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A note on magnetoconvection in a vertical enclosure
Pergamon
Int J Noion-Linear Mechanrcs, Vol. 31, No. 3, pp. 371. 316, 1996 Copyright ‘0 1996 Elsevm Science Ltd Pnnted m Great Bntam. All rights reserved GfJZO-7462196 $15.00 + 0.00
0020-7462(96)00061-5
A NOTE ON MAGNETOCONVECTION ENCLOSURE
IN A VERTICAL
J. C. Umavathi Department of Mathematics, Krishnadevaraya Nagar,
Postgraduate Centre of Gulbarga University, Sandur 583 119, Bellary, Karnataka, India
(Receiaed 2 May 1994; in reaised firm
20 August
1995)
Abstract-The combined effect of viscous and ohmic dissipations on magnetoconvection in a vertical enclosure heated at the vertical side walls in the presence of applied electric field parallel to gravity and magnetic field normal to gravity is investigated. The coupled non-linear equations governing the motion are solved both analytically valid for small buoyancy parameter N and numerically valid for large N. Solutions for large N reveal a marked change in velocity profile, mass Bow rate, skin friction and rate of heat transfer. These results are presented for various Hartmann number M, electric field loading parameter E and buoyancy parameter N. It is shown in the case of open circuit (i.e. E # 0) that the effect of magnetic field is to increase both the velocity and temperature in contrast with the short circuit case (i.e. E = 0). The results for the case when the walls are maintained at the same temperatures (i.e. T, = 7,) are obtained as a particular case. Copyright 0 1996 Elsevier Science Ltd.
1. INTRODUCTION
Convective heat transfer in a vertical channel or pipe has been an important research topic for the last few decades because of its importance in several technological processes. The earliest systematic studies on this topic were presented by Ostrach [l, 21, and subsequently a number of studies by Gupta [3], Rudraiah et al. [4], Perlmutter and Seigal [S] have been carried out covering different aspects that influence the various mechanisms of heat transfer. The corresponding problem of magnetoconvection (i.e. convection in an electrically conducting fluid in the presence of a magnetic field) has been investigated by Fletcher et al. [6], who obtained analytical solutions valid for small N in the absence of an applied electric field. Further, Fletcher et al. [6] dealt with a problem where the restriction of the walls are maintained at the same temperature (i.e. TI = Tz). Many engineering applications, such as cooling of nuclear reactors, the design of MHD generators, cross-field accelerators, shock tubes and pumps, require natural convection with applied electric field and involve large values of N. Therefore, a detailed study of the influence of magnetic field, applied electric field and buoyancy on magnetoconvection in a vertical channel with walls maintained at different temperatures is necessary. These aspects are considered in this paper and the resulting coupled non-linear equations are solved using the finite difference technique with the successive-over-relaxation method.
2. FORMULATION
OF
THE
PROBLEM
AND
THE
METHOD
OF
SOLUTION
The physical configuration considered in this paper is shown in Fig. 1. It consists of a vertical enclosure formed by two parallel plates of infinite extent separated by a distance 2b. The two plates are maintained at different temperatures T1 and T2. A uniform magnetic field I&, is applied across the channel normal to the plates and the uniform electric field E. is applied in the vertical direction. The viscous conducting fluid rises in the channel driven by buoyancy forces and retarded by magnetic forces. The flow is assumed to be steady, unidirectional and fully developed. For this, the momentum and energy equations in dimensionless form following Fletcher et al. [6] are: d2u dy2Contributed
M2(E + u) + 0 = 0
by K. R. Rajagopal. 371
312
J. C. Umavathi
Fig. 1. Physical
configuration
of flow.
$+N 2 Z+NM2(E+u)2=0
0
(2)
Here the basic equations are made dimensionless using the scales for velocity, temperature and distance, respectively, in the form v[gpb’(T, - TO)] _ ‘, (T, - T,,) and b, p is the thermal expansion coefficient, M=
0 Bob J P
N = pb4g2P2(T1 vK
is the Hartmann
- “)
number
is the buoyancy
parameter
and vEo
E=
is the electric field loading
parameter.
BogB(T1 - To) The above equations
are solved using the boundary
conditions:
u=O
aty=+l
(3)
0=1
aty=l
(4)
8=1+m
sty=-1.
Here m = (T, - T2)/(T1 - To) is the non-dimensional
(5) temperature
parameter.
2.1. Analytical solutions Equations (1) and (2) are coupled non-linear equations because of viscous and ohmic dissipations and it is-difficult, in general, to solve them analytically. When N = 0, implying the neglection of dissipative heating, (1) and (2) become linear and have the following solutions: ug =
1 --E+2M2 M2
5
(1 - cash My sech M) + $
(sinh My cosech M - y)
(6)
When m and E are both zero, solutions (6) and (7) tend to those given by Fletcher et al. [6]. In many practical problems cited above, N cannot be zero, but in some situations it can take
373
A note on magnetoconvection -+--NUr?eriCa( -.4nalYticoI
Fig. 2. Velocity
profiles for different
M.
M
m3_c M:2C -~ _E:_1.0
80.
--E:U)
-10
-0.8
-06
Fig.
-02
3. Temperature
oyo
0.2
OL
profiles for different
o&
N.
small values. For example, for mercury in a channel of half-width 2 cm, and with T1 - To = 20 “C, N takes the value of 0.128. Small values of N (< 1) facilitate finding analytical solutions of (1) and (2) in the form: u = u. + NuI + N2uz + . . .
(8)
8 = B. + NB1 + N2%2 + ’ . .
(9)
where the second and higher order terms on the right-hand side give a correction to uo, B. accounting for the dissipative effects. Substituting (8) and (9) into (1) and (2) and equating the like powers of N to zero, we obtain d2u1 dy2
- M2u1 + %1 = 0
’ + M2(E + uo)2 = 0. The corresponding
boundary
conditions uI=%l=O
(10)
(11)
are aty=fl.
(12)
314
J. C. Umavathi
Solutions to (10) and (11) are found; the expressions are very lengthy and are omitted here. These solutions along with (6) and (7), are computed and the results are shown in Figs 2 and 3. The ratio of mass flow rate in the absence of dissipation MpOand without magnetic and electric field MfO, is
MPO M,(23+mi((I -= M,,
-%)(2+m-ZM’E)).
The mass flow rate Mf per unit channel dimensional form is
(13)
width in the presence
of dissipation
in non-
1
Mf =
udy. s
(14)
-1
The ratio of product of friction factor C, (= gfl(T, - To)B/$&) (= uOD/v) with and without magnetic and electric field is CrRe
to Reynolds
number
Re
M2(2 + m) (15)
Once the velocity distribution non-dimensional form is
is known,
we can determine
the skin-friction,
which in the
(16) It is of practical interest and importance to calculate fluid and the plates. This is given in non-dimensional
the rate of heat transfer form as:
between
the
(17)
2.2. Numerical solutions The analytical solutions obtained in Section 2.1 are valid only for small values of N. In many practical problems mentioned earlier, the values of N are usually large. In that case analytical solutions are difficult, and hence we resort to numerical solutions. The numerical method employed in this section involves solving the non-linear two-point boundary value problem equations (1) and (2) using the finite difference method. Replacing the derivatives with corresponding central difference approximations, we obtain a set of k linear algebraic equations, where k is the number of divisions from - 1 to + 1 which are solved by the successive-over-relaxation method. The values of k are varied successively from k = 25 to 50. The converging results are found for k = 41. Mass flow rate, skin friction and rate of heat transfer defined by (14), (16) and (17) are evaluated numerically using Newton’s quadratic interpolation formula for numerical differentiation and Simpson’s one-third rule for numerical integration.
3. RESULTS
AND
DISCUSSION
In this section the fluid flow and heat transfer results for an electrically conducting fluid flow in a vertical enclosure are discussed in the presence of an applied magnetic field, H,, normal to gravity and an applied electric field, EO, parallel to gravity, considering both viscous and ohmic dissipations. Both analytical and numerical solutions are obtained. Analytical solutions, obtained using a regular perturbation technique, are valid only for small values of the buoyancy parameter N and for numerical solutions, obtained by finite differencing the governing equations, a central differencing scheme is used to evaluate the second derivatives while the non-linear terms are replaced by a suitable linearized approximation. The fast convergence of the scheme is achieved using the successive-over-relaxation method.
A note on magnetoconvection
375
Analytical solutions for velocity and temperature are computed for different values of M and E and the results are represented graphically in Figs 2 and 3. These figures reveal that analytical and numerical solutions are in good agreement. It is seen in Fig. 2 that the effect of increasing M is to accelerate the flow in the positive direction of y for E = - 1 and in the negative direction of y for E = 1 exhibiting a tunneling effect, namely boundary layer nature near the boundaries and a flattening nature in the middle of the channel. Figure 3 displays different temperature distributions for M = 2, with m = 1.0, E = 5 1.0 and for different values of N. It is seen that the effect of increase in N is to increase the temperature. This increase in temperature is higher for E = 1.0 than for E = - 1.0. The direction of flow when E > 0 is opposite to that when E < 0 and hence the present configuration can be used effectively for the flow reversal situation required in many practical problems.
mil.0
Fig. 4. Mass flow rate vs M.
Fig. 5. Skin friction
vs M.
316
J. C. Umavathi
Fig. 6. Rate of heat transfer
vs M.
Mass flow rate Mf, skin friction z and Nusselt number q are computed for different values of M, m, E and N, and the results are presented in Figs 4-6. For E = 0 the effect of increase in M is to decrease MI, while the opposite is true for E = - 1.0, because when E = 0 there is a decrease in momentum, while there is an increase in momentum for E = - 1.0. Skin friction increases with M when E = - 1.0 and dissipation has no effect on z. For E = 0.0, z decreases with an increase in M and the effect of dissipation is to decrease z with an increase in M up to M = 4 and for M > 4, it becomes independent of N. Similar behaviour is obtained for Nusselt number. Finally, we conclude that electromagnetic fields, dissipation (both viscous and ohmic) and temperature parameter m, control heat and momentum transfer. Acknowledgement~The author is grateful to Professor his valuable suggestions, which have greatly improved grateful to the referee for useful comments.
N. Rudraiah, Vice-Chancellor, Gulbarga University, for her understanding of the problem. The author is also
REFERENCES 1. S. Ostrach, Laminar natural convection flow and heat transfer of fluids with and without heat sources in channels with constant wall temperatures. NACATN, 2863 (1952). 2. S. Ostrach, Combined natural and forced convection laminar flows and heat transfer of fluids with and without heat sources in channels with linearly varying wall temperatures. NACATN, 314 (1954). 3. A. S. Gupta, Combined free and forced convection effects on the magnetohydrodynamic flow through a channel. ZAMP 20, 506 (1969). 4. N. Rudraiah, V. Kumidini and W. Unno, Theory on non-linear magnetoconvection and its applications to solar convection problems--I. Publ. Astron. Sot. Jpn 37, 183 (1985). 5. M. Perlmutter and R. Seigal, Heat transfer to an electrically conducting fluid flowing in a channel with a transverse magnetic field. NACATN, D-875 (1961). 6. J. Fletcher Osterle and J. Young, Natural convection between heated vertical plates in a horizontal magnetic field. J. Fluid Mech. 1, 512 (1961).
Int J Noion-Linear Mechanrcs, Vol. 31, No. 3, pp. 371. 316, 1996 Copyright ‘0 1996 Elsevm Science Ltd Pnnted m Great Bntam. All rights reserved GfJZO-7462196 $15.00 + 0.00
0020-7462(96)00061-5
A NOTE ON MAGNETOCONVECTION ENCLOSURE
IN A VERTICAL
J. C. Umavathi Department of Mathematics, Krishnadevaraya Nagar,
Postgraduate Centre of Gulbarga University, Sandur 583 119, Bellary, Karnataka, India
(Receiaed 2 May 1994; in reaised firm
20 August
1995)
Abstract-The combined effect of viscous and ohmic dissipations on magnetoconvection in a vertical enclosure heated at the vertical side walls in the presence of applied electric field parallel to gravity and magnetic field normal to gravity is investigated. The coupled non-linear equations governing the motion are solved both analytically valid for small buoyancy parameter N and numerically valid for large N. Solutions for large N reveal a marked change in velocity profile, mass Bow rate, skin friction and rate of heat transfer. These results are presented for various Hartmann number M, electric field loading parameter E and buoyancy parameter N. It is shown in the case of open circuit (i.e. E # 0) that the effect of magnetic field is to increase both the velocity and temperature in contrast with the short circuit case (i.e. E = 0). The results for the case when the walls are maintained at the same temperatures (i.e. T, = 7,) are obtained as a particular case. Copyright 0 1996 Elsevier Science Ltd.
1. INTRODUCTION
Convective heat transfer in a vertical channel or pipe has been an important research topic for the last few decades because of its importance in several technological processes. The earliest systematic studies on this topic were presented by Ostrach [l, 21, and subsequently a number of studies by Gupta [3], Rudraiah et al. [4], Perlmutter and Seigal [S] have been carried out covering different aspects that influence the various mechanisms of heat transfer. The corresponding problem of magnetoconvection (i.e. convection in an electrically conducting fluid in the presence of a magnetic field) has been investigated by Fletcher et al. [6], who obtained analytical solutions valid for small N in the absence of an applied electric field. Further, Fletcher et al. [6] dealt with a problem where the restriction of the walls are maintained at the same temperature (i.e. TI = Tz). Many engineering applications, such as cooling of nuclear reactors, the design of MHD generators, cross-field accelerators, shock tubes and pumps, require natural convection with applied electric field and involve large values of N. Therefore, a detailed study of the influence of magnetic field, applied electric field and buoyancy on magnetoconvection in a vertical channel with walls maintained at different temperatures is necessary. These aspects are considered in this paper and the resulting coupled non-linear equations are solved using the finite difference technique with the successive-over-relaxation method.
2. FORMULATION
OF
THE
PROBLEM
AND
THE
METHOD
OF
SOLUTION
The physical configuration considered in this paper is shown in Fig. 1. It consists of a vertical enclosure formed by two parallel plates of infinite extent separated by a distance 2b. The two plates are maintained at different temperatures T1 and T2. A uniform magnetic field I&, is applied across the channel normal to the plates and the uniform electric field E. is applied in the vertical direction. The viscous conducting fluid rises in the channel driven by buoyancy forces and retarded by magnetic forces. The flow is assumed to be steady, unidirectional and fully developed. For this, the momentum and energy equations in dimensionless form following Fletcher et al. [6] are: d2u dy2Contributed
M2(E + u) + 0 = 0
by K. R. Rajagopal. 371
312
J. C. Umavathi
Fig. 1. Physical
configuration
of flow.
$+N 2 Z+NM2(E+u)2=0
0
(2)
Here the basic equations are made dimensionless using the scales for velocity, temperature and distance, respectively, in the form v[gpb’(T, - TO)] _ ‘, (T, - T,,) and b, p is the thermal expansion coefficient, M=
0 Bob J P
N = pb4g2P2(T1 vK
is the Hartmann
- “)
number
is the buoyancy
parameter
and vEo
E=
is the electric field loading
parameter.
BogB(T1 - To) The above equations
are solved using the boundary
conditions:
u=O
aty=+l
(3)
0=1
aty=l
(4)
8=1+m
sty=-1.
Here m = (T, - T2)/(T1 - To) is the non-dimensional
(5) temperature
parameter.
2.1. Analytical solutions Equations (1) and (2) are coupled non-linear equations because of viscous and ohmic dissipations and it is-difficult, in general, to solve them analytically. When N = 0, implying the neglection of dissipative heating, (1) and (2) become linear and have the following solutions: ug =
1 --E+2M2 M2
5
(1 - cash My sech M) + $
(sinh My cosech M - y)
(6)
When m and E are both zero, solutions (6) and (7) tend to those given by Fletcher et al. [6]. In many practical problems cited above, N cannot be zero, but in some situations it can take
373
A note on magnetoconvection -+--NUr?eriCa( -.4nalYticoI
Fig. 2. Velocity
profiles for different
M.
M
m3_c M:2C -~ _E:_1.0
80.
--E:U)
-10
-0.8
-06
Fig.
-02
3. Temperature
oyo
0.2
OL
profiles for different
o&
N.
small values. For example, for mercury in a channel of half-width 2 cm, and with T1 - To = 20 “C, N takes the value of 0.128. Small values of N (< 1) facilitate finding analytical solutions of (1) and (2) in the form: u = u. + NuI + N2uz + . . .
(8)
8 = B. + NB1 + N2%2 + ’ . .
(9)
where the second and higher order terms on the right-hand side give a correction to uo, B. accounting for the dissipative effects. Substituting (8) and (9) into (1) and (2) and equating the like powers of N to zero, we obtain d2u1 dy2
- M2u1 + %1 = 0
’ + M2(E + uo)2 = 0. The corresponding
boundary
conditions uI=%l=O
(10)
(11)
are aty=fl.
(12)
314
J. C. Umavathi
Solutions to (10) and (11) are found; the expressions are very lengthy and are omitted here. These solutions along with (6) and (7), are computed and the results are shown in Figs 2 and 3. The ratio of mass flow rate in the absence of dissipation MpOand without magnetic and electric field MfO, is
MPO M,(23+mi((I -= M,,
-%)(2+m-ZM’E)).
The mass flow rate Mf per unit channel dimensional form is
(13)
width in the presence
of dissipation
in non-
1
Mf =
udy. s
(14)
-1
The ratio of product of friction factor C, (= gfl(T, - To)B/$&) (= uOD/v) with and without magnetic and electric field is CrRe
to Reynolds
number
Re
M2(2 + m) (15)
Once the velocity distribution non-dimensional form is
is known,
we can determine
the skin-friction,
which in the
(16) It is of practical interest and importance to calculate fluid and the plates. This is given in non-dimensional
the rate of heat transfer form as:
between
the
(17)
2.2. Numerical solutions The analytical solutions obtained in Section 2.1 are valid only for small values of N. In many practical problems mentioned earlier, the values of N are usually large. In that case analytical solutions are difficult, and hence we resort to numerical solutions. The numerical method employed in this section involves solving the non-linear two-point boundary value problem equations (1) and (2) using the finite difference method. Replacing the derivatives with corresponding central difference approximations, we obtain a set of k linear algebraic equations, where k is the number of divisions from - 1 to + 1 which are solved by the successive-over-relaxation method. The values of k are varied successively from k = 25 to 50. The converging results are found for k = 41. Mass flow rate, skin friction and rate of heat transfer defined by (14), (16) and (17) are evaluated numerically using Newton’s quadratic interpolation formula for numerical differentiation and Simpson’s one-third rule for numerical integration.
3. RESULTS
AND
DISCUSSION
In this section the fluid flow and heat transfer results for an electrically conducting fluid flow in a vertical enclosure are discussed in the presence of an applied magnetic field, H,, normal to gravity and an applied electric field, EO, parallel to gravity, considering both viscous and ohmic dissipations. Both analytical and numerical solutions are obtained. Analytical solutions, obtained using a regular perturbation technique, are valid only for small values of the buoyancy parameter N and for numerical solutions, obtained by finite differencing the governing equations, a central differencing scheme is used to evaluate the second derivatives while the non-linear terms are replaced by a suitable linearized approximation. The fast convergence of the scheme is achieved using the successive-over-relaxation method.
A note on magnetoconvection
375
Analytical solutions for velocity and temperature are computed for different values of M and E and the results are represented graphically in Figs 2 and 3. These figures reveal that analytical and numerical solutions are in good agreement. It is seen in Fig. 2 that the effect of increasing M is to accelerate the flow in the positive direction of y for E = - 1 and in the negative direction of y for E = 1 exhibiting a tunneling effect, namely boundary layer nature near the boundaries and a flattening nature in the middle of the channel. Figure 3 displays different temperature distributions for M = 2, with m = 1.0, E = 5 1.0 and for different values of N. It is seen that the effect of increase in N is to increase the temperature. This increase in temperature is higher for E = 1.0 than for E = - 1.0. The direction of flow when E > 0 is opposite to that when E < 0 and hence the present configuration can be used effectively for the flow reversal situation required in many practical problems.
mil.0
Fig. 4. Mass flow rate vs M.
Fig. 5. Skin friction
vs M.
316
J. C. Umavathi
Fig. 6. Rate of heat transfer
vs M.
Mass flow rate Mf, skin friction z and Nusselt number q are computed for different values of M, m, E and N, and the results are presented in Figs 4-6. For E = 0 the effect of increase in M is to decrease MI, while the opposite is true for E = - 1.0, because when E = 0 there is a decrease in momentum, while there is an increase in momentum for E = - 1.0. Skin friction increases with M when E = - 1.0 and dissipation has no effect on z. For E = 0.0, z decreases with an increase in M and the effect of dissipation is to decrease z with an increase in M up to M = 4 and for M > 4, it becomes independent of N. Similar behaviour is obtained for Nusselt number. Finally, we conclude that electromagnetic fields, dissipation (both viscous and ohmic) and temperature parameter m, control heat and momentum transfer. Acknowledgement~The author is grateful to Professor his valuable suggestions, which have greatly improved grateful to the referee for useful comments.
N. Rudraiah, Vice-Chancellor, Gulbarga University, for her understanding of the problem. The author is also
REFERENCES 1. S. Ostrach, Laminar natural convection flow and heat transfer of fluids with and without heat sources in channels with constant wall temperatures. NACATN, 2863 (1952). 2. S. Ostrach, Combined natural and forced convection laminar flows and heat transfer of fluids with and without heat sources in channels with linearly varying wall temperatures. NACATN, 314 (1954). 3. A. S. Gupta, Combined free and forced convection effects on the magnetohydrodynamic flow through a channel. ZAMP 20, 506 (1969). 4. N. Rudraiah, V. Kumidini and W. Unno, Theory on non-linear magnetoconvection and its applications to solar convection problems--I. Publ. Astron. Sot. Jpn 37, 183 (1985). 5. M. Perlmutter and R. Seigal, Heat transfer to an electrically conducting fluid flowing in a channel with a transverse magnetic field. NACATN, D-875 (1961). 6. J. Fletcher Osterle and J. Young, Natural convection between heated vertical plates in a horizontal magnetic field. J. Fluid Mech. 1, 512 (1961).