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Hydromagnetic convection at a heated semi-infinite vertical plate
Int. J. Engng Sci. Vol. 25, No. 1, pp. 27 35, 1987 Printed in Great Britain
0020 7225/87 $3.00 + 0.00 Pergamon Journals Lid
H Y D R O M A G N E T I C C O N V E C T I O N AT A HEATED SEMIINFINITE VERTICAL PLATE K. V A J R A V E L U Department of Mathematics, University of Central Florida, Orlando, FL 32816, U.S.A. Abstract Numerical results are presented for the transient and steady-state velocity field, temperature field, and heat transfer characteristics. These results are obtained by solving the highly nonlinear partial differentialequations describing the conservation of mass, momentum and energy (with variable fluid properties) by an explicit finite-differencemethod in time-dependent form. Two cases are studied, namely the hydromagnetic (HM) case and the hydrodynamic (HD) case. Values of the velocity components u and v (absolute) in the HD case are larger than those in the HM case. Quite opposite is the phenomenon in the case of the temperature field 0. Further, the velocity and temperature fields are more prominent in the presence of heat sources than in their absence. Numerical results indicate that the presence of magnetic field delays attainment of the steady-state condition. Furthermore, in the hydromagnetic case, it is observed that the variable fluid-property has almost negligible effect on the flow and heat transfer characteristics, which is contrary to that in the hydrodynamic case.
1. I N T R O D U C T I O N F r o m a technological point of view, the free convection study of fluids with known physical properties is always important, for it can reveal hitherto unknown properties of fluids of practical interest. In buoyancy-driven flows, it is usual to assume that all properties of the fluid are constant except the density difference in the body-force term. This term is assumed to depend linearly on the temperature difference. G r a y and Giorgini [1], in a detailed study of the equations governing such flows, concluded that these approximations are valid only if the motive temperature difference is small. Sparrow [2] and Sparrow and Gregg [3] treated perfect gases for different property laws. Using an integral method, Plapp [4] obtained solutions for oil. H a r a [5] obtained solutions for air by a successive approximation method. Minkowycz and Sparrow [6] studied variable property effects in steam. Piau [7, 8] considered the effect of the temperature variation of viscosity and coefficient of thermal expansion for water and viscous liquids. The variations were taken to be linear functions of temperature when the temperature differences are of the order of 20°C. Barrow and Sitharamarao [9] studied the effect of/~ for an isothermal surface in water. Brown [10] extended the work of Barrow and Sitharamarao by including the variation in density with temperature. Ackroyd [11] assessed the effects of the viscous dissipation and pressure terms in the energy equation for perfect gases and liquids for small differences in temperature. Nishikawa and Ito [12] analyzed the effect of variable properties for supercritical water and carbon dioxide. Ito et al. [13] treated a uniform flux surface in oil and supercritical carbon dioxide, using an integral method. Carey and Mollendorf [14] extended Piau's work to various liquids, considering only the variation of viscosity. Clarke [15] considered the effects of variable properties for a vertical isothermal surface with transpiration for a perfect gas. Shaukatullah and Gebhart [16] studied the effect of variable properties on laminar natural-convection boundary-layer flow over a vertical isothermal surface in water. Recently, Vajravelu [17] studied in detail the problem of natural convection at a heated vertical plate with variable fluid properties. In the work [17], the linear temperature dependence of viscosity, thermal conductivity, and density (in the buoyancy-force term) were considered as: p = pi[1 -- f l i ( T - Ti)], # =/~i[1 + a i ( T - Ti)], 27
28
K. VAJRAVELU
k = kill + b i ( T -
Ti)],
where T~ is the initial temperature of the fluid, fl = -(1/p)c~p/c~T, a = (1/l~)C~l~/c~T, and b = (1/k)c3k/c3T. The flow and heat transfer characteristics were found to depend on the dimensionless numbers A [ = ai(T, - Ti)] and B [ = bi(T, - Ti)], which arise, respectively, from the temperature dependence of viscosity and thermal conductivity, in addition to the other usual parameters Pr, the Prandtl number, E, the Eckert number, and ~, the heat source/sink parameter. The contributions of the dimensionless numbers A, B, E and ~ were found to be quite significant. Inspired by this hydrodynamic analysis of the unsteady natural convection problem, the author strongly feels that its hydromagnetic extension would be interesting and have useful applications. The main objective of this paper is to make investigation of hydromagnetic natural convection at a heated vertical plate with variable fluid properties and to solve the highly nonlinear partial-differential equations for the dependent variables u, v and 0, as a function of x, y and t and, in particular, to obtain the steady-state solution, if it exists. The governing equations of the problem are solved numerically, subject to the relevant initial and boundary conditions for the various values of the parameters, for both hydromagnetic and hydrodynamic cases. A comparison is made between the hydromagnetic and hydrodynamic solutions. The contributions of the Hartmann number M z, in particular, and those of the parameters A, B, E and ~, in general, to the flow and heat transfer characteristics are found to be quite significant. 2. F O R M U L A T I O N OF THE PROBLEM Consider a heated vertical plate in an infinite, incompressible, and conducting viscous fluid that is initially cold and at rest (see Fig. 1). The following assumptions are made in the analysis of the problem:
/
/
¢ ~T=~ X
/ /
Fluid (At rest, with T=T i , a t T = O )
/ / / / / / / / / /
///////////////////////, ~- y
T=T i
Fig. 1. Flow configuration.
(i) the flow is laminar, unsteady, and two-dimensional; (ii) the magnetic Reynolds number is small; (iii) the work done by pressure is sufficientl3; small in comparison with the heat flow by conduction and the viscous work (which is justified for moderate and large temperature differences, see Ackroyd I-11]); and (iv) the volumetric heat source/sink term in the energy equation is taken as Q = Q * ( T - TO. Under these assumptions the equations of momentum, mass, and energy in the fluid at all times are described approximately by
Hydromagnetic convection at a heated semi-infinite vertical plate
\~
+
ax + V~
= -pg-
B2aoU + f f ~
29
+ # ~y2 ,
(1)
dU c~V a X + ~Y = 0,
(2)
(#T 8T vdT~ ['aU'~ akaT k82T pCp ~z + U ~ + # y ) = Q * ( r - Ti) + # k ~ ) + #Y aY + a r 2'
(3)
where B o is the transverse magnetic field, ao the coefficient of electric conductivity, and the other symbols have their usual meanings. The initial and b o u n d a r y conditions relevant to the p r o b l e m are U=
V=0, T= T~atX=0,
U=V=O,T= U ~O,T~
Twat Y=O,
(4)
Tias Y ~ oo,
U = V = O , T = T~atz =O. Defining nondimensional variables t = "c(gfliAT)2/3/vl/3,
X = X(gfliAT/v2) 1/3, y = y(gfliAT/v2) 1/3, u = U(vgfliAT) 1/3, V = V(vgfliAT) 1/3, 0 = (T - T0/A T, equations (1)-(3) and the conditions (4) can be written as:
au
c~u c3u au ao AO a2u + U~x + V~y = 0 - MZu + A~y~y + (1 + )~y2,
(5)
au {3y ~xx + ~yy = o,
(6)
pr(C30 t30 ao + \ a t + U~xx + V~y = so + PrE(1 + ao)k Y~
\if-y/
+ (1 +
)~y2,
(7)
u = v = 0,0 = 0 a t x = 0,
u=v=O,O=
l a t y = 0,
u ~O,O-*Oas y ~ oo, u =
v =
0,0
=
0art
=
0,
where M 2 = B~ao vl/3/p(gfliA T) 2/3, the H a r t m a n n number, A = aiA T, the dimensionless number,
B = biAT, the nondimensional number, E = (vgfli AT)Z/3/CoAT, the Eckert number,
(8)
30
K. VAJRAVELU
Pr = #i Cp/ki, the Prandtl number, O~ = Q * v 4 / 3 / k i ( g f l l
A T ) 2/3,
the heat source/sink parameter,
and AT = Tw - T~, the temperature difference. The heat transfer coefficient (or Nusselt number) and the skin friction coefficient at the wall are defined, in nondimensional form, as N u = - h v / k i A T ( v g f l i A T ) 1/a = (1 +
\SY/y=o'
and z* = f*/pi(vgfli A T ) 2/3 = (1 +
\SYA = o"
Let the analysis be restricted to the case of a fluid whose Prandtl number is 0.733. For the temperature of the plate Tw and the fluid temperature taken to be 25°C (35°C, 45°C, Table 1. Fluid Properties (where Pr = 0.733) at Ti = 15°C
Tw (°c) 25 35 45 55
A 0.028 0.056 0.084 0.112
0.024 0.048 0.072 0.096
55°C) and 15°C, respectively, Table 1 (see Batchelor 1-18]) represents the values of the nondimensional parameters A, B and Pr. The numerical results for the flow and heat transfer characteristics are obtained for these values of A, B and Pr only. 3. M E T H O D OF S O L U T I O N The three simultaneous nonlinear partial differential equations (5)-(7), with conditions (8), are now to be solved for the dependent variables u, v and 0 as a function of x, y and t, and in particular, obtain the steady-state solution, if it exists. Although the primary goal is to obtain the steady-state solution, for which both 8u/St and t30/St are zero, the solution is obtained by considering the corresponding unsteady-state problem, already formulated. Successive steps in time can then be regarded as successive approximations toward the steady-state solution. The solution is obtained by numerical integration, and the integrations are carried out on the time-dependent form of the equations by an explicit finite-difference method. The space under investigation is restricted to finite dimensions. A plate of height Xmax = 100 and Ymax= 25 (as corresponding to y = oo) are considered. The velocity field, temperature field, Nusselt number, and skin friction are calculated for the values of A, B and Pr given in Table 1, with M 2 = 0.2, 0.4, 0.6, E = 0.02, and ~ = 0.1, 0, - 0 . 1 for a 100 x 100 grid at t = 0.5, 1.0, 1.5, 2.0. . . . . 80.0, and are summarized in Figs 2-5 for values of t = 10, 20, 30,..., 80. To compare the results of the hydromagnetic case with its hydrodynamic counterpart, numerical values of u, v, O, N u and z* for the hydrodynamic case are also obtained by putting M z = 0 and giving the same set of values to the other parameters. The numerical values thus obtained are presented in Figs 2-5. Examination of the complete results so presented reveal little change in u, v and 0 after t = 50 for all values of the parameters. It should be mentioned here that the results of Vajravelu 1-17] for fluid velocity and fluid temperature are a special case of this analysis for M 2 equal to zero. The deviation of the wall heat transfer parameter of the constant fluid property (CFP) case from the variable fluid property (VFP) counterpart is defined as Nu(VFP) -- Nu(CFP) Nu(CFP)
Hydromagnetic convection at a heated semi-infinite vertical plate
31
M2:0.4
M2= 0
t : 30&40 t=2~
~
~
--t= 40&50 t : 30
m
0
12.5
25.0
0
12.5
Y
25.0
Y (a)
12
M2= 0
tt 30&40
M2=0.4 - ~ t = 4 0 & 5 0 ~ . . ~7
t=30~
!
!
I
50
100 0 X
50 X
!
100
(b)
Fig. 2. Transient and steady-state velocity Profiles for Pr = 0.733, E = 0.02, ~ = 0.1 (with T = 35°C and T~= 15 Ct at (a) x = 20 and (b) y = 5: ( ) transient; (.... ) steady-state.
a n d the percentage, a b b r e v i a t e d as N u D e , is e v a l u a t e d numerically. S o m e of the qualitatively interesting percentages of the d e v i a t i o n of the heat transfer coefficient a n d the skin friction, ~p are shown in T a b l e 2. 4. D I S C U S S I O N O F T H E R E S U L T S E q u a t i o n s (5) (7), subject to c o n d i t i o n s (8), are solved by the finite-difference m e t h o d . The results (with Tw = 35~'C a n d T~ = 15°C) are presented in Figs 2 - 5 for P r = 0.733, E = 0.02 and ~ = 0.! for two cases: (1) the h y d r o d y n a m i c case, a n d (2) the h y d r o m a g n e t i c case. F o r b o t h cases the transient a n d s t e a d y - s t a t e values of the velocity u for M 2 = 0, 0.4 are s u m m a r i z e d in Fig. 2a with fixed x a n d in Fig. 2b with fixed y. F r o m Fig. 2 it can be o b s e r v e d that the values of u have decreased c o n s i d e r a b l y in the presence of m a g n e t i c field. This q u a l i t a t i v e l y agrees with expectations, since the m a g n e t i c field exerts a r e t a r d i n g force on the free-convection flow. F r o m n u m e r i c a l calculations, it is o b s e r v e d that, in b o t h the h y d r o d y n a m i c a n d h y d r o m a g n e t i c cases (with a n d w i t h o u t variable fluid-properties), the values of u have increased c o n s i d e r a b l y in the presence of heat sources. T h e o p p o s i t e p h e n o m e n o n is o b s e r v e d in the presence of heat sinks. H o w e v e r , u has larger values a w a y ES 2 5 : 1 - C
32
K. V A J R A V E L U
1.2
M 2= 0.4
M2=O
o
0.8 t=20
-V
0.4
!
I
12.5
25.0
0
i
12 5
Y
25.0 Y
(a) 0.9
M2=O
M 2 = 0.4
/~..
,--'-
t=40&50
t = 30&40
0.6
tz30 / / , ~
-V
t = 20
0.3 ¢
50
100
0
50
100
X
(b) Fig. 3. T r a n s i e n t and steady-state velocity Profiles for Pr = 0.733, E = 0.02, ~ = 0.1 (with T = 35°C and T i = 15°C) at (a) x = 20 a n d (b) y = 5: ( -) transient; (. . . . ) steady-state.
from the plate in the V F P case than in the C F P case. The reverse is the case near the plate. However, at all times, u has m a x i m u m value only in the C F P case. Observations from Fig. 2 show that u increases monotonically, as nondimensional time increases, up to a certain stage, then decreases slightly, and thereafter remains unchanged. That is, in the presence of magnetic field at t = 50 and in its absence at t = 40, the velocity u reaches the steady-state condition. Physically, it means that, in the case of imposed magnetic field, the time required for the flow to reach steady-state is increased. Figure 3 describes the behavior of velocity component v perpendicular to the plate. The negative values of v at all points in the fluid indicate that v is directed toward the plate. As in the case of u, the values of v (absolute) increase monotonically up to t = 50 in the presence of magnetic field and up to t = 40 in its absence, and thereafter remain unchanged. These steady-state results reveal that v (absolute) is enhanced in the presence of heat sources. However, v (absolute) decreases in the presence of magnetic field. This phenomenon is qualitatively true in both the C F P and V F P cases. Transient and steady-state temperature profiles are shown in Figs 4a and 4b for M z = O, 0,4 and for x and y fixed, respectively. The temperature increases monotonically and reaches the steady-state condition at t = 40 in the presence of magnetic field and at t = 30
H y d r o m a g n e t i c c o n v e c t i o n at a h e a t e d semi-infinite vertical p l a t e
'2 I
M2=O
M2 = 0.4 ~/~t
0.8
33
= 30 &40
t = 20&30
10
!
0.4
12.5
25.0
0
12.5
25.0 Y
Y (a ) 4.5
M2=
M2 = 0.4
0
=20~
t = 30&40 3.0
t = 20&30
t
t"lO 1.5
f
/ 0
0
!
I
50
100
0
I
I
5O
100 x
x
(b) Fig. 4. T r a n s i e n t a n d s t e a d y - s t a t e t e m p e r a t u r e Profiles for Pr = 0.733, E = 0.02, ~ = 0.1 (with T = 35~'C a n d T~ = 15°C) at ( a ) x = 20 a n d ( b ) y = 5:( . . . . . . ) t r a n s i e n t ; ( . . . . ) s t e a d y - s t a t e .
in its absence. F r o m the steady-state value of 0 it can be seen that, unlike in the case of u, the values of 0 are considerably enhanced in the presence of magnetic field. In the hydromagnetic case, the steady-state temperature 0 increases from its value 1 at y -- 0 to a maximum of 1.164 around y = 3.75 and then decreases steadily to its value 0 as y -~ ~ . But, in the hydrodynamic case, the temperature 0 decreases steadily from its value of 1 at y = 0 to its value 0 as y ~ ~ . Figures 5a and 5b, respectively, describe the behavior of the skin friction and rate of heat transfer (Nusselt number) at the plate. From Fig. 5a it is clear that the skin friction at the plate decreases with the magnetic parameter M z (the H a r t m a n n number); however, the skin friction increases with the dimensionless coordinate x. From numerical results, it is further observed that, for all values of ~, the skin friction in the V F P case exceeds that in the C F P case. From Fig. 5b, it is evident that, in the presence of magnetic field, the heat transfer parameter is positive; physically, this means that there is heat flow from the fluid to the plate. Opposite is the case (at x = 20) in the absence of magnetic field. The rate of heat transfer parameter is an increasing function of M 2, t and ~. Further (from numerical results it is observed that), in the presence of heat sources the values of the heat transfer parameter in the V F P case in general exceed those in the C F P case. This behavior
34
K. V A J R A V E L U
X=20
x=6o f
t = 3o&4o
t = 30&40 2 t:20 T*
-"<2__ I
I
0.2
0.4
0
I
I
0.2
0.4
M2
M2
(a) 0.6
=20
X:60
t:30&40
S
0.3 Nu
-0.3 0
0.2
0.4
I 0.2
0
I 0.4 M2
M2
(b) Fig. 5. F l o w a n d h e a t t r a n s f e r c h a r a c t e r i s t i c s at the p l a t e for Pr = 0.733, E = 0.02, ~ = 0.1 (with T = 35°C a n d T~ = 15°C): ( a ) s k i n friction a n d ( b ) N u s s e l t n u m b e r : ( ) t r a n s i e n t ; (. . . . ) s t e a d y - s t a t e .
Table 2. Percentage differences of Nusseh number and skin friction in the C F P case and the VFP case at steady state (T~ = 15°C and T, = 35°C), when Pr = 0.733, E = 0.02, and :~ = 0.l
NUop x/M 2
80 70 60 50 40
r~p
0
0.2
0.4
0
0.2
0.4
4.218 4.372 5.798 6.412 14.781
1.283 1.482 2.087 2.598 3.514
0.066 0.082 0.560 1.256 2.060
3.422 3.644 3.867 4.141 4.435
2.201 2.617 2.953 3.370 3.832
1.084 1.523 2.063 2.600 3.267
is reversed in the presence of heat sinks. Finally, it may be said that the magnetic force is limited to only retardation; thus, separation will never occur, and the heat transfer is never limited (as is verified by the asymptotic trend exhibited in Fig. 5). For possible experimental verification, the percentage differences of the skin friction and heat transfer parameter in the C F P case from those in the VFP case are presented in Table 2.
Hydromagnetic convection at a heated semi-infinite vertical plate
35
5. C O N C L U S I O N S Some of the interesting conclusions of the problem are summarized as follows: 1. The fluid velocity components u and v (absolute) have larger values in the absence of magnetic field than in its presence. But this phenomenon is reversed in the case of fluid temperature 0. 2. In the presence of magnetic field, the fluid velocity components u, v (absolute) and the fluid temperature 0 have larger values away from the plate in the VFP case than in the C F P case. The opposite is true near the plate. 3. In both the hydromagnetic and hydrodynamic cases, u, v (absolute) and 0 increase (decrease) in the presence of heat sources (sinks). But viscous dissipation increases the values of u, v (absolute) and 0. 4. In the absence of magnetic field, the velocity component, v, perpendicular to the plate is always directed toward the plate. This is true even in the presence of magnetic field. 5. In the hydromagnetic case, the steady-state temperature 0 increases from its value 1 at y = 0 to a maximum of 1.164 around y = 3.75 and then decreases steadily to its value 0 as y --* oe. But, in the hydrodynamic case, the temperature decreases steadily from its value of 1 at y = 0 to its value 0 as y --, c~. 6. Percentage-wise, the deviations of the rate of heat transfer in the C F P case from its VFP counterpart decrease as the magnetic parameter M E (the Hartmann number) takes increasing values. That is, the effects of variable fluid properties are negligible in the presence of a strong magnetic field. 7. Magnetic field plays an important role in delaying the velocity and temperature reaching the steady-state condition. Acknowledgements--The author is greatly indebted to Professor Andreas Acrivos for valuable comments and to the referee for constructive criticisms which led to definite improvements in the paper.
REFERENCES [1] D. D. GRAY and A. GIORGINI, Int. J. Heat Mass Transfer 19, 545 (1976). [2] E. M. SPARROW, Free convection with variable properties and variable wall temperature, Ph.D. thesis, Harvard Univ., Cambridge, Mass. (1956). [3] E. M. SPARROW and J. L. GREGG, Trans. ASME 80, 879 (1958). [4] J. E. PLAPP, Laminar free convection with variable fluid properties, Part II. Ph.D. thesis, California Inst. of Technology, Pasadena, Calif. (1957). [5] T. HARA, Bull. JSME 1, 251 (1958). [6] W. J. MINKOWYCZ and E. M. SPARROW, Int. J. Heat Mass Transfer 9, 1145 (1966). [7] J.-M. PIAU, C. R. Acad. Sci. Ser. A 271, 953 (1970). [8] J.-M. PIAU, Int. J. Heat Mass Transfer 17, 465 (1974). [9] H. BARROW and T. L. SITHARAMARAO, Br. Chem. Eng. 16, 704 (1971). [10] A. BROWN, Trans. ASME, J. Heat Transfer 97, 133 (1975). [11] J. A. D. ACKROYD, J. Fluid Mech. 62, 677 (1974). [12] K. NISHIKAWA and T. ITO, Int. J. Heat Mass Transfer 12, 1449 (1969). [13] T. ITO, H. YAMASHITA and K. NISHIKAWA, Proc. 6th Int. Heat Transfer Conf., Tokyo 3, 49 (1974). [14] V. P. CAREY and J. C. MOLLENDORF, Proc. 6th Int. Heat Transfer Conf., Toronto 2, 221 (1978). [15] J. F. CLARKE, J. Fluid Mech. 57, 45 (1973). [16] H. SHAUKATULLAH and B. GEBHART, Numer. Heat Transfer 2, 215 (1979). [17] K. VAJRAVELU, Numer. Heat Transfer 3, 345 (1980). [18] G. K. BATCHELOR, An Introduction to Fluid Dynamics, Cambridge Univ. Press, London (1967). (Revised version received 4 February 1986)
0020 7225/87 $3.00 + 0.00 Pergamon Journals Lid
H Y D R O M A G N E T I C C O N V E C T I O N AT A HEATED SEMIINFINITE VERTICAL PLATE K. V A J R A V E L U Department of Mathematics, University of Central Florida, Orlando, FL 32816, U.S.A. Abstract Numerical results are presented for the transient and steady-state velocity field, temperature field, and heat transfer characteristics. These results are obtained by solving the highly nonlinear partial differentialequations describing the conservation of mass, momentum and energy (with variable fluid properties) by an explicit finite-differencemethod in time-dependent form. Two cases are studied, namely the hydromagnetic (HM) case and the hydrodynamic (HD) case. Values of the velocity components u and v (absolute) in the HD case are larger than those in the HM case. Quite opposite is the phenomenon in the case of the temperature field 0. Further, the velocity and temperature fields are more prominent in the presence of heat sources than in their absence. Numerical results indicate that the presence of magnetic field delays attainment of the steady-state condition. Furthermore, in the hydromagnetic case, it is observed that the variable fluid-property has almost negligible effect on the flow and heat transfer characteristics, which is contrary to that in the hydrodynamic case.
1. I N T R O D U C T I O N F r o m a technological point of view, the free convection study of fluids with known physical properties is always important, for it can reveal hitherto unknown properties of fluids of practical interest. In buoyancy-driven flows, it is usual to assume that all properties of the fluid are constant except the density difference in the body-force term. This term is assumed to depend linearly on the temperature difference. G r a y and Giorgini [1], in a detailed study of the equations governing such flows, concluded that these approximations are valid only if the motive temperature difference is small. Sparrow [2] and Sparrow and Gregg [3] treated perfect gases for different property laws. Using an integral method, Plapp [4] obtained solutions for oil. H a r a [5] obtained solutions for air by a successive approximation method. Minkowycz and Sparrow [6] studied variable property effects in steam. Piau [7, 8] considered the effect of the temperature variation of viscosity and coefficient of thermal expansion for water and viscous liquids. The variations were taken to be linear functions of temperature when the temperature differences are of the order of 20°C. Barrow and Sitharamarao [9] studied the effect of/~ for an isothermal surface in water. Brown [10] extended the work of Barrow and Sitharamarao by including the variation in density with temperature. Ackroyd [11] assessed the effects of the viscous dissipation and pressure terms in the energy equation for perfect gases and liquids for small differences in temperature. Nishikawa and Ito [12] analyzed the effect of variable properties for supercritical water and carbon dioxide. Ito et al. [13] treated a uniform flux surface in oil and supercritical carbon dioxide, using an integral method. Carey and Mollendorf [14] extended Piau's work to various liquids, considering only the variation of viscosity. Clarke [15] considered the effects of variable properties for a vertical isothermal surface with transpiration for a perfect gas. Shaukatullah and Gebhart [16] studied the effect of variable properties on laminar natural-convection boundary-layer flow over a vertical isothermal surface in water. Recently, Vajravelu [17] studied in detail the problem of natural convection at a heated vertical plate with variable fluid properties. In the work [17], the linear temperature dependence of viscosity, thermal conductivity, and density (in the buoyancy-force term) were considered as: p = pi[1 -- f l i ( T - Ti)], # =/~i[1 + a i ( T - Ti)], 27
28
K. VAJRAVELU
k = kill + b i ( T -
Ti)],
where T~ is the initial temperature of the fluid, fl = -(1/p)c~p/c~T, a = (1/l~)C~l~/c~T, and b = (1/k)c3k/c3T. The flow and heat transfer characteristics were found to depend on the dimensionless numbers A [ = ai(T, - Ti)] and B [ = bi(T, - Ti)], which arise, respectively, from the temperature dependence of viscosity and thermal conductivity, in addition to the other usual parameters Pr, the Prandtl number, E, the Eckert number, and ~, the heat source/sink parameter. The contributions of the dimensionless numbers A, B, E and ~ were found to be quite significant. Inspired by this hydrodynamic analysis of the unsteady natural convection problem, the author strongly feels that its hydromagnetic extension would be interesting and have useful applications. The main objective of this paper is to make investigation of hydromagnetic natural convection at a heated vertical plate with variable fluid properties and to solve the highly nonlinear partial-differential equations for the dependent variables u, v and 0, as a function of x, y and t and, in particular, to obtain the steady-state solution, if it exists. The governing equations of the problem are solved numerically, subject to the relevant initial and boundary conditions for the various values of the parameters, for both hydromagnetic and hydrodynamic cases. A comparison is made between the hydromagnetic and hydrodynamic solutions. The contributions of the Hartmann number M z, in particular, and those of the parameters A, B, E and ~, in general, to the flow and heat transfer characteristics are found to be quite significant. 2. F O R M U L A T I O N OF THE PROBLEM Consider a heated vertical plate in an infinite, incompressible, and conducting viscous fluid that is initially cold and at rest (see Fig. 1). The following assumptions are made in the analysis of the problem:
/
/
¢ ~T=~ X
/ /
Fluid (At rest, with T=T i , a t T = O )
/ / / / / / / / / /
///////////////////////, ~- y
T=T i
Fig. 1. Flow configuration.
(i) the flow is laminar, unsteady, and two-dimensional; (ii) the magnetic Reynolds number is small; (iii) the work done by pressure is sufficientl3; small in comparison with the heat flow by conduction and the viscous work (which is justified for moderate and large temperature differences, see Ackroyd I-11]); and (iv) the volumetric heat source/sink term in the energy equation is taken as Q = Q * ( T - TO. Under these assumptions the equations of momentum, mass, and energy in the fluid at all times are described approximately by
Hydromagnetic convection at a heated semi-infinite vertical plate
\~
+
ax + V~
= -pg-
B2aoU + f f ~
29
+ # ~y2 ,
(1)
dU c~V a X + ~Y = 0,
(2)
(#T 8T vdT~ ['aU'~ akaT k82T pCp ~z + U ~ + # y ) = Q * ( r - Ti) + # k ~ ) + #Y aY + a r 2'
(3)
where B o is the transverse magnetic field, ao the coefficient of electric conductivity, and the other symbols have their usual meanings. The initial and b o u n d a r y conditions relevant to the p r o b l e m are U=
V=0, T= T~atX=0,
U=V=O,T= U ~O,T~
Twat Y=O,
(4)
Tias Y ~ oo,
U = V = O , T = T~atz =O. Defining nondimensional variables t = "c(gfliAT)2/3/vl/3,
X = X(gfliAT/v2) 1/3, y = y(gfliAT/v2) 1/3, u = U(vgfliAT) 1/3, V = V(vgfliAT) 1/3, 0 = (T - T0/A T, equations (1)-(3) and the conditions (4) can be written as:
au
c~u c3u au ao AO a2u + U~x + V~y = 0 - MZu + A~y~y + (1 + )~y2,
(5)
au {3y ~xx + ~yy = o,
(6)
pr(C30 t30 ao + \ a t + U~xx + V~y = so + PrE(1 + ao)k Y~
\if-y/
+ (1 +
)~y2,
(7)
u = v = 0,0 = 0 a t x = 0,
u=v=O,O=
l a t y = 0,
u ~O,O-*Oas y ~ oo, u =
v =
0,0
=
0art
=
0,
where M 2 = B~ao vl/3/p(gfliA T) 2/3, the H a r t m a n n number, A = aiA T, the dimensionless number,
B = biAT, the nondimensional number, E = (vgfli AT)Z/3/CoAT, the Eckert number,
(8)
30
K. VAJRAVELU
Pr = #i Cp/ki, the Prandtl number, O~ = Q * v 4 / 3 / k i ( g f l l
A T ) 2/3,
the heat source/sink parameter,
and AT = Tw - T~, the temperature difference. The heat transfer coefficient (or Nusselt number) and the skin friction coefficient at the wall are defined, in nondimensional form, as N u = - h v / k i A T ( v g f l i A T ) 1/a = (1 +
\SY/y=o'
and z* = f*/pi(vgfli A T ) 2/3 = (1 +
\SYA = o"
Let the analysis be restricted to the case of a fluid whose Prandtl number is 0.733. For the temperature of the plate Tw and the fluid temperature taken to be 25°C (35°C, 45°C, Table 1. Fluid Properties (where Pr = 0.733) at Ti = 15°C
Tw (°c) 25 35 45 55
A 0.028 0.056 0.084 0.112
0.024 0.048 0.072 0.096
55°C) and 15°C, respectively, Table 1 (see Batchelor 1-18]) represents the values of the nondimensional parameters A, B and Pr. The numerical results for the flow and heat transfer characteristics are obtained for these values of A, B and Pr only. 3. M E T H O D OF S O L U T I O N The three simultaneous nonlinear partial differential equations (5)-(7), with conditions (8), are now to be solved for the dependent variables u, v and 0 as a function of x, y and t, and in particular, obtain the steady-state solution, if it exists. Although the primary goal is to obtain the steady-state solution, for which both 8u/St and t30/St are zero, the solution is obtained by considering the corresponding unsteady-state problem, already formulated. Successive steps in time can then be regarded as successive approximations toward the steady-state solution. The solution is obtained by numerical integration, and the integrations are carried out on the time-dependent form of the equations by an explicit finite-difference method. The space under investigation is restricted to finite dimensions. A plate of height Xmax = 100 and Ymax= 25 (as corresponding to y = oo) are considered. The velocity field, temperature field, Nusselt number, and skin friction are calculated for the values of A, B and Pr given in Table 1, with M 2 = 0.2, 0.4, 0.6, E = 0.02, and ~ = 0.1, 0, - 0 . 1 for a 100 x 100 grid at t = 0.5, 1.0, 1.5, 2.0. . . . . 80.0, and are summarized in Figs 2-5 for values of t = 10, 20, 30,..., 80. To compare the results of the hydromagnetic case with its hydrodynamic counterpart, numerical values of u, v, O, N u and z* for the hydrodynamic case are also obtained by putting M z = 0 and giving the same set of values to the other parameters. The numerical values thus obtained are presented in Figs 2-5. Examination of the complete results so presented reveal little change in u, v and 0 after t = 50 for all values of the parameters. It should be mentioned here that the results of Vajravelu 1-17] for fluid velocity and fluid temperature are a special case of this analysis for M 2 equal to zero. The deviation of the wall heat transfer parameter of the constant fluid property (CFP) case from the variable fluid property (VFP) counterpart is defined as Nu(VFP) -- Nu(CFP) Nu(CFP)
Hydromagnetic convection at a heated semi-infinite vertical plate
31
M2:0.4
M2= 0
t : 30&40 t=2~
~
~
--t= 40&50 t : 30
m
0
12.5
25.0
0
12.5
Y
25.0
Y (a)
12
M2= 0
tt 30&40
M2=0.4 - ~ t = 4 0 & 5 0 ~ . . ~7
t=30~
!
!
I
50
100 0 X
50 X
!
100
(b)
Fig. 2. Transient and steady-state velocity Profiles for Pr = 0.733, E = 0.02, ~ = 0.1 (with T = 35°C and T~= 15 Ct at (a) x = 20 and (b) y = 5: ( ) transient; (.... ) steady-state.
a n d the percentage, a b b r e v i a t e d as N u D e , is e v a l u a t e d numerically. S o m e of the qualitatively interesting percentages of the d e v i a t i o n of the heat transfer coefficient a n d the skin friction, ~p are shown in T a b l e 2. 4. D I S C U S S I O N O F T H E R E S U L T S E q u a t i o n s (5) (7), subject to c o n d i t i o n s (8), are solved by the finite-difference m e t h o d . The results (with Tw = 35~'C a n d T~ = 15°C) are presented in Figs 2 - 5 for P r = 0.733, E = 0.02 and ~ = 0.! for two cases: (1) the h y d r o d y n a m i c case, a n d (2) the h y d r o m a g n e t i c case. F o r b o t h cases the transient a n d s t e a d y - s t a t e values of the velocity u for M 2 = 0, 0.4 are s u m m a r i z e d in Fig. 2a with fixed x a n d in Fig. 2b with fixed y. F r o m Fig. 2 it can be o b s e r v e d that the values of u have decreased c o n s i d e r a b l y in the presence of m a g n e t i c field. This q u a l i t a t i v e l y agrees with expectations, since the m a g n e t i c field exerts a r e t a r d i n g force on the free-convection flow. F r o m n u m e r i c a l calculations, it is o b s e r v e d that, in b o t h the h y d r o d y n a m i c a n d h y d r o m a g n e t i c cases (with a n d w i t h o u t variable fluid-properties), the values of u have increased c o n s i d e r a b l y in the presence of heat sources. T h e o p p o s i t e p h e n o m e n o n is o b s e r v e d in the presence of heat sinks. H o w e v e r , u has larger values a w a y ES 2 5 : 1 - C
32
K. V A J R A V E L U
1.2
M 2= 0.4
M2=O
o
0.8 t=20
-V
0.4
!
I
12.5
25.0
0
i
12 5
Y
25.0 Y
(a) 0.9
M2=O
M 2 = 0.4
/~..
,--'-
t=40&50
t = 30&40
0.6
tz30 / / , ~
-V
t = 20
0.3 ¢
50
100
0
50
100
X
(b) Fig. 3. T r a n s i e n t and steady-state velocity Profiles for Pr = 0.733, E = 0.02, ~ = 0.1 (with T = 35°C and T i = 15°C) at (a) x = 20 a n d (b) y = 5: ( -) transient; (. . . . ) steady-state.
from the plate in the V F P case than in the C F P case. The reverse is the case near the plate. However, at all times, u has m a x i m u m value only in the C F P case. Observations from Fig. 2 show that u increases monotonically, as nondimensional time increases, up to a certain stage, then decreases slightly, and thereafter remains unchanged. That is, in the presence of magnetic field at t = 50 and in its absence at t = 40, the velocity u reaches the steady-state condition. Physically, it means that, in the case of imposed magnetic field, the time required for the flow to reach steady-state is increased. Figure 3 describes the behavior of velocity component v perpendicular to the plate. The negative values of v at all points in the fluid indicate that v is directed toward the plate. As in the case of u, the values of v (absolute) increase monotonically up to t = 50 in the presence of magnetic field and up to t = 40 in its absence, and thereafter remain unchanged. These steady-state results reveal that v (absolute) is enhanced in the presence of heat sources. However, v (absolute) decreases in the presence of magnetic field. This phenomenon is qualitatively true in both the C F P and V F P cases. Transient and steady-state temperature profiles are shown in Figs 4a and 4b for M z = O, 0,4 and for x and y fixed, respectively. The temperature increases monotonically and reaches the steady-state condition at t = 40 in the presence of magnetic field and at t = 30
H y d r o m a g n e t i c c o n v e c t i o n at a h e a t e d semi-infinite vertical p l a t e
'2 I
M2=O
M2 = 0.4 ~/~t
0.8
33
= 30 &40
t = 20&30
10
!
0.4
12.5
25.0
0
12.5
25.0 Y
Y (a ) 4.5
M2=
M2 = 0.4
0
=20~
t = 30&40 3.0
t = 20&30
t
t"lO 1.5
f
/ 0
0
!
I
50
100
0
I
I
5O
100 x
x
(b) Fig. 4. T r a n s i e n t a n d s t e a d y - s t a t e t e m p e r a t u r e Profiles for Pr = 0.733, E = 0.02, ~ = 0.1 (with T = 35~'C a n d T~ = 15°C) at ( a ) x = 20 a n d ( b ) y = 5:( . . . . . . ) t r a n s i e n t ; ( . . . . ) s t e a d y - s t a t e .
in its absence. F r o m the steady-state value of 0 it can be seen that, unlike in the case of u, the values of 0 are considerably enhanced in the presence of magnetic field. In the hydromagnetic case, the steady-state temperature 0 increases from its value 1 at y -- 0 to a maximum of 1.164 around y = 3.75 and then decreases steadily to its value 0 as y -~ ~ . But, in the hydrodynamic case, the temperature 0 decreases steadily from its value of 1 at y = 0 to its value 0 as y ~ ~ . Figures 5a and 5b, respectively, describe the behavior of the skin friction and rate of heat transfer (Nusselt number) at the plate. From Fig. 5a it is clear that the skin friction at the plate decreases with the magnetic parameter M z (the H a r t m a n n number); however, the skin friction increases with the dimensionless coordinate x. From numerical results, it is further observed that, for all values of ~, the skin friction in the V F P case exceeds that in the C F P case. From Fig. 5b, it is evident that, in the presence of magnetic field, the heat transfer parameter is positive; physically, this means that there is heat flow from the fluid to the plate. Opposite is the case (at x = 20) in the absence of magnetic field. The rate of heat transfer parameter is an increasing function of M 2, t and ~. Further (from numerical results it is observed that), in the presence of heat sources the values of the heat transfer parameter in the V F P case in general exceed those in the C F P case. This behavior
34
K. V A J R A V E L U
X=20
x=6o f
t = 3o&4o
t = 30&40 2 t:20 T*
-"<2__ I
I
0.2
0.4
0
I
I
0.2
0.4
M2
M2
(a) 0.6
=20
X:60
t:30&40
S
0.3 Nu
-0.3 0
0.2
0.4
I 0.2
0
I 0.4 M2
M2
(b) Fig. 5. F l o w a n d h e a t t r a n s f e r c h a r a c t e r i s t i c s at the p l a t e for Pr = 0.733, E = 0.02, ~ = 0.1 (with T = 35°C a n d T~ = 15°C): ( a ) s k i n friction a n d ( b ) N u s s e l t n u m b e r : ( ) t r a n s i e n t ; (. . . . ) s t e a d y - s t a t e .
Table 2. Percentage differences of Nusseh number and skin friction in the C F P case and the VFP case at steady state (T~ = 15°C and T, = 35°C), when Pr = 0.733, E = 0.02, and :~ = 0.l
NUop x/M 2
80 70 60 50 40
r~p
0
0.2
0.4
0
0.2
0.4
4.218 4.372 5.798 6.412 14.781
1.283 1.482 2.087 2.598 3.514
0.066 0.082 0.560 1.256 2.060
3.422 3.644 3.867 4.141 4.435
2.201 2.617 2.953 3.370 3.832
1.084 1.523 2.063 2.600 3.267
is reversed in the presence of heat sinks. Finally, it may be said that the magnetic force is limited to only retardation; thus, separation will never occur, and the heat transfer is never limited (as is verified by the asymptotic trend exhibited in Fig. 5). For possible experimental verification, the percentage differences of the skin friction and heat transfer parameter in the C F P case from those in the VFP case are presented in Table 2.
Hydromagnetic convection at a heated semi-infinite vertical plate
35
5. C O N C L U S I O N S Some of the interesting conclusions of the problem are summarized as follows: 1. The fluid velocity components u and v (absolute) have larger values in the absence of magnetic field than in its presence. But this phenomenon is reversed in the case of fluid temperature 0. 2. In the presence of magnetic field, the fluid velocity components u, v (absolute) and the fluid temperature 0 have larger values away from the plate in the VFP case than in the C F P case. The opposite is true near the plate. 3. In both the hydromagnetic and hydrodynamic cases, u, v (absolute) and 0 increase (decrease) in the presence of heat sources (sinks). But viscous dissipation increases the values of u, v (absolute) and 0. 4. In the absence of magnetic field, the velocity component, v, perpendicular to the plate is always directed toward the plate. This is true even in the presence of magnetic field. 5. In the hydromagnetic case, the steady-state temperature 0 increases from its value 1 at y = 0 to a maximum of 1.164 around y = 3.75 and then decreases steadily to its value 0 as y --* oe. But, in the hydrodynamic case, the temperature decreases steadily from its value of 1 at y = 0 to its value 0 as y --, c~. 6. Percentage-wise, the deviations of the rate of heat transfer in the C F P case from its VFP counterpart decrease as the magnetic parameter M E (the Hartmann number) takes increasing values. That is, the effects of variable fluid properties are negligible in the presence of a strong magnetic field. 7. Magnetic field plays an important role in delaying the velocity and temperature reaching the steady-state condition. Acknowledgements--The author is greatly indebted to Professor Andreas Acrivos for valuable comments and to the referee for constructive criticisms which led to definite improvements in the paper.
REFERENCES [1] D. D. GRAY and A. GIORGINI, Int. J. Heat Mass Transfer 19, 545 (1976). [2] E. M. SPARROW, Free convection with variable properties and variable wall temperature, Ph.D. thesis, Harvard Univ., Cambridge, Mass. (1956). [3] E. M. SPARROW and J. L. GREGG, Trans. ASME 80, 879 (1958). [4] J. E. PLAPP, Laminar free convection with variable fluid properties, Part II. Ph.D. thesis, California Inst. of Technology, Pasadena, Calif. (1957). [5] T. HARA, Bull. JSME 1, 251 (1958). [6] W. J. MINKOWYCZ and E. M. SPARROW, Int. J. Heat Mass Transfer 9, 1145 (1966). [7] J.-M. PIAU, C. R. Acad. Sci. Ser. A 271, 953 (1970). [8] J.-M. PIAU, Int. J. Heat Mass Transfer 17, 465 (1974). [9] H. BARROW and T. L. SITHARAMARAO, Br. Chem. Eng. 16, 704 (1971). [10] A. BROWN, Trans. ASME, J. Heat Transfer 97, 133 (1975). [11] J. A. D. ACKROYD, J. Fluid Mech. 62, 677 (1974). [12] K. NISHIKAWA and T. ITO, Int. J. Heat Mass Transfer 12, 1449 (1969). [13] T. ITO, H. YAMASHITA and K. NISHIKAWA, Proc. 6th Int. Heat Transfer Conf., Tokyo 3, 49 (1974). [14] V. P. CAREY and J. C. MOLLENDORF, Proc. 6th Int. Heat Transfer Conf., Toronto 2, 221 (1978). [15] J. F. CLARKE, J. Fluid Mech. 57, 45 (1973). [16] H. SHAUKATULLAH and B. GEBHART, Numer. Heat Transfer 2, 215 (1979). [17] K. VAJRAVELU, Numer. Heat Transfer 3, 345 (1980). [18] G. K. BATCHELOR, An Introduction to Fluid Dynamics, Cambridge Univ. Press, London (1967). (Revised version received 4 February 1986)