Flow of a micropolar fluid past a continuously moving plate by the presence of radiation

PERGAMON

International

Journal

of Heat and Mass Transfer

41 (1998)

2865-2866

Technical Note Flow of a micropolar fluid past a continuously by the presence of radiation

moving plate

A. Raptis* University

qf loannina.

Department

qf Mathematics,

1. Introduction Eringen [I] first derived the constitutive equations for fluids with micro-structures. The boundary layer flow of such a micropolar fluid past a semi-infinite plate has been studied by Peddieson and McNitt [2], whereas a similarity solution for boundary layer flow near a stagnation point was presented by Ebert [3]. On taking into account the gyration vector normal to the xy-plane and the micro-inertia effects, the boundary layer Row of micropolar fluids past a semi-inlinite plate was studied by Ahmadi [4]. Willson [5] obtained the solution in the stagnation region of the micropolar fluid. The flow of a micropolar fluid past a wedge was studied by Nath [6]. Takhar and Soundalgekar [7] studied the heat transfer aspect of the flow of a micropolar fluid past a semi-infinite plate. Recently Perdikis and Raptis [S] studied the heat of a micropolar fluid by the presence of radiation. In all these paper studies, the plate was assumed to be stationary and the micropolar fluid moved over this plate. Another situation commonly observed is the flow of a micropolar stationary fluid past a continuously moving plate. It is now proposed to study the flow of a micropolar fluid past a moving plate by the presence of radiation.

2. Analysis We consider a steady two dimensional flow of a micropolar fluid past a continuously moving plate with a constant velocity. The origin is located at the spot through which the plate is drawn in the fluid medium, the x-axis is chosen along the plate and yaxis is taken normal to it. The fluid is considered to be a gray, absorbing-emitting radiation but non-scattering medium and the Rosseland approximation is used to describe the radiative heat flux in the energy equation. The radiative heat flux in the .x-direction is considered negligible in comparison to the ,y-direction.

Ioannina

Then under the usual boundary layer approximation, the flow and heat transfer by the presence of radiation are governed by the following equations : (1)

(3)

and the boundary

conditions

are given by

u = u,,

T=

o=O

u-0,

Ii = 0, T+T,,

author

0017~9310;98 $19.00 a 1998 Elsevier PII:SOO17--9310(98)0000664

Ltd. All rights

reserved

as

at

?‘-t

j’ = 0

XL.

4a* (‘T4 q, = -p-3;

(6)

with u* the Stefan-Boltzmann constant and k* the mean absorption coefficient. We assume that the temperature differences within the flow are sufficiently small such that p may be expressed as a linear function of temperature. This is accomplished by expanding Y in a Taylor series about T, and neglecting higher-order terms, thus

By using Science

17-o

T,,

(5) Here U. 11, are the velocity components along X, y coordinates respectively, Y = (r~+ .S)/p is the apparent kinematic viscosity, p the coefficient of dynamic viscosity, S a constant characteristic of the fluid, p the density, c the microrotation component, K, = S/p (K, > 0) the coupling constant, G, ( > 0) the microrotation constant, T the temperature of the fluid, L’,, the specific heat at constant pressure, k the thermal conductivity, yr the radiative heat flux, (lo the uniform velocity oftheplate, T,, the temperature of the plate and T, the temperature of the fluid far away from the plate. By using Rosselant approximation we have

T” 2 4T: *Corresponding

451 IO, Greecr

T-3T4,. (6) and (7) equation

(7) (4) gives

2866 dT dT u-++tr-=--+I’ 2.Y aY

16a*i-‘, + __3pc$i*

On introducing

into equations

the following

(I).

f’“+ff”+Kg’

transformations

(2), (3) and (8) we get

= 0

G,q”-2(2,q+I”)

= 0

with the corresponding f’(0)

= 0.

boundary

= 1.

O(K)

conditions

H(0) = 1.

= 0.

g(7_)

4

6

n

profiles.

(II1

g(O) = 0.

are shown in Fig. I when P = 7, E = 0.02, G‘ = 2 and K = 0.1. We observe from this figure that an mcrease in radiation parameter leads to a decrease of the temperature.

= 0. Acknowledgement

Here E = -L$r P SL P = “2 K = :

Fig. 1. Temperature

(IO)

(3!V+4)H”+3NP/H’+3IVP~~~

f’(s)

2

(9)

= 0

f(0) = 0.

c?‘T --iv’

I

(Eckert

number)

(Prandtl

number)

(couphng

constant

G‘=- G 1C’,, I’Y

(mlcrorotation

,y= lc!c

(radiation

I would like to thank Assoc. Professor in numerical calculations.

S. Galanis

for his help

References

parameter) parameter)

parameter).

40* T; The system of equations (9)-(11) under the boundary conditions (12) is solved numerically and the temperature profiles

[I] Ermgen AC. In1 J Engng Sci lY64;2:205. [2] Peddieson J, McNitt RPM. Recent Adv Engng 1970;5:405. [3] Ebert F. Chem Eng J 197X5:85. [4] Ahmadi A. Int J Engng Sci 1(976;14:639. [5] Wilson AJ. Camb Phil Sot 1970:67:969. [h] Nath G. Reol Acta lY75;14:850. [7] Takhar HS. Soudalgekar VM. Reol Acta 1980;19:525. [8] Perdikis C, Raptis A. Heat Mass Transfer 1996:31:3X1.

Sci