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FLOW AND HEAT TRANSFER OF A MICROPOLAR FLUID IN AN AXISYMMETRIC STAGNATION FLOW ON A CYLINDER I. A. HASSANIEN and A. A. SALAMA Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt (Received 3 April 1995)

Abstract-An analysis is presented for the steady state boundary layer flow and heat transfer of a micropolar fluid in the vicinity of an axisymmetric stagnation point on a cylinder. The governing momentum, angular momentum and heat transfer equations have been solved numerically using a procedure based on an expansion of Chebyshev polynomials approximation. Missing values of the velocity, microrotation function and heat transfer are illustrated for a range of values of material parameters and Reynolds numbers. The results have been compared with the results of the corresponding flow of a Newtonian fluid. It is observed that micropolar fluids display drag reduction when compared to Newtonian fluids. Boundary layer

Axisymmetric stagnation flow

Micropolar fluid

Chebyshev approximation

NOMENCLATURE A= a = B= f = g = h= P = j = R, = r= u= v= w= z= r) = p = v= s= A, 1 =

Constant used in equation (6) (A = Urn/a for cylinder) Radius of cylinder Curvature parameter Dimensionless stream function Dimensionless microrotation Dimensionless azimuthal velocity Pressure Microinertia per unit mass Reynolds number (R, = Aa*2v) Coordinate normal to cylindrical surface Velocity component in r-direction Velocity component in angular direction Velocity component in z-direction Coordinate parallel to wall Dimensionless coordinate Dynamic viscosity Kinematic viscosity Fluid density Dimensionless material properties

Subscripts o = Conditions at wall co = Conditions far away from wall

INTRODUCTION

In recent years, the dynamics of micropolar fluids, originated from the theory of Eringen [l, 23 has been a popular area of research. The equations governing the flow of a micropolar fluid involve a microrotation vector and a gyration parameter in addition to the classical velocity vector field. This theory may be applied to explain the flow of colloidal solutions, liquid crystals, fluid with additives, suspension solutions, animal blood, etc. The boundary layer concept in such fluids was first studied by Peddieson and McNitt [3] and Willson [4]. While extending the theory of micropolar 301

302

HASSANIEN and SALAMA:

HEAT TRANSFER OF A MICROF’OLAR FLUID

fluids, Eringen [5] developed the theory of thermomicro fluids, taking into account the effect of microelements of fluids on both the kinematics and conduction of heat. The boundary layer flow of a micropolar fluid over a semi-infinite plate was examined by Ahmadi [6]. The problem of two-dimensional stagnation flow of the axisymmetric or unsymmetric case of a Newtonian fluid has received considerable attention from many investigators [7-131. Recently, Hassanien and Gorla [14] studied the boundary layer flow of a micropolar fluid near the stagnation point on a horizontal cylinder. In the present study, we have analysed the steady state heat transfer of an axisymmetric stagnation flow of a micropolar fluid on an infinite circular cylinder. The flow is assumed to be laminar and incompressible. A procedure based on Chebyshev approximations is used for solving the non-linear differential equations governing the flow and heat transfer. Our method defines the unknown wall shear and couple stresses and the rate of heat transfer at the wall directly without the need for any correction interpolation method. The temperature field is obtained for the isothermal wall condition for a range of material parameters and Reynolds numbers. Numerical results are presented, discussed and compared with known results for a Newtonian fluid. Engineering applications of the problem are found in certain industrial cooling processes.

GOVERNING

EQUATIONS

Let us consider a steady, laminar, incompressible flow at an axisymmetric stagnation point on an infinite circular cylinder. A model of the flow with the coordinate system is shown in Fig. 1. The flow is axisymmetric about the z-axis and also symmetric to the z = 0 plane. The stagnation line is at z = 0 and r = a. The cylindrical co-ordinates are (r, 8, z) and the corresponding velocity components are given by (u, v, w). The temperature of the free stream fluid is taken as T,. The equations expressing conservation of mass, momentum, angular momentum and energy with the boundary layer approximation are given by: Mass:

&$rtv)+&)=O Momentum:

(2)

(3)

Fig. 1. Coordinate system and flow development.

HASSANIEN and SALAMA:

HEAT TRANSFER OF A MICROF’OLAR FLUID

303

(4) Angular momentum:

Energy:

(6) We assume that the flow approaches the potential flow as r + co. The boundary conditions are as follows: whenr=a;

u=O,

0~0,

w=O,

T=

Aa

when r--,co;

v =-,

w=2Az,

r

T,,

T-T,.

(7)

Proceeding with the analysis, we define: r 2 3

“=a

0 u = - Aaq -“‘.f(q),

v = - Au~-“~.~(~), w = 2Az *f’(?j),

e =-,

T,-

T

Tw - Tm

&lr P’

If we substitute the definitions (8) into equations (l)-(6), equation (1) is satisfied identically, and equations (2)-(6), after simplification, become: (1 + A)(q_P”+f”) +

$ (g + t/g’) + Ml

+_#-” -.f”)

= 0

(9)

0 qh”+R,jh’=O Q’g”

P=P+P

+ 2qg’) - A.B(qf

(10)

N+ w/2) -

SvAf” 8vrA’p - 4A2(f ‘)2+ .z + a4

Pttf ‘g - 2tlfg’ -fg)

I[ -

= 0

(11)

A2 a”f 2 -+4vAf’-A2a4

tte n+ [i + (&Pr)f]O’ = 0.

(13)

304

HASSANIEN and SALAMA:

HEAT TRANSFER OF A MICROPOLAR FLUID

In the above equations, a prime denotes differentiation with respect to q only. The transformed boundary conditions are given by: f(1) =f’(l)

= 0, f’(co) = 1,

h(1) = 0,

h(co) = 1,

g(1) = - 2f”(l),

g(a)

= 0,

and e(1) = 1, 8(co) = 0.

(14)

The wall stress may be written as:

?v=

1 [(p+k$+kN

.

(15)

r-l?

The local friction coefficient is given by: +[4(1 + A)f”(l) + As(l)], e

(16)

The wall couple stress may be written as (17) The local wall heat flux can be written by Fourier’s law as

(18) The local heat transfer coefficient is given by

h(z)=-fL=

Tw- T.m

-$8’(l).

(19)

The local Nusselt number then becomes Nu,=-

O)*z k

=

_2

s “2

0 4

8 ‘(1)

(20)

where

We notice that the above equations (9)-(13) reduce to the corresponding Newtonian fluid [13] for vanishing microrotation and k = 0. NUMERICAL

equation

for a

SOLUTION

Consider the boundary layer equations (9)-(13) together with the boundary conditions (14). The domainis 1

+[(G$(,+

l)+ l]~~)+(~),[l~~+,.-,]=o,

(21)

HASSANIEN

and SALAMA:

HEAT TRANSFER

OF A MICROPOLAR

FLUID

305

(22)

~[(~)(x+1)+1]{[(~)(*+1)+1lg.+2(E?-+~

[email protected]++

1)+ l][jtf+(vy;]}

-(3(5$){2[(3++

I)+

Ij,fk

(23)

(24) together with the boundary conditions f(-l)=f’(-l)=O,

f’(l)=?,

h(-l)=O, g(-I)=

-2 (

&

Q-l)=

h(l)=

>

1,

*f”(_ l),

g(1) = 0,

1, f?(l)=O,

(25)

where the prime denotes differentiation with respect to X. Our technique is accomplished by starting with the Chebyshev approximation for the highest order derivatives, f”‘, h”, g”, and 8 Mand generating approximations to the lower order derivatives f”, f’, f, h’, h, g’, g, 0 ’ and 8 through integration of the approximation of the highest order derivatives. Let F(x) =f”‘(x), H(x) = h”(x), G(x) = g”(x) and, 0 = 8 “(x), then using the boundary conditions (25), the lower order derivatives can be approximated in the following way:

(264 (26W

(W

e,=i

I-0

dB,q+d:,

8’i [email protected],+del;

WV

HASSANIEN and SALAMA:

306

HEAT TRANSFER OF A MICROPOLAR FLUID

for all i = O(l)N, where

4

’

dh_(%+

, --9

l)

2

&I = f,

df=(l-xi) I -9

&$ = b, - fb$,

2

&I = - f,

(27)

where b!!) = (% - %)* bii bf’ = (x, - xi)b,; rl 2

i = O(l)N,

and b, are the elements of the matrix B, as given in Ref. [lo]. By using equation (26), equations (21x24) are transformed non-linear equations in the highest derivatives

-(+)(~){2[(~)~x,+

I)+

l](jollj4+A)

into the following system of

HASSANIEN and SALAMA:

HEAT TRANSFER OF A MICROPOLAR

FLUID

307

Table 1. Values of R; @f”( 1) and R; ID8 ‘( 1) for various values of R, with Pr = 1.O in the case of a Newtonian fluid Present results

R, 0.01 0.10 0.20 1.00 10.00 100.0 1000.0

R; “*f”( 1) Ref. [13]

3.155141 1.946369 1.757651 1.484185 1.316429 1.259641 1.232588

Exact [ 111

3.15514 1.946369

1.7577 1.484185 1.316427 1.259642 1.232588

1.7577 1.484185 1.31643

R;“*ll’(l) Present results

Ref. [13]

2.392574 1.203489

2.229295 1.200518

0.797001 0.646798 0.595212 0.578357

0.796878 0.646796 0.595211 0.570400

Table 2. Values for f”( I), h’( 1), g’( 1) and 6 ‘( 1) for various values of A, I and R.whenPr=landB=O.l A

1

0.5

0.5

5.0

0.5

5.0

5.0

R. 0.01 0.10 1.00 10.00 100.00 0.01 0.10 1.00 10.00 100.00 0.01 0.10 1.00 10.00 100.00

f”(1)

h’(l)

0.3173 0.5836 1.3778 3.8208 11.5045 0.2895 0.4721 1.0022 2.6302 7.6622 0.2935 0.4332 0.8812 2.2853 6.7128

0.0273 0.0752 0.3534 1.4789 5.2511 0.1014 0.1156 0.3072 1.3107 4.6541 0.1031 0.0748 0.3038 1.2981 4.5600

g’(1) 0.6348 1.4118 5.1370 30.9106 253.2580 0.985 1 1.0765 3.4622 19.3275 150.0580 0.9850 0.9125 2.3026 9.2616 60.9468

O’(l) -

0.3420 0.6675 1.5902 4.4204 - 13.3210 - 0.0333 - 0.3024 - 1.4592 - 3.9534 11.7391 - 0.3337 - 0.6208 - 1.4205 - 3.8356 - 11.3979

(274

This system is solved using Newton’s iteration. RESULTS

AND DISCUSSION

The considered micropolar fluid flow and heat transfer problem has been solved numerically for a range of values of A, rl, B and R, . The values of the nondimensional parameters A, 3,and B chosen in the present investigation are based on the inequalities p > 0, k 3 0, y 3 0 and j > 0 for the Table 3. Values for f”( I), h’(l), g’( 1) and 0 ‘(1) for various values of A, 1 and R, when Pr = 1 and B = 10.0 A 0.5

A 0.5

R, 0.01 0.10 1.00 10.00 100.00

f”(1) 0.4412 0.5964 1.3639 3.8132 11.5020

h’(1) 0.1014 0.1166 0.3446 1.4787 5.2543

g’(1) 0.8992 1.3656 4.7133 30.1340 252.4160

O’(l) -

0.3419 0.6664 1.5874 4.4188 - 13.3204

HASSANIEN and SALAMA:

308

HEAT TRANSFER OF A MICROPOLAR

FLUID

0.8

0.6

I f 0.4

2 Re=lO.O 3 Re=l.O 4 Re=O.l

0.2

5 Re=O.Ol 0 1

1.5

2

2.5

3

3.5

q

4

4.5

5

5.5

6

Fig. 2. Stream velocity distribution for A = 1 = 0.5 and B = 0.1.

micropolar fluid (see Eringen [2]). Double precision arithmetic was used in all the computations. For each Reynolds number, a value of qrn was chosen such that the obtained numerical solutions exhibited no change up to the sixth significant figure when computations were repeated with higher values of qa,. In many practical applications, it is usually the surface characteristics, such as wall shear stress, wall couple stress and rate of wall heat transfer, that are of importance. These quantities may be directly evaluated using the data in Tables l-3. Figures 2-5 illustrate the distribution of dimensionless streamwise velocity, azimuthal velocity, angular velocity and temperature distribution within the boundary layer for A = I = 0.5, B = 0.1 and Pr = 1.O. The Reynolds number was varied from 0.01 to 100. We notice that the boundary layer thickness decreases with increasing values of R,. As would be expected, the temperature inside the boundary layer decreases with the increase in micropolar effects, which results in cooling of the fluid.

2 Re=lO.O 3 Re=l.O 4 Re=O.l 5 Re=O.Ol

1

1.5

2

fl

2.5

3

Fig. 3. Azimuthal velocity distribution for A = 1 = 0.5 and B = 0.1.

3.5

HASSANIEN and SALAMA:

HEAT TRANSFER OF A MICROPOLAR

FLUID

309

-2.5 g -3

-A

-4.5 -5 1

1.5

2

2.5

3

rl

3.5

4

4.5

5

5.5

6

Fig. 4. Angular velocity distribution for A = rZ= 0.5 and B = 0.1.

In order to assess the accuracy of our method, we have compared the present results for skin friction coefficient, as well as Nusselt number, with the corresponding results for a Newtonian fluid [l 1, 131. Values of R, "*f"(l) and R,‘/*fJ ‘(1) versus R, are shown in Table 1. Moreover, we observe that the values of R, “*f”( 1) and R, ‘/*8 ‘(1) for Re = 100 agree with the two-dimensional stagnation point. Table 2 presents numerical results for the missing wall values of the velocity and temperature functions. The micropolar properties A, 1, and B, as well as the Reynolds number R,, were chosen as prescribable parameters. It is noticed that, as the value of Reynolds number, R,, increases, the wall shear and couple stresses and rate of heat transfer increase. Table 3 shows the same results for B = 10.0. The results indicate that the curvature parameter B has no significant influence on the wall shear stress, the wall couple stress and the rate of heat 1 Re=lOO.O 2 Re=lO.O 3 Re=l.O 4 Re=O.l 5 Re=O.Ol

1

1.5

2

2.5

3

3.5 q

4

4.5

5

Fig. 5. Temperature distribution for A = L = 0.5, B = 0.1 and Pr = 1.0.

5.5

6

HASSANJEN and SALAMA:

310

HEAT TRANSFER OF A MICROPGLAR

FLUID

transfer at the wall. In comparison to a Newtonian fluid, the micropolar fluid displays a reduction in drag and heat transfer. The friction factor is higher for a Newtonian fluid when compared to micropolar fluids. REFERENCES 1. 2. 3. 4.

C. Eringen, ht. J. Engng.Sci. 2, 205 (1964). C. Eringen, J. Math. Mech. 16, 1 (1966). Peddieson and R. P. McNitt, Recent Ado. Engng. Sci. 5, 405 (1970). J. Willson, Proc. Comb. Phil. Sot. 67, 469 (1970). C. Eringen, J. Math. Anal. Appl. 38, 480 (1972). G. Ahmadi, Int. J. Engng. Sci. 16, 639 (1976). K. Hiemenz, Danglers J. 32, 321 (1911). F. Homann, Z. Angew. Math. Mech. 16, 153 (1936). L. Howarth, Phi. Magazine 42, 1433 (1951). A. Davey, J. Fhdd Mech. 10, 593 (1961). C. Y. Wang, Q. Appl. Math. 32, 207 (1974). G. S. Guram and A. C. Smith, Utilitas Mathematics 9, 147 (1976). R. S. R. Gorla, Appl. Sci. Res. 32, 541 (1976). I. A. Hassanien and R. S. R. Gorla, Int. J. Engng. Sci. 28, 15 (1990). S. E. El-Gendi, Computer J. 12, 282 (1969).

A. A. A. A. 5. A.

6.

7. 8. 9. 10. 11. 12. 13. 14.

15.

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