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Free convection effects on MHD Stokes problem for a vertical plate
Nuclear Engineering and Design 51 (1979) 403-407 © North-Holland Publishing Company
403
FREE CONVECTION EFFECTS ON MHD STOKES PROBLEM FOR A VERTICAL PLATE V.M. SOUNDALGEKAR Department of Mathematics, Indian Institute of Technology, Powai, Bombay 400076, India S.K. GUPTA Indian Meteorological Department Pune (5), India and R.N. ARANAKE Walchand College of Engineering, Sangli, India Received 30 July 1978
An exact analysis of MHD Stokes problem (also Rayleigh's problem) for the flow of an electrically conducting, incompressible, viscous fluid past an impulsivelystarted vertical plate, under the action of transversely applied magnetic field, is carried out. The heat due to viscous dissipation and the induced magnetic field are assumed to be negligible. The effects of the external heating or cooling of the plate by the free convection currents are studied. It is observed that reverse flow occurs when the plate is being heated by the free convection currents. The skin-friction increases owing to greater heating of the plate and decreases due to greater cooling of the plate.
1. Introduction One of the simple solutions of the Navier-Stokes equation was first given by Stokes [1] for the case of the flow of an incompressible viscous fluid past an infinite horizontal plate moving impulsively in its own plane. Hence it is known as Stokes' first problem. It is also known as Rayleigh's problem in the literature. This study has been extended to the flow past bodies of different shapes, moving impulsively in their own plane. Notable amongst them are those by Stewartson [2], Hall [3] and Dennis [4] etc. In references [2], [3], the analytical study of Rayleigh's problem for a semi-infinite plate was made, whereas in ref. [3], the problem was studied by the infinite difference method. But the study of the flow past an infinite vertical plate, moving impulsively in its own plane, has not been presented for a long time. Now in this type of flow, two types of physical situations can be considered: (i) the temperature of the plate and the fluid extending to infinity is equal
(ii) the plate temperature differs from the temperature of the fluid at infinity. From the physical and technological point of view, the second case is more important. In this case, if the difference between the plate temperature Tw and that of the fluid at infinity T" viz(Tw - T ' ) is appreciably large, there is a flow of free convection currents in the neighbourhood of the plate and the free convection currents do affect the flow past an impulsively started vertical plate. This study of an important physical phenomenon was recently presented by Soundalgekar [5]. In this paper, Soundalgekar studied the effects of the free convection currents on the velocity field in a quantitative manner. Illingworth [6] also studied Stokes' problem for a vertical plate, by the method of successive approximation, for a compressible fluid and this was extended to the compressible flow past an infinite plate with time.dependent velocity and temperature by Elliott [7]. In both references [6,7] the problem was solved mathematically without any discussion of the physical situation.
404
V.M. Soundalgekar et al. / Convection effects on MHD stokes problem
In recent years, the effects of the transverse magnetic field on the flow of an incompressible, viscous, electrically conducting fluid, have also been studied extensively by many research workers. The magnetohydrodynamic aspect of Stokes' or Rayleigh's problem, on neglecting the induced magnetic field, was first presented by Rossow [8] in case of a horizontal plate. How does a transversely applied magnetic field affect the flow of an electrically conducting, viscous incompressible fluid past an impulsively started vertical plate? This indeed is the motivation of the present study. In section 2, the mathematical analysis is presented under proper assumptions and followed by a discussion. In section 3, the conclusions are set out.
u'->O,T'~T"
asy' ~ oo.
(3)
On introducing the following non-dimensional quantities y ' = yUo/v ; t = t'U~o/V ; u = u'/Uo ; T'-T~ T " - T~ ; P = l~Cp/k the Prandtl number,
0
G = vg/3(Tw - T ' ) / U ~ , the Grashof number ; M 2 = oB~ vlo'~o,
(4)
the magnetic field parameter in eqs. (1), (2) and (3); we have ~u/at = a2u/~y 2 + GO - M=u ,
(5)
and
Here, the x'-axis is taken along the plate in the vertical direction and the y'-axis is taken normal to it. An uniformly transverse magnetic field is applied along the y'-axis. It is also assumed that the induced magnetic field is negligible. All the fluid properties are assumed to be constant except that the influence of the density variation with temperature is considered only in the body force term. This is usual Boussinesq's approximation. Then under these assumptions, the problem is governed by the following equations. au' ~t ' - g / 3 ( T ' -
T')+v
~2u' OY2
(6)
e a o / o t = a20/~y = ,
2. Mathematical analysis
oB~ u I , P'
(1)
for u = 1 ,0 = 1 asy = 0 , u=0,0=0asy~
oo.
(7)
These are the coupled equations whose solutions can be obtained by the usual Laplace-transform technique and are given as follows: 0 = erfc(r/x/P),
(8)
f o r P < 1, 1 u = ~(1 - G/M2)[e-2nMx/Terfc(~ - MX/7)
+ e2nMvqerfc(r~ + MX/7)] - (G/2M 2) e bt [e-=n Px/~-/erfc(rlX/ff- x / ~ )
OT'
p'Cp ~ ;
~2T'
= k Oy,~.
(2)
Here u' is the velocity of the fluid, T' the temperature of the fluid near the plate, T ' , the temperature of the fluid far away from the plate, g the acceleration due to gravity, 13the coefficient of volume expansion, v, the kinematic viscosity, o the scalar electrical conductivity, Bo the applied uniform magnetic field and P' is the density of the fluid, k the thermal conductivity and t' is the time. In eq. (2), the heat due to viscous dissipation is assumed to be negligible. This is possible when the velocity is small. The boundary conditions are: u' = Uo, T' = Tw asy' = 0 ,
+ e2nPx/T~erfc(r/x/ff+ x/rb~)] + (G/2M z) erfc(~/ff) + (G/2M 2) e bt [e-anx/b-~erfc(r/- X/-P-~)
+ eZm'/PK/erfc(r/+ x/~b-t)] ,
(9)
where b =M2/(p - 1) and r/=y/2x/~-;
forP= 1 u = ½(1 - G / M 2 ) [e-2nMx/Terfc(r/-- Mx,/t-) + eZnMX/Terfc(r/+MVT)] + ( G / M 2) erfc(rrV/ff). (10) Now, for all electrical conducting fluids, P is either less than unity or P ~ 1. For P < 1, the argument of
405
V.M. Soundalgekar et al. / Convection effects on MHD stokes problem
the complementary error function in eq. (10) is imaginary and hence for calculation purposes, this has to be separated into real and imaginary parts. This can be done, following Strand [9], as follows:
oI
0.8
0.6
,
F o r x > 0 , y ~> 0 ,
G -6 -G -6 -6
t 0.5 0.5 0.5 1.0
M 2 2 /, 2
-4
0.5
2
P 0.5 0.7 0.5 0.5 0.5
I
II nl IV V
r
o -0.2 az
erfc(x + iy)
-
= e -2ixy ~
(xy)n[pn(X) - i(n + 1)Pn+l(X)]
n=O
where 2 Vn+I(X) = (2n + 1) x/q
n=0,1,2
Fig. 1.
(11)
= e-2ixy(o(x, y ) ,
F
e-X2
L(n + 1)! x =n+1 -
V'~ n +1
vn(x)]
....
and Vo(X) = erfc(x)
Hence, with the help of eq. (11), the velocity profiles for any value of P can be calculated. Here, to show the effects of G and t, we have calculated the velocity and temperature for P = 1,0.5, 0.7 and t = 0.5, 1, 1.5, 2 and for realistic purpose, all the real values of G are chosen as they are interesting from the physical point of view, Now, the free convection currents exist because of the temperature difference (Tw T ' ) and hence (Tw - T ' ) may be positive, zero or negative. Then the value of the Grashof number G (=vg[3)/(T" - T ' ) / U ~ ) will assume positive, zero or negative values. From the physical point of view, G < 0 corresponds to an externally heated plate as the free convection currents are carried towards the plate. Then G > 0 corresponds to an externally cooled plate and G = 0 then corresponds to the absence of free convection currents. On figs. 1 and 2, the velocity profiles are shown for P < 1. The curves on fig. 1 correspond to the velocity profiles in the presence of the plate being heated by the free convection currents. We conclude from the nature of these curves that they are of reverse type. An increase in P leads to an increase in the velocity when the values of G, t and M are constant. An increase in M also leads to an increase in the velocity. This leads us to conclude that the reverse type of flow may be avoided, in the presence of the
plate being heated by the free convection currents, by increasing the strength of the magnetic field. As time is increased, the velocity still decreases when the value of G, M and P are constant. Greater heating of the plate by the free convection currents also causes a decrease in the velocity. Quantitatively, for M = 2, P = 0.5, t = 0.5, there is a 100% fall in the velocity at r/= 1 when the plate is heated such that G is increased from 4 to 6. On fig. 2, the velocity profiles are shown in the case of the plate being cooled by the free convection currents. These all being positive, we conclude that there is a non-reverse type of flow. Here, an increase in P leads to a decrease in the velocity when the values of G, t and M are constant. An increase in M or t leads to a decrease in the velocity. But greater cooling of the plate by the free convection currents causes a rise in the velocity. Quantitatively, for t = 0.5, M = 2, P = 0.5, there is a 76% rise in the velocity at r/= 1 when G is increased from 2 to 4 owing to greater cooling of the plate. On figs. 3 and 4, the velocity profiles are shown for fluids whose Prandtl number is unity. The trend is the same as described above. Knowing the velocity field, we can calculate the
1.0
G t M - 6 0.5 2 I -6 0.5 4 IT - 6 1.0 2 Ill -4 0,5 2 IV
0.8 uO'G 0.4 0.2 0
I.,0
-0.2 - 0,1
Fig. 2.
2;0
V.M. Soundalgekar et al. / Convection effects on MHD stokes problem
406 G 0 2 2 2 2 Z.
1.0
t 0,5 (35 0.5 0.5 1.0 0.5
M 2 2 2 4 2 2
P 0.5 0-5 0.7 0.5 0.5 0.5
I IT IE tV V Vl
G -6 -6 -~ 2 2 6
| G
~4 2 4 2 2 ~ z.
f
O,B
-III
IV
0.6
I II III IV v VT II I
- - v
Uoz
vi
0.;
IV 1.0 tl
2.0 0
ols
Fig. 3.'
~ot its
2~0
Fig. 5.
skin-friction. It is defined as r '=
-lz(Ou'/Oy')y'
= 0,
(12)
and in virtue of eq. (4), eq. (12) reduces to (13)
r = r'/pU~o = -(au/ay)y=o.
Substituting eqs. (9) and (10) in eq. (13), we have for P
of the plate), an increase in M leads to an increase in the skin-friction whereas owing to greater heating of the plate by the free convection currents, there is also a rise in the skin-friction. There is also a rise in the skin-friction when M is increased for G > 0 but owing to greater cooling of the plate, there is a fall in the value of the skin-friction. Same is the trend of the skin-friction for P = 1.
G +(l_~2)(~meM2t+Merf(MvC[)
T=M2x/
3. Conclusions
C eb,(v,p eb, _ d , ~ ) , M2~r~,
(14)
and for P = 1,
m:v/~ •
(is) is plotted in figs. 5 and 6 for P = 0.7 and 1 respectively. We conclude from fig. 5 that the value of the skin-friction is more when the plate is being heated by the free convection currents. For G < 0 (heating
G 0
1.0k
2 2 ~
|\ 0.81\ u
t
m
G -6 -6 -~ 2 2
M
0.5 2
2 05
1. When the plate is being heated by free convection currents, the velocity profiles are of the reverse type. 2. On greater heating of the plate, the velocity decreases wheras on greater cooling of the plate, the velocity increases. 3. On greater heating of the plate, the skin-friction increases and on greater cooling of the plate, it decreases.
2
1.0 2 1.0 z; 0.5 2
I E 1"11 IV v
M 2 ~ 2 2 4 4 11
"T ?
~ - ~ - ~
C_v L-1T[ EVI
2 -IV
l 0.5
Fig. 4.
I IT ITT IV V VI II
l.O + 1.5
Fig. 6.
2-0
V.M. Soundalgekar et al. / Convection effects on MHD stokes problem
Acknowledgements
x', y ' = co-ordinate axes
We are grateful to the Atomic Energy Commission of India for the financial support to carry out this research.
v o p' 0 erfc r!
Nomenclature B0 cp g G K M P T' T" Tw u' Uo
= magnetic induction = specific heat at constant pressure -- acceleration due to gravity --- grashof number (vg/~(Tw - T')/Uao) --- thermal conductivity = magnetic field parameter = Prandtl number ~Cp/K = temperature of fluid = temperature of fluid away from plate = temperature of the plate = velocity component = velocity of the plate
407
= coefficient of volume expansion = kinematic viscosity -- electrical conductivity = density = non-dimensional temperature = complementary error function -- skin-friction
References [1] G.G. Stokes, Trans. Cambridge PhiL Soc. 9 (1851) 8. [2] K. Stewartson, Quart. J. Appl. Math. Mech. 4 (1951) 182. [3] M.G. Hail, Proc. Royal Soc. London, 310 A (1969) 401. [4] S.C.R. Dennis, J. Inst. Math. Appl. 10 (1972) 105. [5] V.M. Soundalgekar, J. Heat Transfer 99 C (1977) 499. [6] C.R. Illingworth, Proc. Cambridge Phil. Soc. 46 (1950) 603. [7] L. Elliot, Z. Angew. Math. Mech. 49 (1969) 647. [8] V.J. Rossow, Nat. Advisory Comm. Aeron. (NACA) Rept. 1358 (1958). [9] O.N. Strand, Math. Computatoins 19 (1965) 127.
403
FREE CONVECTION EFFECTS ON MHD STOKES PROBLEM FOR A VERTICAL PLATE V.M. SOUNDALGEKAR Department of Mathematics, Indian Institute of Technology, Powai, Bombay 400076, India S.K. GUPTA Indian Meteorological Department Pune (5), India and R.N. ARANAKE Walchand College of Engineering, Sangli, India Received 30 July 1978
An exact analysis of MHD Stokes problem (also Rayleigh's problem) for the flow of an electrically conducting, incompressible, viscous fluid past an impulsivelystarted vertical plate, under the action of transversely applied magnetic field, is carried out. The heat due to viscous dissipation and the induced magnetic field are assumed to be negligible. The effects of the external heating or cooling of the plate by the free convection currents are studied. It is observed that reverse flow occurs when the plate is being heated by the free convection currents. The skin-friction increases owing to greater heating of the plate and decreases due to greater cooling of the plate.
1. Introduction One of the simple solutions of the Navier-Stokes equation was first given by Stokes [1] for the case of the flow of an incompressible viscous fluid past an infinite horizontal plate moving impulsively in its own plane. Hence it is known as Stokes' first problem. It is also known as Rayleigh's problem in the literature. This study has been extended to the flow past bodies of different shapes, moving impulsively in their own plane. Notable amongst them are those by Stewartson [2], Hall [3] and Dennis [4] etc. In references [2], [3], the analytical study of Rayleigh's problem for a semi-infinite plate was made, whereas in ref. [3], the problem was studied by the infinite difference method. But the study of the flow past an infinite vertical plate, moving impulsively in its own plane, has not been presented for a long time. Now in this type of flow, two types of physical situations can be considered: (i) the temperature of the plate and the fluid extending to infinity is equal
(ii) the plate temperature differs from the temperature of the fluid at infinity. From the physical and technological point of view, the second case is more important. In this case, if the difference between the plate temperature Tw and that of the fluid at infinity T" viz(Tw - T ' ) is appreciably large, there is a flow of free convection currents in the neighbourhood of the plate and the free convection currents do affect the flow past an impulsively started vertical plate. This study of an important physical phenomenon was recently presented by Soundalgekar [5]. In this paper, Soundalgekar studied the effects of the free convection currents on the velocity field in a quantitative manner. Illingworth [6] also studied Stokes' problem for a vertical plate, by the method of successive approximation, for a compressible fluid and this was extended to the compressible flow past an infinite plate with time.dependent velocity and temperature by Elliott [7]. In both references [6,7] the problem was solved mathematically without any discussion of the physical situation.
404
V.M. Soundalgekar et al. / Convection effects on MHD stokes problem
In recent years, the effects of the transverse magnetic field on the flow of an incompressible, viscous, electrically conducting fluid, have also been studied extensively by many research workers. The magnetohydrodynamic aspect of Stokes' or Rayleigh's problem, on neglecting the induced magnetic field, was first presented by Rossow [8] in case of a horizontal plate. How does a transversely applied magnetic field affect the flow of an electrically conducting, viscous incompressible fluid past an impulsively started vertical plate? This indeed is the motivation of the present study. In section 2, the mathematical analysis is presented under proper assumptions and followed by a discussion. In section 3, the conclusions are set out.
u'->O,T'~T"
asy' ~ oo.
(3)
On introducing the following non-dimensional quantities y ' = yUo/v ; t = t'U~o/V ; u = u'/Uo ; T'-T~ T " - T~ ; P = l~Cp/k the Prandtl number,
0
G = vg/3(Tw - T ' ) / U ~ , the Grashof number ; M 2 = oB~ vlo'~o,
(4)
the magnetic field parameter in eqs. (1), (2) and (3); we have ~u/at = a2u/~y 2 + GO - M=u ,
(5)
and
Here, the x'-axis is taken along the plate in the vertical direction and the y'-axis is taken normal to it. An uniformly transverse magnetic field is applied along the y'-axis. It is also assumed that the induced magnetic field is negligible. All the fluid properties are assumed to be constant except that the influence of the density variation with temperature is considered only in the body force term. This is usual Boussinesq's approximation. Then under these assumptions, the problem is governed by the following equations. au' ~t ' - g / 3 ( T ' -
T')+v
~2u' OY2
(6)
e a o / o t = a20/~y = ,
2. Mathematical analysis
oB~ u I , P'
(1)
for u = 1 ,0 = 1 asy = 0 , u=0,0=0asy~
oo.
(7)
These are the coupled equations whose solutions can be obtained by the usual Laplace-transform technique and are given as follows: 0 = erfc(r/x/P),
(8)
f o r P < 1, 1 u = ~(1 - G/M2)[e-2nMx/Terfc(~ - MX/7)
+ e2nMvqerfc(r~ + MX/7)] - (G/2M 2) e bt [e-=n Px/~-/erfc(rlX/ff- x / ~ )
OT'
p'Cp ~ ;
~2T'
= k Oy,~.
(2)
Here u' is the velocity of the fluid, T' the temperature of the fluid near the plate, T ' , the temperature of the fluid far away from the plate, g the acceleration due to gravity, 13the coefficient of volume expansion, v, the kinematic viscosity, o the scalar electrical conductivity, Bo the applied uniform magnetic field and P' is the density of the fluid, k the thermal conductivity and t' is the time. In eq. (2), the heat due to viscous dissipation is assumed to be negligible. This is possible when the velocity is small. The boundary conditions are: u' = Uo, T' = Tw asy' = 0 ,
+ e2nPx/T~erfc(r/x/ff+ x/rb~)] + (G/2M z) erfc(~/ff) + (G/2M 2) e bt [e-anx/b-~erfc(r/- X/-P-~)
+ eZm'/PK/erfc(r/+ x/~b-t)] ,
(9)
where b =M2/(p - 1) and r/=y/2x/~-;
forP= 1 u = ½(1 - G / M 2 ) [e-2nMx/Terfc(r/-- Mx,/t-) + eZnMX/Terfc(r/+MVT)] + ( G / M 2) erfc(rrV/ff). (10) Now, for all electrical conducting fluids, P is either less than unity or P ~ 1. For P < 1, the argument of
405
V.M. Soundalgekar et al. / Convection effects on MHD stokes problem
the complementary error function in eq. (10) is imaginary and hence for calculation purposes, this has to be separated into real and imaginary parts. This can be done, following Strand [9], as follows:
oI
0.8
0.6
,
F o r x > 0 , y ~> 0 ,
G -6 -G -6 -6
t 0.5 0.5 0.5 1.0
M 2 2 /, 2
-4
0.5
2
P 0.5 0.7 0.5 0.5 0.5
I
II nl IV V
r
o -0.2 az
erfc(x + iy)
-
= e -2ixy ~
(xy)n[pn(X) - i(n + 1)Pn+l(X)]
n=O
where 2 Vn+I(X) = (2n + 1) x/q
n=0,1,2
Fig. 1.
(11)
= e-2ixy(o(x, y ) ,
F
e-X2
L(n + 1)! x =n+1 -
V'~ n +1
vn(x)]
....
and Vo(X) = erfc(x)
Hence, with the help of eq. (11), the velocity profiles for any value of P can be calculated. Here, to show the effects of G and t, we have calculated the velocity and temperature for P = 1,0.5, 0.7 and t = 0.5, 1, 1.5, 2 and for realistic purpose, all the real values of G are chosen as they are interesting from the physical point of view, Now, the free convection currents exist because of the temperature difference (Tw T ' ) and hence (Tw - T ' ) may be positive, zero or negative. Then the value of the Grashof number G (=vg[3)/(T" - T ' ) / U ~ ) will assume positive, zero or negative values. From the physical point of view, G < 0 corresponds to an externally heated plate as the free convection currents are carried towards the plate. Then G > 0 corresponds to an externally cooled plate and G = 0 then corresponds to the absence of free convection currents. On figs. 1 and 2, the velocity profiles are shown for P < 1. The curves on fig. 1 correspond to the velocity profiles in the presence of the plate being heated by the free convection currents. We conclude from the nature of these curves that they are of reverse type. An increase in P leads to an increase in the velocity when the values of G, t and M are constant. An increase in M also leads to an increase in the velocity. This leads us to conclude that the reverse type of flow may be avoided, in the presence of the
plate being heated by the free convection currents, by increasing the strength of the magnetic field. As time is increased, the velocity still decreases when the value of G, M and P are constant. Greater heating of the plate by the free convection currents also causes a decrease in the velocity. Quantitatively, for M = 2, P = 0.5, t = 0.5, there is a 100% fall in the velocity at r/= 1 when the plate is heated such that G is increased from 4 to 6. On fig. 2, the velocity profiles are shown in the case of the plate being cooled by the free convection currents. These all being positive, we conclude that there is a non-reverse type of flow. Here, an increase in P leads to a decrease in the velocity when the values of G, t and M are constant. An increase in M or t leads to a decrease in the velocity. But greater cooling of the plate by the free convection currents causes a rise in the velocity. Quantitatively, for t = 0.5, M = 2, P = 0.5, there is a 76% rise in the velocity at r/= 1 when G is increased from 2 to 4 owing to greater cooling of the plate. On figs. 3 and 4, the velocity profiles are shown for fluids whose Prandtl number is unity. The trend is the same as described above. Knowing the velocity field, we can calculate the
1.0
G t M - 6 0.5 2 I -6 0.5 4 IT - 6 1.0 2 Ill -4 0,5 2 IV
0.8 uO'G 0.4 0.2 0
I.,0
-0.2 - 0,1
Fig. 2.
2;0
V.M. Soundalgekar et al. / Convection effects on MHD stokes problem
406 G 0 2 2 2 2 Z.
1.0
t 0,5 (35 0.5 0.5 1.0 0.5
M 2 2 2 4 2 2
P 0.5 0-5 0.7 0.5 0.5 0.5
I IT IE tV V Vl
G -6 -6 -~ 2 2 6
| G
~4 2 4 2 2 ~ z.
f
O,B
-III
IV
0.6
I II III IV v VT II I
- - v
Uoz
vi
0.;
IV 1.0 tl
2.0 0
ols
Fig. 3.'
~ot its
2~0
Fig. 5.
skin-friction. It is defined as r '=
-lz(Ou'/Oy')y'
= 0,
(12)
and in virtue of eq. (4), eq. (12) reduces to (13)
r = r'/pU~o = -(au/ay)y=o.
Substituting eqs. (9) and (10) in eq. (13), we have for P
of the plate), an increase in M leads to an increase in the skin-friction whereas owing to greater heating of the plate by the free convection currents, there is also a rise in the skin-friction. There is also a rise in the skin-friction when M is increased for G > 0 but owing to greater cooling of the plate, there is a fall in the value of the skin-friction. Same is the trend of the skin-friction for P = 1.
G +(l_~2)(~meM2t+Merf(MvC[)
T=M2x/
3. Conclusions
C eb,(v,p eb, _ d , ~ ) , M2~r~,
(14)
and for P = 1,
m:v/~ •
(is) is plotted in figs. 5 and 6 for P = 0.7 and 1 respectively. We conclude from fig. 5 that the value of the skin-friction is more when the plate is being heated by the free convection currents. For G < 0 (heating
G 0
1.0k
2 2 ~
|\ 0.81\ u
t
m
G -6 -6 -~ 2 2
M
0.5 2
2 05
1. When the plate is being heated by free convection currents, the velocity profiles are of the reverse type. 2. On greater heating of the plate, the velocity decreases wheras on greater cooling of the plate, the velocity increases. 3. On greater heating of the plate, the skin-friction increases and on greater cooling of the plate, it decreases.
2
1.0 2 1.0 z; 0.5 2
I E 1"11 IV v
M 2 ~ 2 2 4 4 11
"T ?
~ - ~ - ~
C_v L-1T[ EVI
2 -IV
l 0.5
Fig. 4.
I IT ITT IV V VI II
l.O + 1.5
Fig. 6.
2-0
V.M. Soundalgekar et al. / Convection effects on MHD stokes problem
Acknowledgements
x', y ' = co-ordinate axes
We are grateful to the Atomic Energy Commission of India for the financial support to carry out this research.
v o p' 0 erfc r!
Nomenclature B0 cp g G K M P T' T" Tw u' Uo
= magnetic induction = specific heat at constant pressure -- acceleration due to gravity --- grashof number (vg/~(Tw - T')/Uao) --- thermal conductivity = magnetic field parameter = Prandtl number ~Cp/K = temperature of fluid = temperature of fluid away from plate = temperature of the plate = velocity component = velocity of the plate
407
= coefficient of volume expansion = kinematic viscosity -- electrical conductivity = density = non-dimensional temperature = complementary error function -- skin-friction
References [1] G.G. Stokes, Trans. Cambridge PhiL Soc. 9 (1851) 8. [2] K. Stewartson, Quart. J. Appl. Math. Mech. 4 (1951) 182. [3] M.G. Hail, Proc. Royal Soc. London, 310 A (1969) 401. [4] S.C.R. Dennis, J. Inst. Math. Appl. 10 (1972) 105. [5] V.M. Soundalgekar, J. Heat Transfer 99 C (1977) 499. [6] C.R. Illingworth, Proc. Cambridge Phil. Soc. 46 (1950) 603. [7] L. Elliot, Z. Angew. Math. Mech. 49 (1969) 647. [8] V.J. Rossow, Nat. Advisory Comm. Aeron. (NACA) Rept. 1358 (1958). [9] O.N. Strand, Math. Computatoins 19 (1965) 127.