahmadi, Int. J. Eng. Sci., 14(1976)639.

[11] K.A. Kline, Int. J. Eng. Sci., 15(1977)131.

[12] R.S.R. Gorla, Int. J. Eng. Sci., 26(1988)385.

Gamal M. Abdel-Rahman (1) and Noura S. Al-sudais (2)

(1) Department of Mathematics, Faculty of Science, Benha University, 13518 Benha, Egypt (Princess Norah Bint Abdelrahman University, Riyadh) (2) Department of Mathematics, Faculty of Science, Princess Norah Bint Abdelrahman University, Riyadh, KSA

Table 1: Numerical values of -1/2 [([R.sub.e]).sup.1/2] [C.sub.f] and
[N.sub.u][([R.sub.e]).sup.1/2] at the plate surface with K, m, Hand G

K m H G -1/2 [([R.sub.e]) [N.sub.u] [([R.sub.e])
 .sup.1/2] [C.sub.f] .sup.1/2]

1.0 1.0 1.0 0.0 1.73205
2.0 2.00004
3.0 2.44981
1.0 1.0 1.0 0.0 1.73205 1.00000
 1/3 1.29436 0.8165
 0.0 1.00748 0.70711
1.0 1.0 0.0 0.0 1.22554
 3.0 2.44949
 5.0 3.00000
1.0 1.0 1.0 -0.1 1.50573
 0.0 1.73205
 0.1 1.66154