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Hall effects on combined free and forced convective hydromagnetic flow through a channel
hf. J. Engng SC;.. 1976, Vol. 14. pp. 285-292.
Pergamon Press.
Printed in Great Britain
HALL EFFECTS ON COMBINED FREE AND FORCED CONVECTIVE HYDROMAGNETIC FLOW THROUGH A CHANNEL B. S. MAZUMDER,
A. S. GUPTA and N. DATTA
Mathematics Department. Indian Institute of Technology, Kharagpur. India
(Communicated
by S. I. PAI)
Abstract-Hydromagnetic free and forced convection in a parallel plate channel permeated by a transverse magnetic field has been considered taking Hall effects into account. When there is a uniform axial temperature variation along the walls. the primary flow shows incipient flow reversal at the upper plate for an increase in temperature along that plate. Similarly flow reversal at the lower plate occurs for a decrease in temperature along that plate. Hall currents are found to exert a destabilizing influence on the flow. The skin-friction for the cross-flow increases with the Hall parameter. The induced magnetic field and the heat transfer characteristics in the flow are also determined.
I. INTRODUCTION
MHD FREE convective flow past a semi-infinite hot vertical plate in the presence of a transverse magnetic field was investigated by Gupta[l], Cramer[2], Singh and Cowling[3] and Riley[4]. These authors found that increase in magnetic field causes a reduction in the local wall shear and heat transfer rate. Combined free and forced convection of an electrically conducting liquid in a horizontal channel permeated by a transverse magnetic field was studied by Gupta[5] who assumed uniform axial temperature variation along the channel walls. This study has some bearing on the problem of cooling of nuclear reactors where liquid metals (which are electrically conducting) are used as coolant. All these studies, however, do not take Hall effects into account. It is well-known (Cowling[6]) that these effects would be important when the strength of the magnetic field is very large. The purpose of the present investigation is to extend the analysis of Gupta[5] to include Hall effects and to study these effects on the flow and heat transfer characteristics. We believe that our study will be of some use in the problem of cooling of nuclear reactors where very strong magnetic fields are used.
2. MATHEMATICAL
PROBLEM
AND ITS SOLUTION
Consider the flow of an electrically conducting liquid between two horizontal parallel plates (distant 2L apart) such that x* and y* axes are along and transverse to the plates and suppose that a strong uniform magnetic field H, acts along y*-axis. At a large distance from the entry section the flow will be fully developed and in the steady state all the physical variables (except pressure) depend on y * only. The equation V*H* = 0 gives H; = constant = H, everywhere in the flow. The continuity equation V.q* = 0 and the no slip conditions at the plates give U* = 0 everywhere, where (u *, u *, \t’*) are the components of the velocity q*. It is well-known that the introduction of Hall effects produces cross-flow (see Sherman and Sutton[7]). The equations of momentum for the fully developed steady flow in rationalised MKS units are
O=
d2u* aP* -i)x*+~dy*2+~,Hnd~*,
dH1
dH: dHT+H!_ H;Fdv* o= -$-PA’-pe dy*
where z * axis is perpendicular
(1)
1’
to x * y * plane and the other symbols have their usual meaning. 285
(2)
286
B. S.MAZUMDER
Maxwell’s equations
etal.
are
along with Ohm’s law including
VxE*=O,
(4)
VxH*=j*
(9
Hall effects (Cowling161) given by
j* +g
0
j*
x
H* = r[E* + peq* X H*],
(6)
where j*, w, T and (T denote the current density, electron Larmor frequency, electron collision time and the electrical conductivity respectively. In writing (6), the ion slip effects arising out of imperfect coupling between ions and neutrals as well as the electron pressure gradient are neglected. Eliminating E* from (4) to (6) and taking the x* and z* components of the resulting equation, we have
d*H: / o1 d’HTz -
du* WHO dy*,
A_
dy*’
dy*
(7)
d2HT d=H: dw* - wr ----T = upeHo dy*, dy@ dy* the y*-component of the equation being identically satisfied. If we assume uniform axial temperature variation along the channel walls, we may take the temperature in the flow as T-z-“=Nx*+c#zJ(y*),
where N is a constant.
Using (9) and the equation
(9) of state
p = poll - P(T - To)1 in (2) and integrating, p*=
(10)
we get
-POgy*+P,g~Nx*y*+pog~
4 dy*-$H9*+HT*)+F’(x*), I
(11)
where po, To and p denote the density and temperature of a reference state and the coefficient of volume expansion respectively. Eliminating p * from (1) and (11) and introducing the dimensionless quantities
(12) we find
d2U+Me!s_Gy= -1,
dy=
(13)
dy
where MZ = /~:Ho~L~u/p,u, Since P, >O, the positive and negative respectively along the channel walls.
values
G = /3gNL4/v2Px. of N correspond
(14) to heating
and cooling
281
Hall effects on convective hydromagnetic flow
Further
(3) reduces to
(15)
Equation
(15) multiplied
by i( =d-
1) when added to (13) gives 2 dC’+M2dh_Gy
=
dy
-
1,
(16)
dy
where u = u + iw, Similarly combining
(17)
(7) and (8) and using (12) we have 1 dU -~-(1 + ion) dy ’
d2h dy’= Elimination
h = H, + iH,.
(18)
of h from (16) and (18) leads to d3U M2 7___.dL!_G,o dy
which upon integration
(l+ iox)
(19)
dy
yields
(20) where C, is a constant
and
M,‘=MZ
(21)
1 + iwr’
The no-slip conditions
at the plates y = +l are U(k1) = 0
and since the plates are assumed electrically are
non-conducting,
(22) the magnetic boundary
conditions
h(+l)=O. Solutions
U(y)=
of (16) and (20) satisfying
l
Ml sinh M,
(23)
(22) and (23) are
* (cash M, -cash
M,y) +
G - (sinh Ml y - y sinh M,), Ml2 sinh Ml (24)
h(y)
= +[;(y2
- 1) + M,
si!hM,(cosh
+M2 si!h M I
(sinh M,y - y sinh Ml).
Assuming Ml = a + i/3 in (21) and substituting
1JE.S Vol. 14. No. 3-D
M, -cd
1
MIY 1
(25)
in (24) and (25), we get upon separating (24) and
288
B. S. MAZUMDER
(25)
into real and imaginary
u = (a,z!
et al.
parts and using (17)
azzJ . [al(cosh (Ycos /3 -cash
ay cos py) + az(sinh (Ysin p -- sinh (mysin By)]
+ (a,~~a,~~[ a3 .( sm ’ h ay cos fly - y sinh cy cos p) + a,(cosh ay sin py - y sin p . cash a)], (26) w = (a,2:
,_+n,(sinh 7
+ to,2fa4+
(Ysin /3 - sinh ay sin py) - a:(cosh (Ycos p - cash ay cos py )]
a3 .( cosh ay sin py - y cash a sin p) - a,(sinh ay cos py - y sinh (Ycos p)], (27)
G H.\ =M’
1 3
[
,(Y--
1
{a,(cosh (Ycos ,B -cash
I)+ta,z+a2zJ
+ a?(sinh LYsin p - sinh a? sin oy
ay cos py)
)}I
+ sinh cy cos P(sinh cuy cos by - y sinh cy cos p) + cash (Ysin P(cosh ay sin fly - y cash (Ysin p) M’(sinh’ cy cos’ p + cash’ (Ysin’ 0) (28) H: = Mz(~,?+ ,zZ)[a4sinh
(Ysin p - sinh ay sin py) - &(cosh (Ycos
p
-cash
av
cos
py)]
+ sinh (Ycos /3(cosh cry sin by - y cash cx sin p) - cash (Ysin p(sinh ay cos /3y - y sinh (Ycos p) M’(sinh’ (Yco? /I + coshZ (Ysin’ p) (2% where u,=cusinhcucosp-Pcoshasinp, nz = p sinh (Ycos p + (Ycash (Ysin p. a?=(Cy’-P’)sinhacosP-2apcoshasinP. a,=2a/3 ff=
sinha
cos/?+(a’-p’)cosha
M[l +(I +o’+~]“’ [2( I + u2T2)y
sinp,
’ (30)
The dimensionless
shear stress at the upper and lower plates, respectively
are given by
(31)
(32) where [(Y’sinhacosha-P3sinPcosP+ap’coshcrsinh(Y-cu’PsinPcosP F(a,p)=(n’ya 3 4 2)
Further the dimensionless
shear stress for the secondary
flow at the upper plate is the same as
Hall effects on convective hydromagnetic
289
flow
that at the lower plate and is given by G [2@(sinh2 (Ycos’ /3 + cash’ (Ysir? p) Y=t, = (a: + a:) (33) When buoyancy forces are absent (G = 0), eqs (31) and (32) show that the skin-friction for the primary flow neither vanishes on the upper nor on the lower plate. However, when G = 0, the skin-friction for the cross-flow vanishes at both the plates y = + 1. Thus we arrive at the interesting conclusion that in the absence of the buoyancy forces, the cross-flow due to Hall currents shows incipient flow reversal although the primary flow does not. On the other hand, eqn (31) shows that in the presence of buoyancy forces the primary flow at the upper plate y = 1 shows incipient flow reversal when G = G,,,, such that (a: + ad’)
Gcrit = [(cY’+ /~‘)(cI sinh (Ycash (Y- /3 sin p cos 0) - (cu’- /3’)(sinhz (Ycos’ /3 + cash’ (Ysin’ p)] (34) We have computed GCri,for several values of the Hall parameter OT with A4 = 10 as shown in the table below. Table 1. M = 10
0.50
1.oo
1.50 2.00 2.50 3.00
10.7633 IO.0132 9.2268 8.5322 7.9438 74468
This shows that the critical value of G is positive and the definition of G in (14) shows that incipient primary flow reversal occurs at the upper plate when there is an increase in temperature along that plate (G > 0). Since G,,,, decreases with increase in the Hall parameter, it follows that Hall currents exert a destabilizing influence on the flow. Further, it can be seen from eqn (32) that the primary flow at the lower plate shows incipient flow reversal when G = G&, where GA,,,= - Gcrit,
(35)
G,,i, being given by (34). Thus flow reversal at the lower plate occurs when there is a decrease in temperature along that plate (G < 0). We have also computed from (33) the skin-friction at the plates y = ? 1 for the cross-flow w(y) when G # 0 and the results are shown in the following table. Table 2. M = 10, G = 1 (dw/dy),==, 0.50
I .oo 1.50 2.00 2.50 3.00
-0.01929 -0.03551 -0.04836 -0.05862 -0.06701 -0.07401
This shows that for a fixed Hartmann number, the magnitude of the skin-friction for the cross-flow increases with the Hall parameter. We have plotted u(y) and -w(y) in the case when the flow does not separate for several values of the Hall parameter or with G = 1.0 and M = 5,10. Figure 1 shows that for fixed M,
B. S. MAZUMDER
et ul.
‘3Or
-
M=5*0
----
,1=10.0
I
-.6
I
-‘4
L
-,2
I
0.0
I
I
I
I
.2
.4
.6
‘6
I.0
Y-
Fig. 1. Profiles of non-dimensional
primary velocity u(r) for G = 1.0.
u(y) increases with increase in WT,while Fig. 2 shows a similar result for the cross-flow -w(y). The flattening effect of the magnetic field on the velocity profile is clearly discernible in these figures. Having found U(y), h(y) can be obtained from (16) after using the boundary conditions (23). When h(y) is separated into real and imaginary parts, H, and Hz can be found (see (28) and (29)). We have plotted H,(y) and K(y) in Figs. 3 and 4 for several values of w7 with A4 = 10, and G = 1 and 5. These figures indicate that the disposition of the induced magnetic field is in accord with the pulling of the lines of force by the convection in the channel. 3. HEAT
The equation
of energy including
TRANSFER
viscous and ohmic dissipation
is
u*K=KaZT+J_[(~>‘+(~>‘]+~[(~s+(~~], dy
8.x”
*’
PC,,
‘IO -
t
s
‘08-
FO,.
Fig. 2. Profiles of non-dimensional
secondary velocity w(y) for G = 1.0.
(36)
Hall effects on convective hydromagnetic
291
flow
-G = I.0 ----wz5.0
-.025 t
Fig. 3. Profiles of the non-dimensional
induced magnetic field H, for M = 10.0.
-
G =
I.0
----
1,
5’0
=
0-O
-.I N
t 0
x
-a2 --_-___----___
: -,3
-'4
!
-10
-.*
I
-6
I
I
I
0.0
-.2
-“j
1
I
1
1
,2
'4
'6
'8
1
1'0
T-
Fig. 4. Profiles of non-dimensional
induced magnetic field H2 for M = 10.0.
where K is the thermal diffusivity, c, is the specific heat and the temperature T is given by (9). Using (9) and (12), the above equation can be put in dimensionless form as
(37) where Pr is the Prandtl number
u/K and
K,=“3p,
c,KNL3’
L
t9= NLP,
(38)
292
B. S. MAZUMDER
et al.
and the over bar denotes a complex conjugate. As for the temperature boundary conditions we take the reference temperature T,, in (9) in such a manner that the temperature of the lower wall y = - 1 is T, + Nx*. Hence (38) implies that l9(-l)=O along with
(40) where iV, may be regarded as the wall temperature parameter. The temperature distribution B(y) can now be obtained by integrating (37) after substituting the expressions for U(y), h(y) and u (y ) from (24), (25) and (26) and making use of the boundary conditions (39) and (40). We omit the details of calculation as they are quite cumbersome. We present below the values of the dimensionless heat transfer coefficient d0/dy at the plates y = * 1 for several values of wr and M forG=l.O,Pr=landK,=0.5. Table 3. Valuesof (dO/d.v),_, and(dO/dyLl
(dO/dyL M 5
10 15
w7
for G = 1. (dti/dy),=-,
1.0
1.5
2.0
1.0
1.5
2.0
0.59914 0.55171 0.53482
0.61081 0.55911 0.53793
0.62545 0.56569 0.54151
0.42777 0.45453 0.4673 1
0.42398 0.45371 0.46484
0.423 11 0.45182 0.46266
It may be seen that for fixed M, the rate of heat transfer at the upper plate increases while that at the lower plate decreases with increase in the Hall parameter or. Further for fixed UT, the rate of heat transfer at the upper plate decreases while that at the lower plate increases with increase in h4. 4. CONCLUDING
REMARKS
We find from the foregoing analysis that the inclusion of Hall currents exerts a profound influence on the flow and heat transfer characteristics. It would be interesting to study the flow in the entrance region of the channel, since the buoyancy, Lorentz forces as well as Hall currents will influence the upper and lower boundary layers differently. The acceleration of the central core will inevitably have a stabilizing influence on the system. REFERENCES [l] [2] [3] [4] [5] [6] [7]
A. S. GUPTA, Appl. Sci. Res. (A), 9. 319 (1960). K. R. CRAMER, Trans. ASME, Paper No. 62-HT-22 (1962). K. R. SINGH and T. G. COWLING, Q. J. Mech. appl. Math. 16, 1 (1963). N. RILEY. J. fluid Mech. 18, 577 (1964). A. S. GUPTA. Zeit. ang. Math. Phys. 20, 506 (1969). T. G. COWLING, Magnetohydrodynamics. Interscience, New York (1957). A. SHERMAN and G. W. SUTTON, Magnetohydrodynamics (Edited by A. B. CAMBEL). Northwestern Press, Evanston, Illinois (1%2).
(Receiued 24 March 1975)
University
Pergamon Press.
Printed in Great Britain
HALL EFFECTS ON COMBINED FREE AND FORCED CONVECTIVE HYDROMAGNETIC FLOW THROUGH A CHANNEL B. S. MAZUMDER,
A. S. GUPTA and N. DATTA
Mathematics Department. Indian Institute of Technology, Kharagpur. India
(Communicated
by S. I. PAI)
Abstract-Hydromagnetic free and forced convection in a parallel plate channel permeated by a transverse magnetic field has been considered taking Hall effects into account. When there is a uniform axial temperature variation along the walls. the primary flow shows incipient flow reversal at the upper plate for an increase in temperature along that plate. Similarly flow reversal at the lower plate occurs for a decrease in temperature along that plate. Hall currents are found to exert a destabilizing influence on the flow. The skin-friction for the cross-flow increases with the Hall parameter. The induced magnetic field and the heat transfer characteristics in the flow are also determined.
I. INTRODUCTION
MHD FREE convective flow past a semi-infinite hot vertical plate in the presence of a transverse magnetic field was investigated by Gupta[l], Cramer[2], Singh and Cowling[3] and Riley[4]. These authors found that increase in magnetic field causes a reduction in the local wall shear and heat transfer rate. Combined free and forced convection of an electrically conducting liquid in a horizontal channel permeated by a transverse magnetic field was studied by Gupta[5] who assumed uniform axial temperature variation along the channel walls. This study has some bearing on the problem of cooling of nuclear reactors where liquid metals (which are electrically conducting) are used as coolant. All these studies, however, do not take Hall effects into account. It is well-known (Cowling[6]) that these effects would be important when the strength of the magnetic field is very large. The purpose of the present investigation is to extend the analysis of Gupta[5] to include Hall effects and to study these effects on the flow and heat transfer characteristics. We believe that our study will be of some use in the problem of cooling of nuclear reactors where very strong magnetic fields are used.
2. MATHEMATICAL
PROBLEM
AND ITS SOLUTION
Consider the flow of an electrically conducting liquid between two horizontal parallel plates (distant 2L apart) such that x* and y* axes are along and transverse to the plates and suppose that a strong uniform magnetic field H, acts along y*-axis. At a large distance from the entry section the flow will be fully developed and in the steady state all the physical variables (except pressure) depend on y * only. The equation V*H* = 0 gives H; = constant = H, everywhere in the flow. The continuity equation V.q* = 0 and the no slip conditions at the plates give U* = 0 everywhere, where (u *, u *, \t’*) are the components of the velocity q*. It is well-known that the introduction of Hall effects produces cross-flow (see Sherman and Sutton[7]). The equations of momentum for the fully developed steady flow in rationalised MKS units are
O=
d2u* aP* -i)x*+~dy*2+~,Hnd~*,
dH1
dH: dHT+H!_ H;Fdv* o= -$-PA’-pe dy*
where z * axis is perpendicular
(1)
1’
to x * y * plane and the other symbols have their usual meaning. 285
(2)
286
B. S.MAZUMDER
Maxwell’s equations
etal.
are
along with Ohm’s law including
VxE*=O,
(4)
VxH*=j*
(9
Hall effects (Cowling161) given by
j* +g
0
j*
x
H* = r[E* + peq* X H*],
(6)
where j*, w, T and (T denote the current density, electron Larmor frequency, electron collision time and the electrical conductivity respectively. In writing (6), the ion slip effects arising out of imperfect coupling between ions and neutrals as well as the electron pressure gradient are neglected. Eliminating E* from (4) to (6) and taking the x* and z* components of the resulting equation, we have
d*H: / o1 d’HTz -
du* WHO dy*,
A_
dy*’
dy*
(7)
d2HT d=H: dw* - wr ----T = upeHo dy*, dy@ dy* the y*-component of the equation being identically satisfied. If we assume uniform axial temperature variation along the channel walls, we may take the temperature in the flow as T-z-“=Nx*+c#zJ(y*),
where N is a constant.
Using (9) and the equation
(9) of state
p = poll - P(T - To)1 in (2) and integrating, p*=
(10)
we get
-POgy*+P,g~Nx*y*+pog~
4 dy*-$H9*+HT*)+F’(x*), I
(11)
where po, To and p denote the density and temperature of a reference state and the coefficient of volume expansion respectively. Eliminating p * from (1) and (11) and introducing the dimensionless quantities
(12) we find
d2U+Me!s_Gy= -1,
dy=
(13)
dy
where MZ = /~:Ho~L~u/p,u, Since P, >O, the positive and negative respectively along the channel walls.
values
G = /3gNL4/v2Px. of N correspond
(14) to heating
and cooling
281
Hall effects on convective hydromagnetic flow
Further
(3) reduces to
(15)
Equation
(15) multiplied
by i( =d-
1) when added to (13) gives 2 dC’+M2dh_Gy
=
dy
-
1,
(16)
dy
where u = u + iw, Similarly combining
(17)
(7) and (8) and using (12) we have 1 dU -~-(1 + ion) dy ’
d2h dy’= Elimination
h = H, + iH,.
(18)
of h from (16) and (18) leads to d3U M2 7___.dL!_G,o dy
which upon integration
(l+ iox)
(19)
dy
yields
(20) where C, is a constant
and
M,‘=MZ
(21)
1 + iwr’
The no-slip conditions
at the plates y = +l are U(k1) = 0
and since the plates are assumed electrically are
non-conducting,
(22) the magnetic boundary
conditions
h(+l)=O. Solutions
U(y)=
of (16) and (20) satisfying
l
Ml sinh M,
(23)
(22) and (23) are
* (cash M, -cash
M,y) +
G - (sinh Ml y - y sinh M,), Ml2 sinh Ml (24)
h(y)
= +[;(y2
- 1) + M,
si!hM,(cosh
+M2 si!h M I
(sinh M,y - y sinh Ml).
Assuming Ml = a + i/3 in (21) and substituting
1JE.S Vol. 14. No. 3-D
M, -cd
1
MIY 1
(25)
in (24) and (25), we get upon separating (24) and
288
B. S. MAZUMDER
(25)
into real and imaginary
u = (a,z!
et al.
parts and using (17)
azzJ . [al(cosh (Ycos /3 -cash
ay cos py) + az(sinh (Ysin p -- sinh (mysin By)]
+ (a,~~a,~~[ a3 .( sm ’ h ay cos fly - y sinh cy cos p) + a,(cosh ay sin py - y sin p . cash a)], (26) w = (a,2:
,_+n,(sinh 7
+ to,2fa4+
(Ysin /3 - sinh ay sin py) - a:(cosh (Ycos p - cash ay cos py )]
a3 .( cosh ay sin py - y cash a sin p) - a,(sinh ay cos py - y sinh (Ycos p)], (27)
G H.\ =M’
1 3
[
,(Y--
1
{a,(cosh (Ycos ,B -cash
I)+ta,z+a2zJ
+ a?(sinh LYsin p - sinh a? sin oy
ay cos py)
)}I
+ sinh cy cos P(sinh cuy cos by - y sinh cy cos p) + cash (Ysin P(cosh ay sin fly - y cash (Ysin p) M’(sinh’ cy cos’ p + cash’ (Ysin’ 0) (28) H: = Mz(~,?+ ,zZ)[a4sinh
(Ysin p - sinh ay sin py) - &(cosh (Ycos
p
-cash
av
cos
py)]
+ sinh (Ycos /3(cosh cry sin by - y cash cx sin p) - cash (Ysin p(sinh ay cos /3y - y sinh (Ycos p) M’(sinh’ (Yco? /I + coshZ (Ysin’ p) (2% where u,=cusinhcucosp-Pcoshasinp, nz = p sinh (Ycos p + (Ycash (Ysin p. a?=(Cy’-P’)sinhacosP-2apcoshasinP. a,=2a/3 ff=
sinha
cos/?+(a’-p’)cosha
M[l +(I +o’+~]“’ [2( I + u2T2)y
sinp,
’ (30)
The dimensionless
shear stress at the upper and lower plates, respectively
are given by
(31)
(32) where [(Y’sinhacosha-P3sinPcosP+ap’coshcrsinh(Y-cu’PsinPcosP F(a,p)=(n’ya 3 4 2)
Further the dimensionless
shear stress for the secondary
flow at the upper plate is the same as
Hall effects on convective hydromagnetic
289
flow
that at the lower plate and is given by G [2@(sinh2 (Ycos’ /3 + cash’ (Ysir? p) Y=t, = (a: + a:) (33) When buoyancy forces are absent (G = 0), eqs (31) and (32) show that the skin-friction for the primary flow neither vanishes on the upper nor on the lower plate. However, when G = 0, the skin-friction for the cross-flow vanishes at both the plates y = + 1. Thus we arrive at the interesting conclusion that in the absence of the buoyancy forces, the cross-flow due to Hall currents shows incipient flow reversal although the primary flow does not. On the other hand, eqn (31) shows that in the presence of buoyancy forces the primary flow at the upper plate y = 1 shows incipient flow reversal when G = G,,,, such that (a: + ad’)
Gcrit = [(cY’+ /~‘)(cI sinh (Ycash (Y- /3 sin p cos 0) - (cu’- /3’)(sinhz (Ycos’ /3 + cash’ (Ysin’ p)] (34) We have computed GCri,for several values of the Hall parameter OT with A4 = 10 as shown in the table below. Table 1. M = 10
0.50
1.oo
1.50 2.00 2.50 3.00
10.7633 IO.0132 9.2268 8.5322 7.9438 74468
This shows that the critical value of G is positive and the definition of G in (14) shows that incipient primary flow reversal occurs at the upper plate when there is an increase in temperature along that plate (G > 0). Since G,,,, decreases with increase in the Hall parameter, it follows that Hall currents exert a destabilizing influence on the flow. Further, it can be seen from eqn (32) that the primary flow at the lower plate shows incipient flow reversal when G = G&, where GA,,,= - Gcrit,
(35)
G,,i, being given by (34). Thus flow reversal at the lower plate occurs when there is a decrease in temperature along that plate (G < 0). We have also computed from (33) the skin-friction at the plates y = ? 1 for the cross-flow w(y) when G # 0 and the results are shown in the following table. Table 2. M = 10, G = 1 (dw/dy),==, 0.50
I .oo 1.50 2.00 2.50 3.00
-0.01929 -0.03551 -0.04836 -0.05862 -0.06701 -0.07401
This shows that for a fixed Hartmann number, the magnitude of the skin-friction for the cross-flow increases with the Hall parameter. We have plotted u(y) and -w(y) in the case when the flow does not separate for several values of the Hall parameter or with G = 1.0 and M = 5,10. Figure 1 shows that for fixed M,
B. S. MAZUMDER
et ul.
‘3Or
-
M=5*0
----
,1=10.0
I
-.6
I
-‘4
L
-,2
I
0.0
I
I
I
I
.2
.4
.6
‘6
I.0
Y-
Fig. 1. Profiles of non-dimensional
primary velocity u(r) for G = 1.0.
u(y) increases with increase in WT,while Fig. 2 shows a similar result for the cross-flow -w(y). The flattening effect of the magnetic field on the velocity profile is clearly discernible in these figures. Having found U(y), h(y) can be obtained from (16) after using the boundary conditions (23). When h(y) is separated into real and imaginary parts, H, and Hz can be found (see (28) and (29)). We have plotted H,(y) and K(y) in Figs. 3 and 4 for several values of w7 with A4 = 10, and G = 1 and 5. These figures indicate that the disposition of the induced magnetic field is in accord with the pulling of the lines of force by the convection in the channel. 3. HEAT
The equation
of energy including
TRANSFER
viscous and ohmic dissipation
is
u*K=KaZT+J_[(~>‘+(~>‘]+~[(~s+(~~], dy
8.x”
*’
PC,,
‘IO -
t
s
‘08-
FO,.
Fig. 2. Profiles of non-dimensional
secondary velocity w(y) for G = 1.0.
(36)
Hall effects on convective hydromagnetic
291
flow
-G = I.0 ----wz5.0
-.025 t
Fig. 3. Profiles of the non-dimensional
induced magnetic field H, for M = 10.0.
-
G =
I.0
----
1,
5’0
=
0-O
-.I N
t 0
x
-a2 --_-___----___
: -,3
-'4
!
-10
-.*
I
-6
I
I
I
0.0
-.2
-“j
1
I
1
1
,2
'4
'6
'8
1
1'0
T-
Fig. 4. Profiles of non-dimensional
induced magnetic field H2 for M = 10.0.
where K is the thermal diffusivity, c, is the specific heat and the temperature T is given by (9). Using (9) and (12), the above equation can be put in dimensionless form as
(37) where Pr is the Prandtl number
u/K and
K,=“3p,
c,KNL3’
L
t9= NLP,
(38)
292
B. S. MAZUMDER
et al.
and the over bar denotes a complex conjugate. As for the temperature boundary conditions we take the reference temperature T,, in (9) in such a manner that the temperature of the lower wall y = - 1 is T, + Nx*. Hence (38) implies that l9(-l)=O along with
(40) where iV, may be regarded as the wall temperature parameter. The temperature distribution B(y) can now be obtained by integrating (37) after substituting the expressions for U(y), h(y) and u (y ) from (24), (25) and (26) and making use of the boundary conditions (39) and (40). We omit the details of calculation as they are quite cumbersome. We present below the values of the dimensionless heat transfer coefficient d0/dy at the plates y = * 1 for several values of wr and M forG=l.O,Pr=landK,=0.5. Table 3. Valuesof (dO/d.v),_, and(dO/dyLl
(dO/dyL M 5
10 15
w7
for G = 1. (dti/dy),=-,
1.0
1.5
2.0
1.0
1.5
2.0
0.59914 0.55171 0.53482
0.61081 0.55911 0.53793
0.62545 0.56569 0.54151
0.42777 0.45453 0.4673 1
0.42398 0.45371 0.46484
0.423 11 0.45182 0.46266
It may be seen that for fixed M, the rate of heat transfer at the upper plate increases while that at the lower plate decreases with increase in the Hall parameter or. Further for fixed UT, the rate of heat transfer at the upper plate decreases while that at the lower plate increases with increase in h4. 4. CONCLUDING
REMARKS
We find from the foregoing analysis that the inclusion of Hall currents exerts a profound influence on the flow and heat transfer characteristics. It would be interesting to study the flow in the entrance region of the channel, since the buoyancy, Lorentz forces as well as Hall currents will influence the upper and lower boundary layers differently. The acceleration of the central core will inevitably have a stabilizing influence on the system. REFERENCES [l] [2] [3] [4] [5] [6] [7]
A. S. GUPTA, Appl. Sci. Res. (A), 9. 319 (1960). K. R. CRAMER, Trans. ASME, Paper No. 62-HT-22 (1962). K. R. SINGH and T. G. COWLING, Q. J. Mech. appl. Math. 16, 1 (1963). N. RILEY. J. fluid Mech. 18, 577 (1964). A. S. GUPTA. Zeit. ang. Math. Phys. 20, 506 (1969). T. G. COWLING, Magnetohydrodynamics. Interscience, New York (1957). A. SHERMAN and G. W. SUTTON, Magnetohydrodynamics (Edited by A. B. CAMBEL). Northwestern Press, Evanston, Illinois (1%2).
(Receiued 24 March 1975)
University