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Thermal instability of compressible fluids with hall currents and suspended particles in porous medium
Vol. 31, No. 7, pp. 1053-1060, 1993 Printed in Great Britain. All rights reserved
CKJ20-7225/93$6.00 + 0.00 Copyright @ 1993 PergamonPressLtd
ht. 1. Engng Sci.
THERMAL INSTABILITY OF COMPRESSIBLE FLUIDS WITH HALL CURRENTS AND SUSPENDED PARTICLES IN POROUS MEDIUM R. C. SHARMA?
and URVASHI
GUPTA
Department of Mathematics, Himachal Pradesh University, Shimla-171005, India (Communicated by E. S. SUHUBi) Ah&ad-The thermal instability of compressible fluids in porous medium is considered to include the effects of Hall currents and suspended particles. The dispersion relation has been obtained. For the stationary convection, Hall currents and suspended particles are found to have destabilizing effects whereas compressibility and magnetic field have stabilizing effects on the system. The medium permeability, however, has stabilizing and destabilizing effects on thermal instability in contrast to its destabilizing effect in the absence of magnetic field. The magnetic field and Hall currents are found to introduce oscillatory modes in the system.
1. INTRODUCTION
The theory of Benard convection (thermal instability) in fluid layer heated from below under varying assumptions of hydromagnetics has been given by Chandrasekhar [l]. Gupta [2] studied the problem of thermal instability in the presence of Hall currents and found that Hall currents have a destabilizing effect on the thermal instability of a horizontal layer of a conducting fluid in the presence of a uniform vertical magnetic field. Chandra [3] observed that in an air layer, convection occurred at much lower gradients than predicted if the layer depth was <7 mm and called this motion, “columnar instability”. However, for layers deeper than 10 mm, a Bdnard-type cellular convection was observed. Scanlon and Segel [4] investigated some of the continuum effects of particles on BCnard convection and found that the critical Rayleigh number was reduced solely because the heat capacity of the pure gas was supplemented by that of the particles. Sharma ef al. [5] have studied the effect of suspended particles on the onset of BCnard convection in the presence of magnetic field and rotation separately. In the above studies, the medium has been considered to be non-porous and the fluid as incompressible. Lapwood [6] studied the stability of convective flow in hydrodynamics in a porous medium using Rayleigh’s procedure. The gross effect, when the fluid slowly percolates through the pores of the rock, is represented by Darcy’s law. When the fluids are compressible, the equations governing the system become quite complicated. To simplify the set of equations, Spiegel and Veronis (71 made the assumptions: (i) the depth of fluid layer is much smaller than the scale height as defined by them and (ii) the fluctuations in temperature, pressure and density, introduced due to motion, do not exceed their total static variations. Under these assumptions the flow equations were found to be the same as those for incompressible fluids except that the static temperature gradient is replaced by its excess over the adiabatic. Motivated by the fact that knowledge regarding fluid-particle mixtures is not commensurate with their industrial and scientific importance, we set out to study the effects of suspended particles and Hall currents on thermal instability of compressible fluids in porous medium. The physical properties of comets, meteorites and interplanetary dust strongly suggest importance of porosity in astrophysical context [8]. The problem may find its usefulness in geophysics (e.g. Earth’s molten core) and astrophysics where Hall currents, suspended particles, compressibility and porosity are important. tTo whom all correspondence should be addressed.
1053
1054
R. C. SHARMA
2. FORMULATION
and U. GUFTA
OF THE PROBLEM
Let us consider an infinite horizontal, compressible fluid-particle layer of thickness d bounded by the planes z = 0 and z = d in the presence of uniform vertical magnetic field H (O,O, H) in porous medium. This layer is heated from below such that a steady adverse temperature gradient /3(= (d7’ld z 1) is maintained. The equations of motion and continuity for the fluid are
1
KN
=-Vp-~~l-fu+T(v-u)+~(VxH)xH,
,+;(uV)u
(1)
1
e$+V.(pu)=O, where p, cc, p and u (u, u, w) denote, respectively, the density, the viscosity, the pressure and the velocity of the pure fluid. v (I, r, s) and N (2, I) denote the velocity and number density of the suspended particles, e is medium porosity, kl is medium permeability, pc is magnetic permeability, g is acceleration due to gravity, f = (x, y, z), 3c= (O,O, 1) and K = 6xpp1’, ‘I’ being the particle radius, is the Stokes’ drag coefficient. Assuming uniform particle size, spherical shape and small relative velocities between the fluid and particles, the presence of particles adds an extra force term, in the equations of motion (l), proportional to the velocity difference between particles and fluid. The force exerted by the fluid on the particles is equal and opposite to that exerted by the particles on the fluid. The buoyancy force on the particles is negligibly small. Interparticle reactions are ignored for we assume that the distances between particles are quite large compared with their diameters. If mN is the mass of particles per unit volume, then the equations of motion and continuity for the particles are mN $++‘)v
I
= KN(u-v),
++V(Nv)=O. Let C,, C,, C,,, T and 4 denote, respectively, the heat capacity of fluid at constant volume, the heat capacity of fluid at constant pressure, the heat capacity of particles, the temperature and the “effective thermal conductivity” of the pure fluid. Assuming that the particles and the fluid are in thermal equilibrium, the equation of heat conduction gives W3
+ P,W
- e)l g
+~C”~u*V)T+~NC~~
(
c;~+vV
>
T=qV2T,
t-51
where ps, C, are the density and the heat capacity of the solid matrix respectively. The Maxwell’s equations in the presence of Hall currents give V*H=O,
(6)
where 77, c, N’ and e denote, respectively, the resistivity, the speed of light, the electron number density and the charge of an electron. The initial state of the system is taken to be a quiescent layer (no settling) with a uniform partide distribution No and is given by u=(O,O,O), T = T(z),
P = P(Z),
H =
(0, 0, H),
p = p(z)
v = (090, O), and
N = NO, a constant,
(8) (9)
Thermal instability of compressible fluids
105.5
where, following Spiegel and Veronis [7], we have T(z) = --/I2 + T,, P(Z) = Pm P(Z) = Pl# - %(T - Tm)+ K&P - Prn)l~
Spiegel and Veronis [7] expressed any state variable say X, in the form
x = x, + X,(z)
+ X’(x, y,
2,
(10)
t),
where X, stands for the constant space distribution of X, X0 is the variation in X in the absence of motion and X’(x, y, z, t) stands for the fluctuations in X due to the motion of the fluid. Thus, o,,, and pm stand for the constant space distribution of p and p and p0 and Tostand for the density and temperature of the fluid at the lower boundary. Following Spiegel and Veronis’s [7] assumptions and results for compressible fluids, the flow equations are found to be the same as those of incompressible fluids except that the static temperature gradient /I is replaced by its excess over the adiabatic (/? -g/Cp).
3. PERTURBATION
EQUATIONS
Let dp, 6p, 8, u(u, v, w), v (I, r, s), II (h,, It,,, h,) and N denote the perturbations in fluid velocity, particle velocity, magnetic field H and particle pressure, density, temperature, number density No respectively. Then the linearized hydromagnetic perturbation equations of the fluid-particle layer, under Spiegel and Veronis’s [7] assumptions, are l&l --_=
Eat
-$vsp
F(v-“)+$
-g
m
m
v*u=o,
(V x h) x H, m
(11) (12)
av
mlv,,t=KN,(u-v),
.z+
V . (N,,v) = 0,
(13) (14)
(1%
(E+hC);=(+)(W+hS)+Kv*8, P
V.h=O, r$=Vx(uxH)+eqV*h-&Vx[(Vxh)XH],
(16) (17)
where a;, = UT, = a; say, v = pip,, K = q/pm& and g/C,, v and K stand for the adiabatic gradient, kinematic viscosity and thermal diffusivity respectively. Also,
-pt h_fC
G ’
R. C. SHARMA
1056
The linearized dimensionless perturbation
and U. GUITA
equations are
($?3), N,1~=-~6p-~u+o(f-u)+N,
(18)
(~-~), N,‘~=-asp-~,+o(r-a)+No paW= p’ at
--$p-$w+N&+w(s-w), du+dv+aw=,, ax
ay
dz
(23) (24) (25)
(26) (27) (E + hc) $-f = ($+v
+ hs) + V28,
where Np, = W/K is the modified Prandtl number, N, = rv/r_r is the modified magnetic Prandtl number, Nn = gabd4/vK is the Rayleigh number, No = ~LeH2d2/4~p,vr) is the Chandrasekhar number, M = EN/N,, Ml = cH/4nN’eq is the Hall parameter, o = KN,-,d2/pDIvE, t = mK/Kd2, f = mNo/p, = zwNp, is the mass fraction, G = C,/3lg and P = klld2. Here physical variables have been scaled using d, d2/K, K/d, pvK/d2, /3d and HK/~ as the length, time, velocity, pressure, temperature and magnetic field scale factors respectively. Consider the case of two free boundaries and the medium adjoining the fluid is non-conducting. The boundary conditions appropriate for the problem are [l, 61 a2W
w=jg =e=o,
atz=Oandl,
and h,, h,, h, are continuous with an external vacuum field. Here 5;= (&/ax - du/dy) and 5 = (i3h,/dx - dh,/dy) are the z-components current density respectively. After eliminating v and 6p, equations (18)-(28) can be expressed as
(30) of vorticity and
(31) (32) (33) (34) (35)
Thermal instability of compressible fluids
1057
where
a2
( _A+Fd>
LI=&,’ v2=
tat2
a2
32 2+-p ay
ax2+
4. THE
at
9
a2
F=f
+l,
v2_dz+ Iax2
DISPERSION
L,=r-g+l,
A=h+l.
82 2’
RELATION
Analysing the disturbances in terms of normal modes by seeking solutions whose dependence on x, y and t is given by [w, 8, h,, h 5]= [w(z), 0(z), K(z), Z(z),
-WI1 x exp(uv + qy
+ nt),
(36)
where n is the growth rate and EY= (acz+ CY;)‘”is the wave number of the disturbance. Equations (31)-(35), with the help of expression (36) become (D2 - ou”)W = L,N,(D2
- a?)DK - L2NRd0,
(37)
Z = L2NoDX,
(38)
- (D2 - &)]X = K’DZ f M@2[NW N,-,‘Q
az)DK,
(39)
[N&N,% - (D2 - &‘)]K = l-‘DW - M,DX,
(40)
L&E + he)n - (02 -
(41)
where D = d/dz, Lt = N,‘(rn2 f Fn) and L2 = WI+ 1. Eliminating Z, X, K and 0 between equations (37)-(41), we obtain [(02 - lr2) - (E + h+](D2
- a2)W + L2No[(D2 - 2) - (E + he)n] -1
N,N,‘n)M;%-’
+ L,No
&f-‘E-2D2
L)2 I
x M,(D2 - o?)D2 + M;‘{(D2 - a”) - N,N,‘nj2 x e-‘M;‘{(D2 - cr2)- N~N~~~~~2]-1(~2 - g)W
+ L2No
-’
= (y)NRa2(m
+ ii)
(42)
Using the boundary conditions (29) and (30), it can be shown that all the even order derivatives of W vanish at the boundaries. Hence the proper solution of equation (42) characterizing the lowest mode is W = W, sin az,
(43)
where W. is a constant. Substituting the solution (43) in equation (42), we obtain the dispersion relation NR= (
G
(nr2+ &“)[(n’+ 2) + (E + he)n] > + $::;:;rn)M;‘r-’
+::;o,I,,
[(L, +$) + L2NQ(LI+$)-I
XM;'E-~J? M,~2(~2f~2)+M;1{(112+(y2)+NhNp11n}2 I[ -1 + L2N,(L,
+$)-’
E-‘M,‘{(n2
+ &) + N,,JV,‘n}n2
I
1
.
(44
1058
R. C. SHARMA and U. GUPTA 5.
STATIONARY
CONVECTION
When the instability sets in as stationary convection, by n = 0 and the dispersion relation (44) reduces to
the marginal state will be characterized
1
(45)
G accounting
for
N,~*E-'{(n* + CY*) + Nope%*}
(n*+ ~*){M:~*+(Jc*+ CY*)+N~PE-~JG*} ' For fixed values of P, No, Ml and A, let the non-dimensional compressibility effects be also kept as fixed; then we find that
number
where NCRand K denote, respectively, the critical Rayleigh numbers in the absence and presence of compressibility. Since critical Rayleigh number is positive and finite, so G > 1 and we obtain a stabilizing effect of compressibility, i.e. its effect is to postpone the onset of thermal convection in fluid-particle layer. To study the effects of suspended particles, medium permeability, magnetic field and Hall currents, we examine the natures of dN,/&, dNR/dP, dN,/dNo and dN,ldM, respectively. Equation (45) gives Nojd*e-‘{(jd* + (u2) + NQP~-'n2} (n~+~~~M~~~+(~~+~)+N~P~-~~~~
1 '
(47)
which is negative. The effect of suspended particles is, thus, destabilizing on thermal instability of compressible fluid-particle layer in the presence of Hall currents through porous medium. It is evident from equation (45) that 1 -i+
P
1
(Nalc2~-')'M:~" (n*+ cu'){Mfn'+(n*+ d)+N,Pc'rc*}* '
(48)
which is positive if
P{MIn-(n2+d)1n}>
(n*+ ti)'n{M:~2+(n2+2)) , N&-h*
and is negative if
(n*+ d)'n{M:n2+(n2+ d)} P{M,r--((j~*+d)~~}< N,E-Y Thus, the medium permeability has both stabilizing and destabilizing effects depending upon In the absence of magnetic field, the medium the values of the various parameters. permeability has a destabilizing effect since for this case
which is always negative. The medium permeability, thus, succeeds in stabilizing the thermal instability of compressible fluid-particle layer for certain wave numbers in the presence of magnetic field, which was unstable in the absence of magnetic field. Equation (45) yields
(n* + (U~)JC’E-’ a’A
{M$r*+(n*+ ~2)-tNQP~-1n2}-1
x[{(n*+~?)+N,PE-'~*}+M:~~~N~PE-~{M:~*+(I~*+ CI?)+N~PE%*}-~], (50) which is always positive. This shows that magnetic field has a stabilizing effect on the system.
Thermal instability of compressible fluids
1059
Equation (45) also yields G dNR ;ir?i?;=-‘G-1 (
(3t.*+ ~?)N~e-~&r~{(~* >
+ at2) + NoP&r*}
ouZfi{M$r2 + (Jr* + a2) + N*PCW}2
’
(51)
which is always negative. Therefore, the effect of Hall currents is destabilizing on the thermal convection in compressible fluid-particle layer in porous medium.
6. EXISTENCE
OF OSCILLATORY
MODES
Multiplying equation (37) by W*, the complex conjugate of W, integrating over the range of z and using equations (38)-(47) together with the boundary conditions (29) and (30), we obtain
(53) which are all positive definite. Putting IZ= ino, where no is real, in equation (52) and equating the ima~nary parts on both sides, we obtain either no-
-
0,
(54)
or J+
-r-2
-Zs)-N,~t(l, +Z6) KN,;'fiF-f >Z,+ N,E Np2Np;'tl(Z2
- Np;'Z?F+;)Z4+NRm2($){T17+(E+h~)Z~}] ( X
N,'(Z?
+ 1-
F) - ;]Z, f NogN,N;,‘fi(Z2
- 4)
x { tz,-I” (E f hE)&}]-‘.
(55)
In the absence of magnetic field
+
-
ti
Np;L~F-~)Z,+N~~2(~){~Z,+(E+h~)zs)]
(56) r2[ (N$(fi
ES 31:t-F
+ 1 - F) - ;}Z, + Nua*(&){rZ,
+ (E + ht)z8)]*
1060
R. C. SHARMA
and U. GUPTA
From equations (54)-(56) we can draw the following conclusions: (1) The presence of magnetic field (and therefore Hall currents) introduces oscillatory modes in the system. (2) In the absence of magnetic field, the expression for n& given by (56) is negative if urn - k,K 9
fk,K
(57)
>
for all positive Nn. Now, since no is real and nz is negative we must have no = 0. This establishes that n is real for positive NR in the absence of magnetic field if the inequality (57) holds true and that the principle of exchange of stabilities is valid for this case. However if the inequality (57) is violated, then oscillatory modes may come into play even in the absence of magnetic field.
REFERENCES [l] [2] [3] [4]
S. CHANDRASEKHAR, Hydrodynamic and Hydromagnetic Stability. Dover, New York (1981). A. S. GUPTA. Rev. Roum. Math. Pures Et. Appl. 12,665 (1%7). K. CHANDRA, Proc. R. Sot. L.ond. AM, 231 (1938). J. W. SCANLON and L. A. SEGEL, Phys. Fluids 16, 1573 (1973). [S] R. C. SHARMA, K. PRAKASH and S. N. DUBE, Acra Phys. Hung. 40,3 (1976). (61 E. R. LAPWOOD, Proc. Comb. Phil. Sot. 44,508 (1948). [7] E. A. SPIEGEL and G. VERONIS, Asfrophys. J. Wl, 442 (1960). [8] J. A. M. McDonnel, Cosmic Dust p. 330. Wiley, Toronto (1978). (Received 7 July 1992; accepted 6 August 1992)
CKJ20-7225/93$6.00 + 0.00 Copyright @ 1993 PergamonPressLtd
ht. 1. Engng Sci.
THERMAL INSTABILITY OF COMPRESSIBLE FLUIDS WITH HALL CURRENTS AND SUSPENDED PARTICLES IN POROUS MEDIUM R. C. SHARMA?
and URVASHI
GUPTA
Department of Mathematics, Himachal Pradesh University, Shimla-171005, India (Communicated by E. S. SUHUBi) Ah&ad-The thermal instability of compressible fluids in porous medium is considered to include the effects of Hall currents and suspended particles. The dispersion relation has been obtained. For the stationary convection, Hall currents and suspended particles are found to have destabilizing effects whereas compressibility and magnetic field have stabilizing effects on the system. The medium permeability, however, has stabilizing and destabilizing effects on thermal instability in contrast to its destabilizing effect in the absence of magnetic field. The magnetic field and Hall currents are found to introduce oscillatory modes in the system.
1. INTRODUCTION
The theory of Benard convection (thermal instability) in fluid layer heated from below under varying assumptions of hydromagnetics has been given by Chandrasekhar [l]. Gupta [2] studied the problem of thermal instability in the presence of Hall currents and found that Hall currents have a destabilizing effect on the thermal instability of a horizontal layer of a conducting fluid in the presence of a uniform vertical magnetic field. Chandra [3] observed that in an air layer, convection occurred at much lower gradients than predicted if the layer depth was <7 mm and called this motion, “columnar instability”. However, for layers deeper than 10 mm, a Bdnard-type cellular convection was observed. Scanlon and Segel [4] investigated some of the continuum effects of particles on BCnard convection and found that the critical Rayleigh number was reduced solely because the heat capacity of the pure gas was supplemented by that of the particles. Sharma ef al. [5] have studied the effect of suspended particles on the onset of BCnard convection in the presence of magnetic field and rotation separately. In the above studies, the medium has been considered to be non-porous and the fluid as incompressible. Lapwood [6] studied the stability of convective flow in hydrodynamics in a porous medium using Rayleigh’s procedure. The gross effect, when the fluid slowly percolates through the pores of the rock, is represented by Darcy’s law. When the fluids are compressible, the equations governing the system become quite complicated. To simplify the set of equations, Spiegel and Veronis (71 made the assumptions: (i) the depth of fluid layer is much smaller than the scale height as defined by them and (ii) the fluctuations in temperature, pressure and density, introduced due to motion, do not exceed their total static variations. Under these assumptions the flow equations were found to be the same as those for incompressible fluids except that the static temperature gradient is replaced by its excess over the adiabatic. Motivated by the fact that knowledge regarding fluid-particle mixtures is not commensurate with their industrial and scientific importance, we set out to study the effects of suspended particles and Hall currents on thermal instability of compressible fluids in porous medium. The physical properties of comets, meteorites and interplanetary dust strongly suggest importance of porosity in astrophysical context [8]. The problem may find its usefulness in geophysics (e.g. Earth’s molten core) and astrophysics where Hall currents, suspended particles, compressibility and porosity are important. tTo whom all correspondence should be addressed.
1053
1054
R. C. SHARMA
2. FORMULATION
and U. GUFTA
OF THE PROBLEM
Let us consider an infinite horizontal, compressible fluid-particle layer of thickness d bounded by the planes z = 0 and z = d in the presence of uniform vertical magnetic field H (O,O, H) in porous medium. This layer is heated from below such that a steady adverse temperature gradient /3(= (d7’ld z 1) is maintained. The equations of motion and continuity for the fluid are
1
KN
=-Vp-~~l-fu+T(v-u)+~(VxH)xH,
,+;(uV)u
(1)
1
e$+V.(pu)=O, where p, cc, p and u (u, u, w) denote, respectively, the density, the viscosity, the pressure and the velocity of the pure fluid. v (I, r, s) and N (2, I) denote the velocity and number density of the suspended particles, e is medium porosity, kl is medium permeability, pc is magnetic permeability, g is acceleration due to gravity, f = (x, y, z), 3c= (O,O, 1) and K = 6xpp1’, ‘I’ being the particle radius, is the Stokes’ drag coefficient. Assuming uniform particle size, spherical shape and small relative velocities between the fluid and particles, the presence of particles adds an extra force term, in the equations of motion (l), proportional to the velocity difference between particles and fluid. The force exerted by the fluid on the particles is equal and opposite to that exerted by the particles on the fluid. The buoyancy force on the particles is negligibly small. Interparticle reactions are ignored for we assume that the distances between particles are quite large compared with their diameters. If mN is the mass of particles per unit volume, then the equations of motion and continuity for the particles are mN $++‘)v
I
= KN(u-v),
++V(Nv)=O. Let C,, C,, C,,, T and 4 denote, respectively, the heat capacity of fluid at constant volume, the heat capacity of fluid at constant pressure, the heat capacity of particles, the temperature and the “effective thermal conductivity” of the pure fluid. Assuming that the particles and the fluid are in thermal equilibrium, the equation of heat conduction gives W3
+ P,W
- e)l g
+~C”~u*V)T+~NC~~
(
c;~+vV
>
T=qV2T,
t-51
where ps, C, are the density and the heat capacity of the solid matrix respectively. The Maxwell’s equations in the presence of Hall currents give V*H=O,
(6)
where 77, c, N’ and e denote, respectively, the resistivity, the speed of light, the electron number density and the charge of an electron. The initial state of the system is taken to be a quiescent layer (no settling) with a uniform partide distribution No and is given by u=(O,O,O), T = T(z),
P = P(Z),
H =
(0, 0, H),
p = p(z)
v = (090, O), and
N = NO, a constant,
(8) (9)
Thermal instability of compressible fluids
105.5
where, following Spiegel and Veronis [7], we have T(z) = --/I2 + T,, P(Z) = Pm P(Z) = Pl# - %(T - Tm)+ K&P - Prn)l~
Spiegel and Veronis [7] expressed any state variable say X, in the form
x = x, + X,(z)
+ X’(x, y,
2,
(10)
t),
where X, stands for the constant space distribution of X, X0 is the variation in X in the absence of motion and X’(x, y, z, t) stands for the fluctuations in X due to the motion of the fluid. Thus, o,,, and pm stand for the constant space distribution of p and p and p0 and Tostand for the density and temperature of the fluid at the lower boundary. Following Spiegel and Veronis’s [7] assumptions and results for compressible fluids, the flow equations are found to be the same as those of incompressible fluids except that the static temperature gradient /I is replaced by its excess over the adiabatic (/? -g/Cp).
3. PERTURBATION
EQUATIONS
Let dp, 6p, 8, u(u, v, w), v (I, r, s), II (h,, It,,, h,) and N denote the perturbations in fluid velocity, particle velocity, magnetic field H and particle pressure, density, temperature, number density No respectively. Then the linearized hydromagnetic perturbation equations of the fluid-particle layer, under Spiegel and Veronis’s [7] assumptions, are l&l --_=
Eat
-$vsp
F(v-“)+$
-g
m
m
v*u=o,
(V x h) x H, m
(11) (12)
av
mlv,,t=KN,(u-v),
.z+
V . (N,,v) = 0,
(13) (14)
(1%
(E+hC);=(+)(W+hS)+Kv*8, P
V.h=O, r$=Vx(uxH)+eqV*h-&Vx[(Vxh)XH],
(16) (17)
where a;, = UT, = a; say, v = pip,, K = q/pm& and g/C,, v and K stand for the adiabatic gradient, kinematic viscosity and thermal diffusivity respectively. Also,
-pt h_fC
G ’
R. C. SHARMA
1056
The linearized dimensionless perturbation
and U. GUITA
equations are
($?3), N,1~=-~6p-~u+o(f-u)+N,
(18)
(~-~), N,‘~=-asp-~,+o(r-a)+No paW= p’ at
--$p-$w+N&+w(s-w), du+dv+aw=,, ax
ay
dz
(23) (24) (25)
(26) (27) (E + hc) $-f = ($+v
+ hs) + V28,
where Np, = W/K is the modified Prandtl number, N, = rv/r_r is the modified magnetic Prandtl number, Nn = gabd4/vK is the Rayleigh number, No = ~LeH2d2/4~p,vr) is the Chandrasekhar number, M = EN/N,, Ml = cH/4nN’eq is the Hall parameter, o = KN,-,d2/pDIvE, t = mK/Kd2, f = mNo/p, = zwNp, is the mass fraction, G = C,/3lg and P = klld2. Here physical variables have been scaled using d, d2/K, K/d, pvK/d2, /3d and HK/~ as the length, time, velocity, pressure, temperature and magnetic field scale factors respectively. Consider the case of two free boundaries and the medium adjoining the fluid is non-conducting. The boundary conditions appropriate for the problem are [l, 61 a2W
w=jg =e=o,
atz=Oandl,
and h,, h,, h, are continuous with an external vacuum field. Here 5;= (&/ax - du/dy) and 5 = (i3h,/dx - dh,/dy) are the z-components current density respectively. After eliminating v and 6p, equations (18)-(28) can be expressed as
(30) of vorticity and
(31) (32) (33) (34) (35)
Thermal instability of compressible fluids
1057
where
a2
( _A+Fd>
LI=&,’ v2=
tat2
a2
32 2+-p ay
ax2+
4. THE
at
9
a2
F=f
+l,
v2_dz+ Iax2
DISPERSION
L,=r-g+l,
A=h+l.
82 2’
RELATION
Analysing the disturbances in terms of normal modes by seeking solutions whose dependence on x, y and t is given by [w, 8, h,, h 5]= [w(z), 0(z), K(z), Z(z),
-WI1 x exp(uv + qy
+ nt),
(36)
where n is the growth rate and EY= (acz+ CY;)‘”is the wave number of the disturbance. Equations (31)-(35), with the help of expression (36) become (D2 - ou”)W = L,N,(D2
- a?)DK - L2NRd0,
(37)
Z = L2NoDX,
(38)
- (D2 - &)]X = K’DZ f M@2[NW N,-,‘Q
az)DK,
(39)
[N&N,% - (D2 - &‘)]K = l-‘DW - M,DX,
(40)
L&E + he)n - (02 -
(41)
where D = d/dz, Lt = N,‘(rn2 f Fn) and L2 = WI+ 1. Eliminating Z, X, K and 0 between equations (37)-(41), we obtain [(02 - lr2) - (E + h+](D2
- a2)W + L2No[(D2 - 2) - (E + he)n] -1
N,N,‘n)M;%-’
+ L,No
&f-‘E-2D2
L)2 I
x M,(D2 - o?)D2 + M;‘{(D2 - a”) - N,N,‘nj2 x e-‘M;‘{(D2 - cr2)- N~N~~~~~2]-1(~2 - g)W
+ L2No
-’
= (y)NRa2(m
+ ii)
(42)
Using the boundary conditions (29) and (30), it can be shown that all the even order derivatives of W vanish at the boundaries. Hence the proper solution of equation (42) characterizing the lowest mode is W = W, sin az,
(43)
where W. is a constant. Substituting the solution (43) in equation (42), we obtain the dispersion relation NR= (
G
(nr2+ &“)[(n’+ 2) + (E + he)n] > + $::;:;rn)M;‘r-’
+::;o,I,,
[(L, +$) + L2NQ(LI+$)-I
XM;'E-~J? M,~2(~2f~2)+M;1{(112+(y2)+NhNp11n}2 I[ -1 + L2N,(L,
+$)-’
E-‘M,‘{(n2
+ &) + N,,JV,‘n}n2
I
1
.
(44
1058
R. C. SHARMA and U. GUPTA 5.
STATIONARY
CONVECTION
When the instability sets in as stationary convection, by n = 0 and the dispersion relation (44) reduces to
the marginal state will be characterized
1
(45)
G accounting
for
N,~*E-'{(n* + CY*) + Nope%*}
(n*+ ~*){M:~*+(Jc*+ CY*)+N~PE-~JG*} ' For fixed values of P, No, Ml and A, let the non-dimensional compressibility effects be also kept as fixed; then we find that
number
where NCRand K denote, respectively, the critical Rayleigh numbers in the absence and presence of compressibility. Since critical Rayleigh number is positive and finite, so G > 1 and we obtain a stabilizing effect of compressibility, i.e. its effect is to postpone the onset of thermal convection in fluid-particle layer. To study the effects of suspended particles, medium permeability, magnetic field and Hall currents, we examine the natures of dN,/&, dNR/dP, dN,/dNo and dN,ldM, respectively. Equation (45) gives Nojd*e-‘{(jd* + (u2) + NQP~-'n2} (n~+~~~M~~~+(~~+~)+N~P~-~~~~
1 '
(47)
which is negative. The effect of suspended particles is, thus, destabilizing on thermal instability of compressible fluid-particle layer in the presence of Hall currents through porous medium. It is evident from equation (45) that 1 -i+
P
1
(Nalc2~-')'M:~" (n*+ cu'){Mfn'+(n*+ d)+N,Pc'rc*}* '
(48)
which is positive if
P{MIn-(n2+d)1n}>
(n*+ ti)'n{M:~2+(n2+2)) , N&-h*
and is negative if
(n*+ d)'n{M:n2+(n2+ d)} P{M,r--((j~*+d)~~}< N,E-Y Thus, the medium permeability has both stabilizing and destabilizing effects depending upon In the absence of magnetic field, the medium the values of the various parameters. permeability has a destabilizing effect since for this case
which is always negative. The medium permeability, thus, succeeds in stabilizing the thermal instability of compressible fluid-particle layer for certain wave numbers in the presence of magnetic field, which was unstable in the absence of magnetic field. Equation (45) yields
(n* + (U~)JC’E-’ a’A
{M$r*+(n*+ ~2)-tNQP~-1n2}-1
x[{(n*+~?)+N,PE-'~*}+M:~~~N~PE-~{M:~*+(I~*+ CI?)+N~PE%*}-~], (50) which is always positive. This shows that magnetic field has a stabilizing effect on the system.
Thermal instability of compressible fluids
1059
Equation (45) also yields G dNR ;ir?i?;=-‘G-1 (
(3t.*+ ~?)N~e-~&r~{(~* >
+ at2) + NoP&r*}
ouZfi{M$r2 + (Jr* + a2) + N*PCW}2
’
(51)
which is always negative. Therefore, the effect of Hall currents is destabilizing on the thermal convection in compressible fluid-particle layer in porous medium.
6. EXISTENCE
OF OSCILLATORY
MODES
Multiplying equation (37) by W*, the complex conjugate of W, integrating over the range of z and using equations (38)-(47) together with the boundary conditions (29) and (30), we obtain
(53) which are all positive definite. Putting IZ= ino, where no is real, in equation (52) and equating the ima~nary parts on both sides, we obtain either no-
-
0,
(54)
or J+
-r-2
-Zs)-N,~t(l, +Z6) KN,;'fiF-f >Z,+ N,E Np2Np;'tl(Z2
- Np;'Z?F+;)Z4+NRm2($){T17+(E+h~)Z~}] ( X
N,'(Z?
+ 1-
F) - ;]Z, f NogN,N;,‘fi(Z2
- 4)
x { tz,-I” (E f hE)&}]-‘.
(55)
In the absence of magnetic field
+
-
ti
Np;L~F-~)Z,+N~~2(~){~Z,+(E+h~)zs)]
(56) r2[ (N$(fi
ES 31:t-F
+ 1 - F) - ;}Z, + Nua*(&){rZ,
+ (E + ht)z8)]*
1060
R. C. SHARMA
and U. GUPTA
From equations (54)-(56) we can draw the following conclusions: (1) The presence of magnetic field (and therefore Hall currents) introduces oscillatory modes in the system. (2) In the absence of magnetic field, the expression for n& given by (56) is negative if urn - k,K 9
fk,K
(57)
>
for all positive Nn. Now, since no is real and nz is negative we must have no = 0. This establishes that n is real for positive NR in the absence of magnetic field if the inequality (57) holds true and that the principle of exchange of stabilities is valid for this case. However if the inequality (57) is violated, then oscillatory modes may come into play even in the absence of magnetic field.
REFERENCES [l] [2] [3] [4]
S. CHANDRASEKHAR, Hydrodynamic and Hydromagnetic Stability. Dover, New York (1981). A. S. GUPTA. Rev. Roum. Math. Pures Et. Appl. 12,665 (1%7). K. CHANDRA, Proc. R. Sot. L.ond. AM, 231 (1938). J. W. SCANLON and L. A. SEGEL, Phys. Fluids 16, 1573 (1973). [S] R. C. SHARMA, K. PRAKASH and S. N. DUBE, Acra Phys. Hung. 40,3 (1976). (61 E. R. LAPWOOD, Proc. Comb. Phil. Sot. 44,508 (1948). [7] E. A. SPIEGEL and G. VERONIS, Asfrophys. J. Wl, 442 (1960). [8] J. A. M. McDonnel, Cosmic Dust p. 330. Wiley, Toronto (1978). (Received 7 July 1992; accepted 6 August 1992)