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Flow of micropolar fluid past a continuously moving plate
hr. J. Engng Sri. Vol. 21, No. 8, pp. 961-965, Printed in Great Britain.
002~7225/83/080961--05$03.00/O 0 1983 Pergamon Press Ltd.
1983
FLOW OF MICROPOLAR FLUID PAST A CONTINUOUSLY MOVING PLATE V. M. SOUNDALGEKAR Gulf Polytechnic, PO Box 32038, Isa Town, Bahrain, Middle East and
Simon engineering Laborato~es,
H. S. TAKHAR University of Manchester, Manchester Ml3 9PL, England
Abstract-A similarity analysis of the flow and heat transfer past a continuously moving semi-infinite plate in a micropolar fluid is presented. The velocity, micro-rotation distribution and the temperature profiles are shown on graphs and the numerical values of the skin friction and the rate of heat transfer are entered in tables. The effects of K (coupling parameter) and G (micro-rotation parameter) are discussed. 1. INTRODUCTION ERINGEN[~] first derived the. constitutive equations for fluids with micro-structures. Simple problems on the flow of such a Auid were studied by a number of researchers and a review of this work was given by Ariman, Turk and Sylvester [2]. The boundary layer flow of such a micropolar fluid past a semi-infinite plate has been studied by Peddieson and McNitt[3] by using a finite difference method, whereas a similarity solution for boundary layer flow near a stagnation point was presented by Ebert[4]. On taking into account the gyration vector normal to the xy-plane and the micro-inertia effects, the boundary layer flow of micropolar fluids past a semi-in~nite plate was studied by Ahmadi [5]. Willson [6] obtained the solution in the stagnation region of the micropolar fluid by the Ka~an-Pohlhausen method. The flow of micropolar fluid past a wedge was studied by Nath [7]. In Refs. [6,7], it was observed that in the 2-dimensional boundary layer flow of micropolar fluids, the microinertia effects can be neglected. Recently Takhar and Soundalgekar[8,9], studied the heat transfer aspect of the flow of micropolar fluids past a semi-infinite plate and a wedge respectively. In all these studies, the body was assumed to be stationa~ and the micro-polar fluid moved past this body. Another situation commonly observed in industrial problems is the flow of a stationary fluid past a continuously moving plate. This problem is different from that of the Blasius[ lo] problem because here the boundary layer thickness decreases with increasing x, the coordinate taken along the plate in the direction of the moving plate, whereas in Blasius problem the boundary layer thickness increases with increasing X. Hence, Sakiadis[l 11,first studied the flow of a Newtonian fluid past a continuously moving plate, whereas its heat transfer aspect was studied by Tsou, Sparrow and Goldstein[l2]. It is now proposed to study the flow of a micropolar fluid past a continuously moving plate and the corresponding heat transfer. In Section 2, the mathematical analysis is presented and in Section 4 the conclusions are set out.
2. MATHEMATICAL
ANALYSIS
Consider a steady, 2-dimensional flow of a micropolar fluid past a continuously moving flat plate, with a constant velocity in a micropolar fluid medium at rest. The origin is located at the spot through which the plate is drawn in the fluid medium and the y-axis is taken normal to it. Then, under the usual boundary layer approximations, the flow and heat transfer in micropolar fluid are governed by the following equations:
962
V. M. SOUNDALGEKAR
G
and H. S, TAKHAR
az,,_26-!!!!_o
1ay*
(2)
aY -
(3)
dT ar a9 v au u-+u-_=~,+_---2 ax ay ay c,0ay
(4)
and the boundary conditions are given by u =
u,,
v
u = 0,
=o,
T= T,,
T=T,,
a=0
(r=O,
at
as
y=O
y-xo.
(5)
Here v = (U + S)/p is the apparent kinematic viscosity, p the coefficient of dynamic viscosity, S a constant characteristic of the fluid, CJ the micro-rotation component, K, = S/p, (K, > 0) the coupling constant, G, (> 0) the microrotation constant, p the density, U,v, the velocity components along x, y coordinates, and U, is the uniform velocity of the plate. Also, T is the temperature of the fluid in the boundary layer, T, the wall temperature of the plate and T, is the temperature of the fluid far away from the plate, K the thermal diffusi~ty, C, the specific heat at constant pressure, and the last term in eqn (4) represents the viscous dissipative heat which is to be taken into account when the Prandtl number of the fluid is large or (T,- T,)is small. On introducing the following transformation
$ = in eqns (l)-(4),
~2VU~X)l’~(~),
t?=T-T,/T,--T,
(6)
we get f”+fl+Kg’=O
(7)
Gg"- 2(2g +s”) = 0
(8)
fY+PrJlf'+PrEcf=*=O and the corresponding
(9)
boundary conditions are
f(O)=O,
f’(O)= 1,
_fYcQ)= 0,
6(O) = 1,
&co) = 0,
g(0) = 0
g(c0) = 0.
(10)
Here Pr = V/K is the Prandtl number, K = KJv, G = G~U~iv-x and E,= U$C,(T,- T,) where K,G,E,are the coupling constant parameter, the micro-rotation parameter and the Eckert number, respectively. These are non-linear ordinary differential equations which are to be solved numerically using two-point boundary value methods on a high speed computer.
3. NUMERICAL
ANALYSIS
The numerical procedure used here solves the two-point boundary-value problems for a system of N ordinary differential equations in the range (X, X,). The system is written
Flow of micropolar fluid past a continuously moving plate
963
as
and the derivatives f;: are evahrated by a procedure that evaluates the derivatives of y, at a general point X. Initially N boundary values of the variable yi must be .Yl,YZ,-.., specified, some of which will be specified at X and some at X, aThe remaining N boundary values are guessed and the procedure corrects them by a form of Newton iteration. Starting from the known and guessed values of y, at X, the procedure integrates the equations forward to a matching point R, using Merson’s method. Similarly starting from Xi it integrates backwards to R. The difference between the forward and backward values of yi at R should be zero for a true solution. The procedure uses a generabzed Newton method to reduce these differences to zero, by calculating corrections to the estimated boundary values. This process is repeated iteratively until convergence is obtained to a specified accuracy. The tests for convergence, and the perturbation of the boundary conditions are carried out in a mixed form, e.g. if the error estimate for y, is ERROR,, we test whether, ABS(ERRORJ -CERRUR, x (I+ ABS yJ. Essentially, this makes the test absolute for yj 6 I and relative for yi B t. Note that convergence is not guaranteed, particularly from a poor starting approximation. A serious difficulty which may arise with boundary value problems is inherent instability. In such cases, integration from one or both ends of the range will produce rapidly increasing solutions which may accasionally lead to overAow before the matching point is reached. The position of the matching point R can be varied to improve the situation; if the solution increases rapidly for forward (or backward) integrations R should be taken at X (or X1); if it increases in both directions, R should be taken between X and X,. If the matching point R is at one of the end points X or Xi, there is no need to estimate the unknown boundary values accurately at this point, as they are not required for integration. Another difficulty which often arises is the case when one end of the range, say Xi, is at infimty. The end-point is approximated by taking finite values for X,, which is obtained by estimating where the solution will reach its asymptotic state. The computing time for integrating the differential equations can sometimes depend critically on the quality of the initiaf guesses of the unknown boundary conditions, the iocations of the matching point and the infinite end-point. The numerical results so obtained are plotted for velocity and micro-rotation on Fig. 1 for different values of K and G. We observe from this figure that the velocity increases with increasing K, whereas the micro-rotation distribution increases with increasing K and decreases with increasing G. In Fig. 2 the temperature profiles are shown. We observe from this figure that an increase in K or G leads to a decrease in the temperatures of the micropolar fluid.
Fig. 1. Velocity and microrotation
distribution.
964
V. M. SOUNDALGEKAR
and H. S. TAKHAR
Fig. 2. Temperature pr&fes,
Pr = 7, E =0.02.
From the velocity field we can study the shear stress, r, as given by
and in view of (6), the skin-friction coefficient C7,is given by
where Re = Uxjv is the Reynolds number. This shows that the skin-friction coefficient does not contain the micro-rotation term in an explicit way. The numerical values off’(O) are calculated and they are entered in Table I. We conclude from this table that an increase in K leads to a decrease in the skin-friction. From the temperature field, we can now study the rate of heat transfer. It is given by 4 = _k!?
(14)
a.Y ,S=O’ In view of eqn (6), eqn (14) reduced to
The numerical values of Q’(O) are entered in Table 2. We observe from Table 2 that an increase in K leads to a decrease in the value of the rate of heat transfer f - P(O)), whereas an increase in G leads to an increase in the value of rate of heat transfer { -S’(O)),
4. CONCLUSIONS I.
The velocity and the micro-rotation distribution increases with increasing K 2. An increase in K or G leads to a decrease in the value of the temperature. 3. The skin-friction decreases with increasing 1% 4. The rate of heat transfer decreases with increasing K and increases with increasing G.
Tabte 2. Vahes __--. K\,G
of {-
2 ___---
0.1 i ,944 0.5 1.801 ____~__-._-.---
#‘@I)) _-.4 I.946
Flow of micropolar
fluid past a continuously
moving
plate
965
REFERENCES [l] [2] [3] [4] [5] [6] [7] [8] [9]
A. C. ERINGEN, Znt. J. Engng Sci. 2, 205 (1964). T. ARIMAN, M. A. TURK and N. D. SYLVESTER, ht. J. Engng Sci. 11, 905 (1973). J. PEDDIESON, Jr. and R. P. M. McNITT, Recenf Adc. Engng Sci. 5, 405 (1970). F. EBERT, Chem. Engng J. 5, 85 (1973). A. AHMADI, Znf. J. Engng Sci. 14, 639 (1976). A. J. WILLSON, Cumb. Phil. Sot. 67, 969 (1970). G. NATH, Rheol. Acta 14, 850 (1975). H. S. TAKHAR and V. M. SOUNDALKEGAR, Rheol. Acfa 19, 525 (1980). H. S. TAKHAR and V. M. SOUNDALGEKAR, Proc. 8th Inf. Cong. Rheology, p. 525. Naples, Italy (1980). IO] H. BLASIUS, Z. Math. u. Phys. 56, 1 (1908). II] B. C. SAKIADIS, A.I.Ch.E. J. 7, 221 (1961). 121 F. K. TSOU, E. M. SPARROW and R. J. GOLDSTEIN, Int. J. Heat Mass Trans. 10, 219 (1967). (Receiced
27 August
1982)
002~7225/83/080961--05$03.00/O 0 1983 Pergamon Press Ltd.
1983
FLOW OF MICROPOLAR FLUID PAST A CONTINUOUSLY MOVING PLATE V. M. SOUNDALGEKAR Gulf Polytechnic, PO Box 32038, Isa Town, Bahrain, Middle East and
Simon engineering Laborato~es,
H. S. TAKHAR University of Manchester, Manchester Ml3 9PL, England
Abstract-A similarity analysis of the flow and heat transfer past a continuously moving semi-infinite plate in a micropolar fluid is presented. The velocity, micro-rotation distribution and the temperature profiles are shown on graphs and the numerical values of the skin friction and the rate of heat transfer are entered in tables. The effects of K (coupling parameter) and G (micro-rotation parameter) are discussed. 1. INTRODUCTION ERINGEN[~] first derived the. constitutive equations for fluids with micro-structures. Simple problems on the flow of such a Auid were studied by a number of researchers and a review of this work was given by Ariman, Turk and Sylvester [2]. The boundary layer flow of such a micropolar fluid past a semi-infinite plate has been studied by Peddieson and McNitt[3] by using a finite difference method, whereas a similarity solution for boundary layer flow near a stagnation point was presented by Ebert[4]. On taking into account the gyration vector normal to the xy-plane and the micro-inertia effects, the boundary layer flow of micropolar fluids past a semi-in~nite plate was studied by Ahmadi [5]. Willson [6] obtained the solution in the stagnation region of the micropolar fluid by the Ka~an-Pohlhausen method. The flow of micropolar fluid past a wedge was studied by Nath [7]. In Refs. [6,7], it was observed that in the 2-dimensional boundary layer flow of micropolar fluids, the microinertia effects can be neglected. Recently Takhar and Soundalgekar[8,9], studied the heat transfer aspect of the flow of micropolar fluids past a semi-infinite plate and a wedge respectively. In all these studies, the body was assumed to be stationa~ and the micro-polar fluid moved past this body. Another situation commonly observed in industrial problems is the flow of a stationary fluid past a continuously moving plate. This problem is different from that of the Blasius[ lo] problem because here the boundary layer thickness decreases with increasing x, the coordinate taken along the plate in the direction of the moving plate, whereas in Blasius problem the boundary layer thickness increases with increasing X. Hence, Sakiadis[l 11,first studied the flow of a Newtonian fluid past a continuously moving plate, whereas its heat transfer aspect was studied by Tsou, Sparrow and Goldstein[l2]. It is now proposed to study the flow of a micropolar fluid past a continuously moving plate and the corresponding heat transfer. In Section 2, the mathematical analysis is presented and in Section 4 the conclusions are set out.
2. MATHEMATICAL
ANALYSIS
Consider a steady, 2-dimensional flow of a micropolar fluid past a continuously moving flat plate, with a constant velocity in a micropolar fluid medium at rest. The origin is located at the spot through which the plate is drawn in the fluid medium and the y-axis is taken normal to it. Then, under the usual boundary layer approximations, the flow and heat transfer in micropolar fluid are governed by the following equations:
962
V. M. SOUNDALGEKAR
G
and H. S, TAKHAR
az,,_26-!!!!_o
1ay*
(2)
aY -
(3)
dT ar a9 v au u-+u-_=~,+_---2 ax ay ay c,0ay
(4)
and the boundary conditions are given by u =
u,,
v
u = 0,
=o,
T= T,,
T=T,,
a=0
(r=O,
at
as
y=O
y-xo.
(5)
Here v = (U + S)/p is the apparent kinematic viscosity, p the coefficient of dynamic viscosity, S a constant characteristic of the fluid, CJ the micro-rotation component, K, = S/p, (K, > 0) the coupling constant, G, (> 0) the microrotation constant, p the density, U,v, the velocity components along x, y coordinates, and U, is the uniform velocity of the plate. Also, T is the temperature of the fluid in the boundary layer, T, the wall temperature of the plate and T, is the temperature of the fluid far away from the plate, K the thermal diffusi~ty, C, the specific heat at constant pressure, and the last term in eqn (4) represents the viscous dissipative heat which is to be taken into account when the Prandtl number of the fluid is large or (T,- T,)is small. On introducing the following transformation
$ = in eqns (l)-(4),
~2VU~X)l’~(~),
t?=T-T,/T,--T,
(6)
we get f”+fl+Kg’=O
(7)
Gg"- 2(2g +s”) = 0
(8)
fY+PrJlf'+PrEcf=*=O and the corresponding
(9)
boundary conditions are
f(O)=O,
f’(O)= 1,
_fYcQ)= 0,
6(O) = 1,
&co) = 0,
g(0) = 0
g(c0) = 0.
(10)
Here Pr = V/K is the Prandtl number, K = KJv, G = G~U~iv-x and E,= U$C,(T,- T,) where K,G,E,are the coupling constant parameter, the micro-rotation parameter and the Eckert number, respectively. These are non-linear ordinary differential equations which are to be solved numerically using two-point boundary value methods on a high speed computer.
3. NUMERICAL
ANALYSIS
The numerical procedure used here solves the two-point boundary-value problems for a system of N ordinary differential equations in the range (X, X,). The system is written
Flow of micropolar fluid past a continuously moving plate
963
as
and the derivatives f;: are evahrated by a procedure that evaluates the derivatives of y, at a general point X. Initially N boundary values of the variable yi must be .Yl,YZ,-.., specified, some of which will be specified at X and some at X, aThe remaining N boundary values are guessed and the procedure corrects them by a form of Newton iteration. Starting from the known and guessed values of y, at X, the procedure integrates the equations forward to a matching point R, using Merson’s method. Similarly starting from Xi it integrates backwards to R. The difference between the forward and backward values of yi at R should be zero for a true solution. The procedure uses a generabzed Newton method to reduce these differences to zero, by calculating corrections to the estimated boundary values. This process is repeated iteratively until convergence is obtained to a specified accuracy. The tests for convergence, and the perturbation of the boundary conditions are carried out in a mixed form, e.g. if the error estimate for y, is ERROR,, we test whether, ABS(ERRORJ -CERRUR, x (I+ ABS yJ. Essentially, this makes the test absolute for yj 6 I and relative for yi B t. Note that convergence is not guaranteed, particularly from a poor starting approximation. A serious difficulty which may arise with boundary value problems is inherent instability. In such cases, integration from one or both ends of the range will produce rapidly increasing solutions which may accasionally lead to overAow before the matching point is reached. The position of the matching point R can be varied to improve the situation; if the solution increases rapidly for forward (or backward) integrations R should be taken at X (or X1); if it increases in both directions, R should be taken between X and X,. If the matching point R is at one of the end points X or Xi, there is no need to estimate the unknown boundary values accurately at this point, as they are not required for integration. Another difficulty which often arises is the case when one end of the range, say Xi, is at infimty. The end-point is approximated by taking finite values for X,, which is obtained by estimating where the solution will reach its asymptotic state. The computing time for integrating the differential equations can sometimes depend critically on the quality of the initiaf guesses of the unknown boundary conditions, the iocations of the matching point and the infinite end-point. The numerical results so obtained are plotted for velocity and micro-rotation on Fig. 1 for different values of K and G. We observe from this figure that the velocity increases with increasing K, whereas the micro-rotation distribution increases with increasing K and decreases with increasing G. In Fig. 2 the temperature profiles are shown. We observe from this figure that an increase in K or G leads to a decrease in the temperatures of the micropolar fluid.
Fig. 1. Velocity and microrotation
distribution.
964
V. M. SOUNDALGEKAR
and H. S. TAKHAR
Fig. 2. Temperature pr&fes,
Pr = 7, E =0.02.
From the velocity field we can study the shear stress, r, as given by
and in view of (6), the skin-friction coefficient C7,is given by
where Re = Uxjv is the Reynolds number. This shows that the skin-friction coefficient does not contain the micro-rotation term in an explicit way. The numerical values off’(O) are calculated and they are entered in Table I. We conclude from this table that an increase in K leads to a decrease in the skin-friction. From the temperature field, we can now study the rate of heat transfer. It is given by 4 = _k!?
(14)
a.Y ,S=O’ In view of eqn (6), eqn (14) reduced to
The numerical values of Q’(O) are entered in Table 2. We observe from Table 2 that an increase in K leads to a decrease in the value of the rate of heat transfer f - P(O)), whereas an increase in G leads to an increase in the value of rate of heat transfer { -S’(O)),
4. CONCLUSIONS I.
The velocity and the micro-rotation distribution increases with increasing K 2. An increase in K or G leads to a decrease in the value of the temperature. 3. The skin-friction decreases with increasing 1% 4. The rate of heat transfer decreases with increasing K and increases with increasing G.
Tabte 2. Vahes __--. K\,G
of {-
2 ___---
0.1 i ,944 0.5 1.801 ____~__-._-.---
#‘@I)) _-.4 I.946
Flow of micropolar
fluid past a continuously
moving
plate
965
REFERENCES [l] [2] [3] [4] [5] [6] [7] [8] [9]
A. C. ERINGEN, Znt. J. Engng Sci. 2, 205 (1964). T. ARIMAN, M. A. TURK and N. D. SYLVESTER, ht. J. Engng Sci. 11, 905 (1973). J. PEDDIESON, Jr. and R. P. M. McNITT, Recenf Adc. Engng Sci. 5, 405 (1970). F. EBERT, Chem. Engng J. 5, 85 (1973). A. AHMADI, Znf. J. Engng Sci. 14, 639 (1976). A. J. WILLSON, Cumb. Phil. Sot. 67, 969 (1970). G. NATH, Rheol. Acta 14, 850 (1975). H. S. TAKHAR and V. M. SOUNDALKEGAR, Rheol. Acfa 19, 525 (1980). H. S. TAKHAR and V. M. SOUNDALGEKAR, Proc. 8th Inf. Cong. Rheology, p. 525. Naples, Italy (1980). IO] H. BLASIUS, Z. Math. u. Phys. 56, 1 (1908). II] B. C. SAKIADIS, A.I.Ch.E. J. 7, 221 (1961). 121 F. K. TSOU, E. M. SPARROW and R. J. GOLDSTEIN, Int. J. Heat Mass Trans. 10, 219 (1967). (Receiced
27 August
1982)