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Effect of rotation on a ferromagnetic fluid heated and soluted from below in the presence dust particles
Applied Mathematics and Computation 177 (2006) 614–628 www.elsevier.com/locate/amc
Effect of rotation on a ferromagnetic fluid heated and soluted from below in the presence dust particles Sunil a
a,*
, Anu Sharma a, R.G. Shandil
b
Department of Applied Sciences, National Institute of Technology (Deemed University), Hamirpur, HP 177 005, India b Department of Mathematics, Himachal Pradesh University, Summer Hill, Shimla, HP 171 005, India
Abstract This paper deals with the theoretical investigation of the effect of rotation on a layer of a ferromagnetic fluid permeated with dust particles, heated and soluted from below subjected to a transverse uniform magnetic field. For a flat fluid layer contained between two free boundaries, an exact solution is obtained using a linearized stability theory and a normal mode analysis method. For the case of stationary convection, non-buoyancy magnetization and dust particles have always a destabilizing effect, whereas stable solute gradient and rotation have a stabilizing effect on the onset of instability. The critical wave number and the critical magnetic thermal Rayleigh number for the onset of instability are also determined numerically for sufficiently large values of buoyancy magnetization M1. Graphs have been plotted by giving numerical values to the parameters, to depict the stability characteristics. It is observed that the critical magnetic thermal Rayleigh number is reduced solely because the heat capacity of clean fluid is supplemented by that of the dust particles. The principle of exchange of stabilities is found to hold true for the ferromagnetic fluid heated from below in the absence of dust particles, stable solute gradient and rotation. The oscillatory modes are introduced due to the presence of the dust particles, stable solute gradient and rotation, which were non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained. 2005 Elsevier Inc. All rights reserved. Keywords: Ferromagnetic fluid; Thermosolutal convection; Dust particles; Rotation; Magnetization
1. Introduction In the last millennium, the investigation on ferromagnetic fluid attracted researchers because of the increase of applications of magnetic fluids. The most famous application of magnetic fluids is the sealing of rotating shafts. This advantage is commonly used in various technical applications like the sealing of hard disc drives, rotating X-ray tubes and rotating vacuum feed-throughs where reliable sealing at low friction is required. The use of magnetic fluids as a heat transfer medium that may be magnetically hold in a certain position is
*
Corresponding author. E-mail addresses: [email protected], [email protected] (Sunil).
0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.09.092
Sunil et al. / Applied Mathematics and Computation 177 (2006) 614–628
615
nowadays the commercially most important branch of ferrofluid manufacturing. The major application in this field is cooling of loudspeakers, enabling a significant increase of the maximum acoustical power without any geometrical changes of the speaker system. The third important field of application of magnetic fluids is their use in biomedical applications. For examples, their use as a contrast medium in X-ray examinations [1] and for positioning tamponade for retinal detachment repair in eye surgery [2] has been reported. Rosensweig [3] has given an authoritative introduction to research on magnetic liquids in his monograph and the study of the effect of magnetization yields interesting information. This magnetization, in general, is function of the magnetic field, temperature and density of the fluid. Any variation of these quantities can induce a change of body force distribution in the fluid. This leads to convection in ferrofluids in the presence of magnetic field gradient. This mechanism is known as ferroconvection, which is similar to Be´nard convection. A detailed account of the theoretical and experimental results of the onset of thermal instability (Be´nard convection) in a fluid (non-magnetic) layer under varying assumptions of hydrodynamics and hydromagnetics has been given in the celebrated monograph by Chandrasekhar [4]. The convective instability of a ferromagnetic fluid for a fluid layer heated from below in the presence of uniform vertical magnetic field has been considered by Finlayson [5]. Thermoconvective stability of ferrofluids without considering buoyancy effects has been investigated by Lalas and Carmi [6], whereas Shliomis [7] analyzed the linearized relation for magnetized perturbed quantities at the limit of instability. The thermal convection in a magnetic fluid has been considered by Zebib [8], whereas the stability of a static magnetic fluid under the action of an external pressure drop has been studied by Polevikov [9]. Schwab et al. [10] investigated experimentally the Finlayson’s problem in the case of a strong magnetic field and detected the onset of convection by plotting the Nusselt number versus the Rayleigh number. Then, the critical Rayleigh number corresponds to a discontinuity in the slope. Later, Stiles and Kagan [11] examined the experimental problem reported by Schwab et al. [10] and generalized the Finlayson’s model assuming that under a strong magnetic field, the rotational viscosity augments the shear viscosity. The Be´nard convection in ferromagnetic fluids has been considered by many authors [12–22]. The ferromagnetic fluid has been considered to be clean in all the above studies. In many situations, the fluid is often not pure but contains suspended/dust particles. Saffman [23] has considered the stability of laminar flow of a dusty gas. Scanlon and Segel [24] have considered the effects of suspended particles on the onset of Be´nard convection, whereas Sharma et al. [25] have studied the effect of suspended particles on the onset of Be´nard convection in hydromagnetics. The suspended particles were thus found to destabilize the layer. Palaniswamy and Purushotham [26] have studied the stability of shear flow of stratified fluids with fine dust and found the effects of fine dust to increase the region of instability. On the other hand, the multiphase fluid systems are concerned with the motion of a liquid or gas containing immiscible inert identical particles. Of all multiphase fluid systems observed in nature, blood flows in arteries, flow in rocket tubes, dust in gas cooling systems to enhance the heat transfer processes, movement of inert solid particles in atmosphere, sand or other particles in sea or ocean beaches are the most common examples of multiphase fluid systems. Naturally studies of these systems are mathematically interesting and physically useful for various good reasons. The effect of dust particles on non-magnetic fluids has been investigated by many authors [27–30]. In the standard Be´nard problem the instability is driven by a density difference caused by a temperature difference between the upper and lower planes bounding the fluid. If the fluid is additionally has salt dissolved in it then there are potentially two destabilizing sources for the density difference, the temperature field and salt field. When there are two effects such as this, the phenomenon of convection which arises is called double-diffusive convection or thermohaline convection. Vaidyanathan et al. [31] studied ferrothermohaline convection in which a horizontal layer of an incompressible ferromagnetic fluid of thickness d in the presence of transverse magnetic field, heated from below and salted from above is considered. They found that the salinity of a ferromagnetic fluid enables the fluid to get destabilized more when it is salted from above. The really interesting situation from both a physical and a mathematical viewpoint arises when the layer is simultaneously heated from below and salted from below. More recently, Sunil et al. [32] have studied the effect of magnetic field dependent viscosity on thermosolutal convection in a ferromagnetic fluid. In view of the above investigations, it is attempted to discuss the effect of rotation on a ferromagnetic fluid heated and soluted from below in the presence of dust particles, subjected to a vertical magnetic field. The present study can serve as a theoretical support for experimental investigations e.g. evaluating the influence
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of impurities in a ferromagnetic fluid on thermal convection phenomena. This problem, to the best of our knowledge, has not been investigated yet. 2. Mathematical formulation of the problem Consider an infinite, horizontal, electrically non-conducting incompressible thin ferromagnetic fluid layer of thickness d, embedded in dust particles, heated and soluted from below (see Fig. 1). A uniform magnetic field H0 acts along the vertical direction which is taken as z-axis. The temperature T and solute concentration C at the bottom and top surfaces z ¼ 12 d are T0, T1 and C0, C1, respectively, and a uniform temperature gradient b ð¼ j dT jÞ and a uniform solute gradient b0 ð¼ j dC jÞ are maintained. The whole system is assumed dz dz to rotate with angular velocity X along the vertical axis, which is taken as z-axis. Both the boundaries are taken to be free and perfect conductors of heat. The gravity field g = (0, 0, g) and uniform vertical magnetic field intensity H = (0, 0, H0) pervade the system. The mathematical equations governing the motion of a ferromagnetic fluid for the above model are as follows: The continuity equation for an incompressible ferromagnetic fluid is r q ¼ 0.
ð1Þ
The momentum equation is o þ ðq rÞ q ¼ rp þ qg þ KN ðqd qÞ þ r ðHBÞ þ lr2 q þ 2q0 ðq XÞ; q0 ot
ð2Þ
where q, q0, q, l and p are the fluid density, reference density, velocity, dynamic viscosity (constant) and pressure of ferromagnetic fluid, respectively; t and B denote, respectively, the time and magnetic induction. qd(x, t) and N(x, t) denote velocity and number density of the dust particles, respectively. x = (x, y, z) and K = 6plg, g being the particle radius, is the Stokes drag coefficient. Assuming a uniform particle size, a spherical shape, and small relative velocities between the fluid and dust particles, the presence of dust particles adds an extra force term in the equations of motion (2), proportional to the velocity difference between the dust particles and 2 the fluid. The effect of rotation contributes two terms: (a) centrifugal force q20 gradjX rj and (b) coriolis z-axis
z=
Ω = (0, 0, Ω)
d 2
T1, C1 g = (0, 0, –g)
Incompressible rotating ferromagnetic fluid with dust particles
H = (0, 0, H0) z=0
z=– y-axis
d 2 T0, C0 Heated and soluted from below Fig. 1. Geometrical configuration.
x-axis
Sunil et al. / Applied Mathematics and Computation 177 (2006) 614–628
617
2
force 2q0(q · X). In Eq. (2), p ¼ p0 12 q0 jX rj is the reduced pressure, whereas p 0 stands for fluid pressure. Two additional simplifications are assumed in Eq. (2): we assume the viscosity is isotropic and independent of the magnetic field intensity. Both approximations simplify the analysis without changing the ultimate conclusion. We also use the Boussinesq approximation by allowing the density to change only in the gravitational body force term. Since density variations are mainly due to variations in temperature and solute concentration, the equation of state for the fluid is given by q ¼ q0 ½1 aðT T a Þ þ a0 ðC C a Þ;
ð3Þ
where a and a 0 thermal expansion coefficient and an analogous solvent coefficient of expansion, respectively. 1Þ Ta is the average temperature given by T a ¼ ðT 0 þT , where T0 and T1 are the constant average temperatures of 2 1Þ the lower and upper surfaces of the layer and Ca is the average concentration given by C a ¼ ðC0 þC , where C0 2 and C1 are the constant average concentrations of the lower and upper surfaces of the layer. Since the force exerted by the fluid on the particles is equal and opposite to that exerted by the particles on the fluid, there must be an extra force term, equal in magnitude but opposite in sign, in the equations of motion for the particles. The buoyancy force on the particles is neglected. Inter-particle reactions are also ignored since we assume that the distances between particles are quite large compared with their diameters. The effects due to pressure, gravity, viscous force on the particles are negligibly small, and therefore ignored. If mN is the mass of particles per unit volume, then the equations of motion and continuity of the dust particles, under the above assumptions, are o mN þ ðqd rÞ qd ¼ KN ðq qd Þ; ð4Þ ot oN þ r ðN qd Þ ¼ 0. ð5Þ ot The equations expressing the conservation of temperature and solute concentration in presence of dust particles are " # oM DT oM DH o þ l0 T þ mNC pt þ qd r T ¼ K 1 r 2 T ; . ð6Þ q0 C V ;H l0 H oT V ;H Dt oT V ;H Dt ot " # oM DC oM DH o þ l0 C þ mNC pt þ qd r C ¼ K 01 r2 C; ð7Þ q0 C V ;H l0 H oC V ;H Dt oC V ;H Dt ot where CV,H, Cpt, K1, K 01 , M and l0 are the specific heat at constant volume and magnetic field, specific heat of dust particles, thermal conductivity, solute conductivity, magnetization [defined by (10)] and magnetic permeability of free space (l0 = 4p · 107 Henry m1), respectively. The partial derivatives of M are material properties which can be evaluated once the magnetic equation of state, such as (11), is known. Since in ferrohydrodynamics it is usual to assume that the free charge and the electric displacement are absent, Maxwell’s equations are employed, r B ¼ 0;
r H ¼ 0;
ð8a; bÞ
where the magnetic induction is given by B ¼ l0 ðH þ MÞ.
ð9Þ
We assume that the magnetization is aligned with the magnetic field, but allow a dependence on the magnitude of the magnetic field, temperature and salinity, so that H ð10Þ M ¼ MðH ; T ; CÞ. H The magnetic equation of state is linearized about the magnetic field, H0, an average temperature, Ta, and the average salinity, Ca, to become M ¼ M 0 þ vðH H 0 Þ K 2 ðT T a Þ þ K 3 ðC C a Þ;
ð11Þ
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where magnetic susceptibility, pyromagnetic coefficient and salinity magnetic coefficient are defined by v ¼ ðoM Þ ; K 2 ¼ ðoM Þ and K 3 ¼ ðoM Þ , respectively. oH H 0 ;T a oT H 0 ;T a oC H 0 ;C a Here H0 is the uniform magnetic field of the fluid layer when placed in an external magnetic field ^ ^ H ¼ H ext 0 k; k is unit vector in the z-direction, H = jHj, M = jMj and M0 = M(H0, Ta, Ca). The basic state is assumed to be quiescent state and is given by q ¼ qb ¼ 0;
qd ¼ ðqd Þb ¼ 0;
C ¼ C b ðzÞ ¼ b0 z þ C a ; b ¼ K 2 bz K 3 b0 z ^ k; Mb ¼ M 0 þ 1þv 1þv
q ¼ qb ðzÞ; T1 T0 ; d
p ¼ pb ðzÞ;
T ¼ T b ðzÞ ¼ bz þ T a ; C1 C0 K 2 bz K 3 b0 z ^ þ b0 ¼ ; Hb ¼ H 0 k; 1þv 1þv d
N ¼ Nb ¼ N0
ð12Þ
and H 0 þ M 0 ¼ H ext 0 .
Only the spatially varying parts of H0 and M0 contribute to the analysis, so that the direction of the external magnetic field is unimportant and the convective phenomenon is the same whether the external magnetic field is parallel or antiparallel to the gravitational force. 3. The perturbation equations and normal mode analysis method Assume small perturbations around the basic state, and let q 0 = (u, v, w), q01 ¼ ð‘; r; sÞ, p 0 , q 0 , h, c, H 0 , M 0 and N 0 denote, respectively, the perturbations in ferromagnetic fluid velocity, particle velocity, pressure, density, temperature, concentration, magnetic field intensity, magnetization and suspended particles number density N0. The change in density q 0 , caused mainly by the perturbations h and c in temperature and concentration, is given by q0 ¼ q0 ðah a0 cÞ.
ð13Þ
Then the linearized perturbation equations of the ferromagnetic fluid become ou op0 oH 01 ou 2 þ l0 ðM 0 þ H 0 Þ þ 2q0 Xv þ lr u mN 0 ; L0 q0 ¼ L0 ot ot ox oz ov op0 oH 0 ov L0 q0 ¼ L0 þ l0 ðM 0 þ H 0 Þ 2 2q0 Xu þ lr2 v mN 0 ; ot ot oy oz ow op0 oH 0 l K 2b 0 ¼ L0 H 3 ð1 þ vÞ K 2 h þ l0 ðM 0 þ H 0 Þ 3 þ lr2 w 0 L0 q0 ot 1þv oz oz þ
l0 K 3 b0 l K 2K 3 0 ow ; fH 03 ð1 þ vÞ þ K 3 cg 0 ðb h þ bcÞ þ gq0 ðah a0 cÞ mN 0 ot 1þv ð1 þ vÞ
ð14Þ
ð15Þ
ð16Þ
ou ov ow þ þ ¼ 0; ð17Þ ox oy oz 0 oh o oU1 l T 0 K 22 b L0 fqC 1 þ mN 0 C pt g l0 T 0 K 2 w þ mN 0 bC pt w; ð18Þ ¼ L0 K 1 r2 h þ qC 1 b 0 ot ot oz ð1 þ vÞ 0 oc o oU2 l0 C 0 K 23 b0 0 2 0 0 0 L0 fqC 1 þ mN 0 C pt g l0 C 0 K 3 ¼ L0 K 1 r c þ qC 1 b w þ mN 0 b0 C pt w; ð19Þ ot ot oz ð1 þ vÞ where qC 1 ¼ q0 C V ;H þ l0 K 2 H 0 ;
qC 01
¼ q0 C V ;H l0 K 3 H 0 ;
L0 ¼
m o þ1 . K ot
ð20Þ
Sunil et al. / Applied Mathematics and Computation 177 (2006) 614–628
Eqs. (10) and (11) yield H 03 þ M 03 ¼ ð1 þ vÞH 03 K 2 h;
9 > > =
H 03 þ M 03 ¼ ð1 þ vÞH 03 þ K 3 c;
> > 0 H 0i þ M 0i ¼ 1 þ M H 0i ði ¼ 1; 2Þ; ; H0
619
ð21Þ
where we have assumed K2bd (1 + v)H0 and K3b 0 d (1 + v)H0 as the analysis is restricted to physical situation in which the magnetization induced by temperature and concentration variations is small compared to that induced by the external magnetic field. Eq. (8b) means we can write H0 ¼ rðU01 U02 Þ, where U01 is the perturbed magnetic potential and U02 is the perturbed magnetic potential analogous to solute. Eliminating u, v, p 0 between Eqs. (14)–(16) and using Eq. (17), we obtain o o 2 L0 q0 lr þ mN 0 r2 w ot ot
l0 K 2 b 2 o 0 l0 K 3 b0 2 o 0 0 0 r ð1 þ vÞ U U2 K 2 h þ U U2 þ K 3 c r ð1 þ vÞ ¼ L0 1þv 1 oz 1 oz 1 1þv 1 l K 2K 3 2 0 of þq0 gr21 ðah a0 cÞ 0 r ðb h þ bcÞ 2q0 X . ð22Þ oz ð1 þ vÞ 1 The vertical component of the vorticity equation is of ow 2 þ lr f ; ½L0 q0 þ mN 0 ¼ L0 2q0 X ot oz where f ¼ ov ou stands for the z-component of the vorticity. ox oy From (21), we have o2 U01 M0 oh ð1 þ vÞ 2 þ 1 þ r21 U01 K 2 ¼ 0; oz oz H0 o2 U02 M0 oc ð1 þ vÞ 2 þ 1 þ r21 U02 K 3 ¼ 0. oz oz H0
ð23Þ
ð24Þ ð25Þ
Now we analyze the perturbations w, h, c, f, U01 , and U02 into two-dimensional periodic waves; and considering disturbances characterized by a particular wave number k. Thus we assume to all quantities describing the perturbation a dependence on x, y, and t of the form
ð26Þ w; h; c; f; U01 ; U02 ¼ ½W ðz; tÞ; Hðz; tÞ; Cðz; tÞ; Zðz; tÞ; U1 ðz; tÞ; U2 ðz; tÞ exp iðk x x þ k y yÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where kx, ky are the wave numbers along the x- and y-directions, respectively, and k ¼ ðk 2x þ k 2y Þ is the resultant wave number. Following the normal mode analysis, the linearized perturbation dimensionless equations become o o 2 2 L0 ðD a Þ þ f ðD2 a2 ÞW ot ot ¼ aR1=2 L0 ½ðM 1 M 4 ÞDU1 ð1 þ M 1 M 4 ÞT
1=2 þ aS 1=2 L0 M 01 M 04 DU2 þ 1 M 01 þ M 04 C L0 T A DZ ; o o 1=2 2 2 L0 ðD a Þ þ f Z ¼ L0 T A DW ; ot ot oT o L0 P r ð1 þ hÞ M 2 DU1 ¼ L0 ðD2 a2 ÞT þ aR1=2 L0 ð1 M 2 Þ þ h W ; ot ot
oC o L0 P s ð1 þ h0 Þ M 02 DU2 ¼ L0 ðD2 a2 ÞC þ aS 1=2 L0 1 M 02 þ h0 W ; ot ot
ð27Þ ð28Þ ð29Þ ð30Þ
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Sunil et al. / Applied Mathematics and Computation 177 (2006) 614–628
D2 U1 a2 M 3 U1 DT ¼ 0; D
2
U2
a
2
M 3 U2
ð31Þ
DC ¼ 0;
ð32Þ
where the following non-dimensional parameters are introduced: t ¼
mt ; d2
R¼
gabd 4 qC 1 ; mK 1
d W ¼ W; m S¼
U1 ¼
ð1 þ vÞK 1 aR1=2 U1 ; K 2 qC 1 bmd 2
ga0 b0 d 4 qC 01 ; mK 01
T ¼
U2 ¼
K 1 aR1=2 H; qC 1 bmd
ð1 þ vÞK 01 aS 1=2 U2 ; K 3 qC 01 b0 md 2 C ¼
K 01 aS 1=2 C; qC 01 b0 md
a ¼ kd;
z z ¼ ; d
l0 K 22 b l0 K 23 b0 ; M 01 ¼ ; M1 ¼ ð1 þ vÞaq0 g ð1 þ vÞa0 q0 g
0 2 1þM H l0 T 0 K 22 l C K l0 K 2 K 3 b0 0 0 0 3 M2 ¼ ; M ; ; M 02 ¼ ; M ¼ ¼ 3 4 ð1 þ vÞ ð1 þ vÞqC 1 ð1 þ vÞqC 01 ð1 þ vÞaq0 g l0 K 2 K 3 b M 4 M 01 K 3 b0 mm o mN 0 ; M ; s ¼ M 04 ¼ ¼ ¼ ¼ ; L ¼ s þ 1 ; f ¼ ; 5 0 0 2 0 ð1 þ vÞa q0 g ot M 1 M 4 K 2b q0 Kd 2 mN 0 C pt mN 0 C pt 2Xd 2 d2 0 Z. h¼ ; h ¼ ; T ¼ and Z ¼ A qC 1 qC 01 m m
D¼
o ; oz
Pr ¼
m qC 1 ; K1
Ps ¼
m qC 0 ; K 01 1
4. Exact solution for free boundaries Here we consider the case where both boundaries are free as well as perfect conductors of heat. The case of two free boundaries is of little physical interest, but it is mathematically important because one can derive an exact solution, whose properties guide our analysis. Here we consider the case of an infinite magnetic susceptibility v and we neglect the deformability of the horizontal surfaces. Thus the exact solution of the system (27)–(32) subject to the boundary conditions W ¼ D2 W ¼ T ¼ C ¼ DU1 ¼ DU2 ¼ 0
at z ¼
1 2
ð33Þ
is written in the form
W ¼ A1 ert cos pz ; T ¼ B1 ert cos pz ; C ¼ F 1 ert cos pz ; DU1 ¼ C 1 ert cos pz ; C 1 rt E1 rt rt U1 ¼ e sin pz ; DU2 ¼ E1 e cos pz ; U2 ¼ e sin pz ; p p
ð34Þ
where A1, B1, C1, E1, F1 are constants and r is the growth rate which is, in general, a complex constant. Substituting Eq. (34) in Eqs. (27)–(32) and dropping asterisks for convenience, we get the following equations: h i fðr þ p2 þ a2 Þð1 þ srÞ þ f rg2 ðp2 þ a2 Þ þ p2 ð1 þ srÞ2 T A A1 aR1=2 fðr þ p2 þ a2 Þð1 þ srÞ þ f rg½ð1 þ srÞð1 þ M 1 M 4 ÞB1 þ aR1=2 fðr þ p2 þ a2 Þð1 þ srÞ þ f rg½ðM 1 M 4 Þð1 þ srÞC 1 þ aS 1=2 fðr þ p2 þ a2 Þð1 þ srÞ þ f rg½ð1 þ srÞð1 M 01 þ M 04 ÞF 1
þ aS 1=2 fðr þ p2 þ a2 Þð1 þ srÞ þ f rg M 01 M 04 ð1 þ srÞ E1 ¼ 0; 1=2
2
ð35Þ
2
½aR fh þ ð1 M 2 Þð1 þ srÞgA1 ½fp þ a þ P r ð1 þ hÞrgð1 þ srÞB1 þ ½P r M 2 rð1 þ srÞC 1 ¼ 0; 1=2 0
aS h þ 1 M 02 ð1 þ srÞ A1 p2 þ a2 þ P s ð1 þ h0 Þr ð1 þ srÞ F 1 þ P s M 02 rð1 þ srÞ E1 ¼ 0;
ð36Þ ð37Þ
Sunil et al. / Applied Mathematics and Computation 177 (2006) 614–628
p2 B1 þ ðp2 þ a2 M 3 ÞC 1 ¼ 0; 2
2
2
p F 1 þ ðp þ a M 3 ÞE1 ¼ 0.
621
ð38Þ ð39Þ
For existence of non-trivial solutions of the above equations, the determinant of the coefficients of A1, B1, C1, E1 and F1 in Eqs. (35)–(39) must vanish. This determinant on simplification yields T 6 r61 þ iT 5 r51 þ T 4 r41 iT 3 r31 T 2 r21 þ iT 1 r1 þ T 0 ¼ 0.
ð40Þ
T 6 ¼ s21 bL2 L3 ;
ð41Þ
T 5 ¼ s21 b2 L1 ðL3 þ L2 Þ þ 2s1 bL2 L3 L6 ; T 4 ¼ s21 L21 b3 þ 2s1 L1 L6 ðL2 þ L3 Þb2 þ ðL26 þ 2s1 bÞL2 L3 b þ x1 s21 S 1 L2 L5 L8 x1 s21 R1 L3 L4 L7 þ s21 T A1 L2 L3 ;
ð42Þ
Here
T3 ¼
2L21 s1 L6 b3
þ b 2L2 L3 L6 þ L1 ðL26 þ 2s1 bÞðL2 þ L3 Þ
ð43Þ
2
þ T A1 f2s1 L2 L3 þ bs21 L1 ðL2 þ L3 Þg þ x1 S 1 s1 L5 ½fh0 þ L8 gL2 þ L6 L8 L2 þ s1 bL1 L8 x1 R1 s1 L4 ½ðh þ L7 ÞL3 þ L6 L7 L3 þ s1 bL1 L7 ;
T 2 ¼ L2 L3 þ L21 L26 þ 2s1 b þ 2L1 L6 ðL2 þ L3 Þ b3 x1 R1 L4 ½s1 bL7 ðL3 þ L1 L6 Þ þ ðh þ L7 Þðs1 bL1 þ L3 L6 Þ þ x1 S 1 L5 ½s1 bL8 ðL2 þ L1 L6 Þ þ ðh0 þ L8 Þðs1 bL1 þ L2 L6 Þ þ T A1 s21 L21 b2 þ 2s1 L1 ðL2 þ L3 Þb þ L2 L3 ;
ð44Þ
ð45Þ
4
T 1 ¼ L1 ½2L1 L6 þ L2 þ L3 b þ L1 T A1 ½2s1 bL1 þ L2 þ L3 b x1 R1 L4 ½bðh þ L7 ÞðL3 þ L1 L6 Þ þ s1 L1 L7 b2 þ x1 S 1 L5 bðh0 þ L8 ÞðL2 þ L1 L6 Þ þ s1 L1 L8 b2 ; T 0 ¼ b2 L1 b3 L1 þ L1 T A1 x1 R1 ðh þ L7 ÞL4 þ x1 S 1 ðh0 þ L8 ÞL5 ;
ð46Þ ð47Þ
a2
where R1 ¼ pR4 ; S 1 ¼ pS4 ; x1 ¼ p2 ; ir1 ¼ pr2 ; s1 ¼ sp2 ; T A1 ¼ TpA2 ; b ¼ ð1 þ x1 Þ, L1 ¼ 1 þ x1 M 3 ; L2 ¼ P r ½fð1 M 2 Þ þ x1 M 3 g þ L1 h;
L3 ¼ P s 1 M 02 þ x1 M 3 þ L1 h0 ; L4 ¼ f1 þ x1 M 3 ð1 þ M 1 M 4 Þg; L5 ¼ f1 þ x1 M 3 ð1 M 01 þ M 04 Þg; L6 ¼ ð1 þ s1 b þ f Þ; L7 ¼ 1 M 2 and L8 ¼ 1 M 02 . 5. The case of stationary convection Here we first consider the case when the instability sets in as stationary convection (and M 2 ffi 0; M 02 ffi 0Þ, the marginal state will be characterized by r1 = 0, then the Rayleigh number is given by n
o n o ð1 þ x1 Þ3 þ T A1 ð1 þ x1 M 3 Þ S 1 h01 ð1 þ x1 M 3 Þ þ x1 M 3 M 01 M15 1 R1 ¼ þ ; ð48Þ x1 h1 fð1 þ x1 M 3 Þ þ x1 M 3 M 1 ð1 M 5 Þg h1 fð1 þ x1 M 3 Þ þ x1 M 3 M 1 ð1 M 5 Þg which expresses the modified Rayleigh number R1 as a function of the dimensionless wave number x1, buoyancy magnetization parameter M1, the non-buoyancy magnetization parameter M3, Taylor number T A1 , solute gradient parameter S1, dust particles parameter h1, dust particles parameter analogous to solute h01 and the ratio of the salinity effect on magnetic field and pyromagnetic coefficient M5. Here we put h1 = (1 + h) and h01 ¼ ð1 þ h0 Þ. In the absence of dust particles, the values of h1 and h01 is one. Since the marginal state dividing stability from instability is stationary, this means that at the onset of instability there is no relative velocity between particles and fluid and hence no particle drag on the fluid. Therefore, the critical Rayleigh number is reduced solely because the heat capacity of the clean fluid is supplemented by that of the suspended (dust) particles.
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To investigate the effects of non-buoyancy magnetization, dust particles, stable solute gradient and rotadR1 dR1 dR1 tion, we examine the behaviour of dM ; ; and dTdRA1 , analytically. 3 dh1 dS 1 1 Eq. (48) yields h
i M 01 3 0 ð1 M Þ M fð1 þ x Þ þ T g þ S h x M 5 1 1 A 1 1 1 1 1 M5 dR1 ¼ ; ð49Þ 2 dM 3 h1 f1 þ x1 M 3 þ x1 M 3 M 1 ð1 M 5 Þg n
o ð1 þ x1 M 3 Þfð1 þ x1 Þ3 þ T A1 g þ x1 S 1 h01 ð1 þ x1 M 3 Þ þ x1 M 3 M 01 M15 1 dR1 ¼ ; ð50Þ dh1 x1 h21 f1 þ x1 M 3 þ x1 M 3 M 1 ð1 M 5 Þg n
o 0 0 1 h 1 þ x M þ x M M 1 1 3 1 3 1 M5 1 dR1 ; ð51Þ ¼ h1 f1 þ x1 M 3 þ x1 M 3 M 1 ð1 M 5 Þg dS 1 dR1 ð1 þ x1 M 3 Þ . ¼ dT A1 x1 h1 f1 þ x1 M 3 þ x1 M 3 M 1 ð1 M 5 Þg
ð52Þ
This shows that, for a stationary convection, the non-buoyancy magnetization and dust particles are found to have a destabilizing effect, whereas stable solute gradient and rotation have a stabilizing effect on the system. For M1 sufficiently large, we obtain the results for the magnetic mechanism n
o 0 0 1 3 S h ð1 þ x M Þ þ x M M 1 1 1 3 1 3 1 M5 1 fð1 þ x1 Þ þ T A1 gð1 þ x1 M 3 Þ þ ; ð53Þ N ¼ R1 M 1 ¼ x1 h1 M 3 ð1 M 5 Þ x21 h1 M 3 ð1 M 5 Þ where N is the magnetic thermal Rayleigh number. As a function of x1, N given by Eq. (53) attains its minimum when 2M 3 x41 þ ð3M 3 þ 1Þx31 fM 3 þ T A1 M 3 þ S 1 h01 þ 3gx1 2ð1 þ T A1 Þ ¼ 0.
ð54Þ
The values of critical wave number for the onset of instability are determined numerically using Newton– dN Raphson method by the condition dx ¼ 0. With x1 determined as a solution of Eq. (54), Eq. (53) will give 1 the required critical magnetic thermal Rayleigh number Nc. The critical magnetic thermal Rayleigh number (Nc), depends on the non-buoyancy magnetization parameter M3, Taylor number T A1 , stable solute gradient S1, dust particles parameter h1 and dust particles parameter analogous to solute h01 . Values of Nc determined in this fashion for various values of M3, T A1 , S1, h1 and h01 are given in Tables 1 and 2 and the results are further illustrated in Figs. 2–5. Figs. 2–5 represent the plots of critical magnetic thermal Rayleigh number Nc versus non-buoyancy magnetization parameter M3 [for various values of T A1 ], dust particles parameter h1 [for various values of S1], stable solute gradient S1 [for various values of M3] and Taylor number T A1 [for various values of h1], respectively. Figs. 2 and 3 illustrate that as non-buoyancy magnetization parameter M3 and dust particle parameter h1 increase, the critical magnetic Rayleigh number (Nc) decreases. Therefore, lower values of Nc are needed for onset of convection with an increase in M3 and h1, hence justifying the destabilizing effect of non-buoyancy magnetization and dust particles. It is evident from Figs. 4 and 5 that stable solute gradient and rotation have always a stabilizing effect on the system. It is also observed from Tables 1 and 2 that in the absence of dust particles ðh1 ¼ 1; h01 ¼ 1Þ, the critical magnetic thermal Rayleigh number is very high however in the presence of dust particles ðh1 > 1; h01 > 1Þ, critical magnetic thermal Rayleigh number is reduced solely because the heat capacity of clean fluid is supplemented by that of the dust particles. 6. The case of oscillatory modes Here we examine the possibility of oscillatory modes, if any, on stability problem due to the presence of stable solute gradient, rotation, dust particles and magnetization. Equating the imaginary parts of Eq. (40), we obtain
Sunil et al. / Applied Mathematics and Computation 177 (2006) 614–628
623
Table 1 Critical magnetic thermal Rayleigh numbers and wave numbers of the unstable modes at marginal stability for the onset of stationary convection for various values of Taylor number, non-buoyancy magnetization and dust parameters T A1
M3
M 01 ¼ 0:1, M5 = 0.1, S1 = 100 h1 = 1, h01 ¼ 1
h1 = 3, h01 ¼ 3
h1 = 5, h01 ¼ 5
h1 = 7, h01 ¼ 7
xc
xc
xc
xc
Nc
Nc
Nc
Nc
0
1 2 3 4 5
3.18 2.50 2.16 1.94 1.79
279.60 256.20 246.99 241.86 238.51
4.75 3.73 3.23 2.91 2.68
252.45 237.92 232.15 228.92 226.80
5.72 4.50 3.90 3.51 3.24
244.39 232.59 227.88 225.25 223.52
6.45 5.08 4.40 3.97 3.67
240.09 229.76 225.64 223.33 221.81
102
1 2 3 4 5
4.41 3.95 3.76 3.65 3.58
316.16 295.30 287.83 283.95 281.57
5.44 4.63 4.28 4.07 3.94
261.12 247.79 242.82 240.18 238.54
6.22 5.18 4.71 4.43 4.25
248.73 237.64 233.43 231.17 229.74
6.86 5.64 5.08 4.75 4.52
242.83 232.99 229.22 227.16 225.86
103
1 2 3 4 5
8.15 7.84 7.73 7.67 7.63
495.06 473.11 465.60 461.81 459.52
8.54 8.06 7.88 7.79 7.73
314.63 303.10 299.10 297.07 295.84
8.90 8.28 8.03 7.90 7.82
278.32 269.02 265.77 264.10 263.09
9.24 8.48 8.18 8.02 7.92
262.62 254.37 251.46 249.96 249.05
104
1 2 3 4 5
17.15 16.90 16.80 16.76 16.73
1313.20 1279.55 1268.21 1262.53 1259.11
17.25 16.95 16.84 16.79 16.75
582.78 569.44 564.95 562.69 561.33
17.35 17.00 16.88 16.81 16.77
436.68 427.42 424.29 422.72 421.77
17.45 17.05 16.91 16.84 16.80
374.06 366.55 364.01 362.73 361.96
105
1 2 3 4 5
36.85 36.60 36.52 36.48 36.45
4990.55 4925.73 4904.03 4893.16 4886.64
36.88 36.62 36.53 36.48 36.46
1806.27 1783.66 1776.10 1772.30 1770.03
36.90 36.63 36.54 36.49 36.46
1169.41 1155.25 1150.51 1148.13 1146.70
36.92 36.64 36.54 36.50 36.47
896.47 885.93 882.40 880.63 879.57
106
1 2 3 4 5
79.37 79.13 79.04 79.00 78.97
21747.05 21612.18 21567.13 21544.58 21531.05
79.38 79.13 79.04 79.00 78.98
7390.69 7345.27 7330.10 7322.50 7317.95
79.38 79.13 79.05 79.00 78.98
4519.42 4491.89 4482.69 4478.09 4475.32
79.39 79.13 79.05 79.00 78.98
3288.88 3269.01 3262.37 3259.05 3257.06
r1 r41 b s21 L1 ðL2 þ L3 Þb þ 2s1 L2 L3 L6 r21 2s1 L21 L6 b3 þ 2L2 L3 L6 þ L1 L26 þ 2s1 b ðL2 þ L3 Þ b2 þT A1 s1 f2L2 L3 þ bs1 L1 ðL2 þ L3 Þg x1 R1 s1 L4 ½fh þ L7 gL3 þ ðL3 L6 þ s1 L1 bÞL7 þ x1 S 1 s1 L5 ½fh0 þ L8 gL2 þðL2 L6 þ s1 L1 bÞL8 i þ L1 ð2L1 L6 þ L2 þ L3 Þb4 þ T A1 L1 ð2s1 L1 b þ L2 þ L3 Þb x1 R1 L4 bfh þ L7 gðL3 þ L1 L6 Þ þ s1 L1 L7 b2 þ x1 S 1 L5 bfh0 þ L8 gðL2 þ L1 L6 Þ þ s1 L1 L8 b2 ¼ 0.
ð55Þ
It is evident from Eq. (55) that r1 may be either zero or non-zero, meaning that the modes may be either nonoscillatory or oscillatory. In the absence of stable solute gradient, rotation and dust particles, we obtain the result (after simplification) as r1 ½P r fL7 þ x1 M 3 g þ 2ð1 þ x1 M 3 Þ ¼ 0. ð56Þ 6 Here the quantity inside the brackets is positive definite because the typical values of M2 are +10 [5]. Hence r1 ¼ 0; ð57Þ which implies that the oscillatory modes are not allowed and the principle of exchange of stabilities is satisfied, in the absence of stable solute gradient, rotation and dust particles. Thus from Eq. (55), we conclude that the
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Table 2 Critical magnetic thermal Rayleigh numbers and wave numbers of the unstable modes at marginal stability for the onset of stationary convection for various values of stable solute gradient, non-buoyancy magnetization and dust parameters S1
M3
M 01 ¼ 0:1, M5 = 0.1, T A1 ¼ 100 h1 = 1, h01 ¼ 1
h1 = 3, h01 ¼ 3
h1 = 5, h01 ¼ 5
h1 = 7, h01 ¼ 7
xc
xc
xc
xc
Nc
Nc
Nc
Nc
0
1 2 3 4 5
3.72 3.53 3.45 3.40 3.38
77.75 69.34 66.44 64.97 64.08
3.72 3.53 3.45 3.40 3.38
25.92 23.11 22.15 21.66 21.36
3.72 3.53 3.45 3.40 3.38
15.55 13.87 13.29 12.99 12.82
3.72 3.53 3.45 3.40 3.38
11.11 9.91 9.49 9.28 9.15
100
1 2 3 4 5
4.41 3.95 3.76 3.65 3.58
316.16 295.30 287.83 283.95 281.57
5.44 4.63 4.28 4.07 3.94
261.12 247.79 242.82 240.18 238.54
6.22 5.18 4.71 4.43 4.25
248.73 237.64 233.43 231.17 229.74
6.86 5.64 5.08 4.75 4.52
242.83 232.99 229.22 227.16 225.86
200
1 2 3 4 5
4.97 4.32 4.03 3.87 3.77
550.93 519.85 508.44 502.45 498.73
6.55 5.42 4.90 4.60 4.39
490.72 469.94 461.99 457.70 454.98
7.65 6.23 5.56 5.16 4.89
475.81 458.47 451.74 448.06 445.71
8.52 6.88 6.11 5.63 5.31
468.35 452.96 446.93 443.62 441.49
300
1 2 3 4 5
5.44 4.63 4.28 4.07 3.94
783.37 743.37 728.47 720.55 715.61
7.40 6.04 5.41 5.03 4.77
717.73 690.73 680.28 674.58 670.94
8.71 7.03 6.23 5.74 5.40
700.49 677.96 669.13 664.26 661.13
9.74 7.81 6.89 6.32 5.93
691.62 671.62 663.74 659.37 656.55
400
1 2 3 4 5
5.85 4.92 4.50 4.26 4.10
1014.14 966.11 948.01 938.33 932.25
8.11 6.57 5.85 5.41 5.11
943.16 910.65 897.97 891.01 886.55
9.58 7.69 6.79 6.23 5.84
923.74 896.62 885.93 880.01 876.19
10.72 8.57 7.53 6.89 6.44
913.58 889.52 879.98 874.68 871.25
500
1 2 3 4 5
6.22 5.18 4.71 4.43 4.25
1243.65 1188.22 1167.16 1155.83 1148.69
8.71 7.03 6.23 5.74 5.40
1167.48 1129.93 1115.21 1107.10 1101.89
10.32 8.26 7.27 6.66 6.23
1146.01 1114.69 1102.31 1095.43 1090.99
11.56 9.22 8.09 7.39 6.89
1134.67 1106.87 1095.83 1089.69 1085.70
10000
1000
Nc 100
Curve 1 Curve 2 Curve 3
10
Curve 4 Curve 5
1 1
2
3
4
5
M3
Fig. 2. The variation of critical magnetic Rayleigh number (Nc) with magnetization parameter (M3) for S1 = 100, M 01 ¼ 0:1, M5 = 0.1, h1 = 3; T A1 ¼ 0 for curve 1, T A1 ¼ 100 for curve 2, T A1 ¼ 1000 for curve 3, T A1 ¼ 10000 for curve 4 and T A1 ¼ 100000 for curve 5.
oscillatory modes are introduced due to the presence of the stable solute gradient, rotation and dust particles, which were non-existent in their absence.
Sunil et al. / Applied Mathematics and Computation 177 (2006) 614–628
625
10000
1000
N c 100 Curve Curve Curve Curve Curve
10
1 1
1 2 3 4 5
2
3
4
5
6
7
h1
Fig. 3. The variation of critical magnetic Rayleigh number (Nc) with dust parameter (h1) for M 01 ¼ 0:1, M5 = 0.1, T A1 ¼ 100, M3 = 1; S1 = 0 for curve 1, S1 = 100 for curve 2, S1 = 200 for curve 3, S1 = 300 for curve 4 and S1 = 400 for curve 5.
1400 Curve 1
1200
Curve 2 Curve 3
1000
Nc
800 600 400 200 0 0
50
100
150
200
250
300
350
400
450
500
S1
Fig. 4. The variation of critical magnetic Rayleigh number (Nc) with stable solute gradient (S1) for M 01 ¼ 0:1, M5 = 0.1, h1 = 3, T A1 ¼ 100; M3 = 1 for curve 1, M3 = 2 for curve 2 and M3 = 5 for curve 3.
20000
15000
Curve Curve Curve Curve
1 2 3 4
Nc
10000
5000
0 100
1000
10000
TA
100000
1000000
1
Fig. 5. The variation of critical magnetic Rayleigh number (Nc) with Taylor number ðT A1 Þ for M 01 ¼ 0:1, M5 = 0.1, M3 = 1, S1 = 100; h1 = 1 for curve 1, h1 = 3 for curve 2, h1 = 5 for curve 3 and h1 = 7 for curve 4.
7. The case of overstability The present section is devoted to find the possibility as to whether instability may occur as overstability. Since we wish to determine the Rayleigh number for the onset of instability via a state of pure oscillations, it suffices to find conditions for which (40) will admit of solutions with r1 real.
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Equating real and imaginary parts of (40) and eliminating R1 between them, we obtain A4 c41 þ A3 c31 þ A2 c21 þ A1 c1 þ A0 ¼ 0;
ð58Þ
where c1 ¼ r21 ; A3 ¼ s41 L21 L7 ðL2 þ L1 Þ þ s41 L23 L7 ðL2 þ L1 Þ b4
þ s31 L2 ffL7 hg L23 þ L21 þ 2s31 L23 L7 ðL2 þ L1 Þ þ s31 L1 L23 h b3 h i 2 þ 2s21 L2 L23 f ffL7 hg þ s21 L23 L7 ðf þ 1Þ ðL2 þ L1 Þ þ s21 L23 fh þ L7 gfL2 þ L1 ð1 þ f Þg b2 h i 2 þ s1 L2 L23 ðf þ 1Þ ffL7 hg þ x1 s41 S 1 L1 L5 L7 L8 ðL2 L3 Þ þ s41 T A1 L23 L7 ðL2 L1 Þ b þ s31 L2 L23 T A1 fh þ L7 f g þ x1 s31 S 1 L2 L3 L5 fhL8 h0 L7 g ; A4 ¼ b2 s41 L23 L7 ðL1 þ L2 Þ þ bs31 L2 L23 ffL7 hg;
ð59Þ ð60Þ
and the coefficients A0, A1, A2, being quite lengthy and not needed in the discussion of overstability, have not been written here. Since r1 is real for overstability, the four values of c1 ð¼ r21 Þ are positive. The sum of roots of (58) is AA34 , and if this is to be negative, then A3 > 0 and A4 > 0. Eqs. (59) and (60) shows that this is clearly impossible if f >
h h0 > ; L7 L8
h ; L7
L2 > L3
and
ð61Þ
L2 > L1 ;
0
h 0 1 i.e. if f > 1M 0 , Pr > Ps, Pr > Ps + PrM2 and hPr > h Ps and P r > h, which implies that 2
ðq0 C V ;H l0 K 3 H 0 Þð1 M 02 Þ > q0 C pt ;
P r > P s þ P rM 2;
hP r > h0 P s
and
1 Pr > ; h
ð62Þ
and the other inequality Pr > Ps being automatically satisfied in view of (62). Thus, for ðq0 C V ;H l0 K 3 H 0 Þð1 M 02 Þ > q0 C pt , Pr > Ps + PrM2, hP r > h0 P s and P r > 1h, overstability cannot occur and the principle of the exchange of stabilities is valid. Hence the above conditions are the sufficient conditions for the non-existence of overstability, the violation of which does not necessarily imply the occurrence of overstability. 8. Discussion of results and conclusions The effect of rotation on a ferromagnetic fluid permeated with dust particles, heated and soluted from below in the presence of uniform vertical magnetic field has been studied. We have investigated the effects of non-buoyancy magnetization, rotation, stable solute gradient and dust particles on the onset of convection. The principal conclusions from the analysis of this paper are as under: (i) For the case of stationary convection for buoyancy magnetization parameter M1 sufficiently small, the non-buoyancy magnetization and dust particles have always a destabilizing effect whereas rotation and stable solute gradient have always a stabilizing effect on the system. (ii) The critical wave numbers and critical magnetic thermal Rayleigh numbers for the onset of instability are also determined numerically for sufficiently large values of buoyancy magnetization parameter M1 and the results are depicted graphically. The effects of governing parameters on the stability of the system are discussed below. • Figs. 2 and 3, display the destabilizing effect of non-buoyancy magnetization and dust particles, respectively. Therefore, lower values of Nc are needed for onset of convection with an increase in M3 and h1. The destabilizing effect of dust particles on non-magnetic fluid is accounted by many authors [24–30] and is found to be valid for a ferromagnetic fluid also. • We have also looked into the effect of stable solute gradient S1. Fig. 4 demonstrates the effect of S1. It is observed that a stable solute gradient delays the onset of convection. This is in contrast to the case
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of ‘‘soluted from above’’ where a unstable solute gradient have a destabilizing effect on the system [31,33]. • Fig. 5 depicts the effect of T A1 . It is observed that rotation T A1 delays the onset of convection. In conclusion, we see that the critical stability parameter, Nc is reduced in the presence of dust particles because the heat capacity of the clean fluid is supplemented by that of the suspended (dust) particles. (iii) The principle of exchange of stabilities is found to hold true for the ferromagnetic fluid heated from below in the absence of dust particles, stable solute gradient and rotation. The oscillatory modes are introduced due to the presence of the dust particles, stable solute gradient and rotation, which were non-existent in their absence. (iv) The conditions ðq0 C V ;H l0 K 3 H 0 Þð1 M 02 Þ > q0 C pt , Pr > Ps + PrM2, hP r > h0 P s and P r > 1h are the sufficient for the non-existence of overstability. Some special cases are as under: • In the absence of magnetic field (and hence in the absence of magnetic parameters i.e. non-magnetic fluid) the above conditions, as expected, reduces to CV > Cpt i.e. the specific heat of fluid at constant volume is greater than the specific heat of dust particles, K 1 < K 01 i.e. the thermal conductivity is less than the solute conductivity and P r > 1h, which is in good agreement with the analytical and previous results obtained earlier [25–27]. • In the absence of magnetic field, stable solute gradient and dust particles, the above conditions reduces to Pr > 1. Thus, in the absence of dust particles, rotation contributes Pr > 1 (Chandrasekhar [4]) as a condition for non-existence of overstability, but in the presence of dust particles, rotation C contributes a modified condition P r > 1h, where h ¼ f Cptv .
Acknowledgement Financial assistance to Dr. Sunil in the form of a Research and Development Project [No. 25(0129)/02/ EMR-II] of the Council of Scientific and Industrial Research (CSIR), New Delhi is gratefully acknowledged. References [1] M.I. Papisov, A. Bogdannov, B. Schaffer Jr., N. Nossiff, T. Shen, R. Weissleder, T.J. Bradey, Colloidal magnetic resonance contrast agents: effect of particle surface on biodistribution, J. Magn. Magn. Mater. 122 (1993) 383–386. [2] J.P. Dailey, J.P. Phillips, C. Li, J.S. Riffle, Synthesis of silicane magnetic fluid for use in eye surgery, J. Magn. Magn. Mater. 194 (1999) 140–148. [3] R.E. Rosensweig, Ferrohydrodynamics, Cambridge University Press, Cambridge, 1985. [4] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publications, New York, 1981. [5] B.A. Finlayson, Convective instability of ferromagnetic fluids, J. Fluid Mech. 40 (4) (1970) 753–767. [6] D.P. Lalas, S. Carmi, Thermoconvective stability of ferrofluids, Phys. Fluids 14 (2) (1971) 436–437. [7] M.I. Shliomis, Magnetic fluids, Soviet Phys. Uspekhi (Engl. trans.) 17 (2) (1974) 153–169. [8] A. Zebib, Thermal convection in a magnetic fluid, J. Fluid. Mech. 321 (1996) 121–136. [9] V.K. Polevikov, Stability of a magnetic fluid under the action of an external pressure drop, Fluid Dyn. 32 (3) (1997) 457–461. [10] L. Schwab, U. Hildebrandt, K. Stierstadt, Magnetic Be´nard convection, J. Magn. Magn. Mater. 39 (1–2) (1983) 113–114. [11] P.J. Stiles, M.J. Kagan, Thermoconvective instability of a horizontal layer of ferrofluid in a strong vertical magnetic field, J. Magn. Magn. Mater. 85 (1–3) (1990) 196–198. [12] N. Rudraiah, G.N. Shekar, Convection in magnetic fluid with internal heat generation, ASME J. Heat Transfer 113 (1991) 122–127. [13] S. Aniss, M. Souhar, J.P. Brancher, Thermal convection in a magnetic fluid in an annular Hele-Shaw cell, J. Magn. Magn. Mater. 122 (1–3) (1993) 319–322. [14] P.G. Siddheshwar, Rayleigh–Be´nard convection in a ferromagnetic fluid with second sound, Jpn. Soc. Mag. Fluids 25 (1993) 32–36. [15] Y. Qin, P.N. Kaloni, Nonlinear stability problem of a ferromagnetic fluid with surface tension effect, Eur. J. Mech. B/ Fluids 13 (1994) 305–321. [16] P.G. Siddheshwar, Convective instability of ferromagnetic fluids bounded by fluid- permeable, magnetic boundaries, J. Magn. Magn. Mater. 149 (1–2) (1995) 148–150. [17] M. Souhar, S. Aniss, J.P. Brancher, Rayleigh–Be´nard convection in a magnetic fluid in an annular Hele-Shaw cell, Int. J. Heat Mass Transfer 42 (1) (1999) 61–72. [18] S. Aniss, M. Belhaq, M. Souhar, Effect of magnetic modulation on the stability of a magnetic liquid layer heated from below, ASMEJ. Heat Transfer 123 (2001) 428–433.
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[19] P.G. Siddheshwar, A. Abraham, Effect of time-periodic boundary temperatures/body force on Rayleigh–Be´nard convection in a ferromagnetic fluid, Acta Mech. 161 (2003) 131–150. [20] Sunil, Divya, R.C. Sharma, Effect of rotation on ferromagnetic fluid heated and soluted from below saturating a porous medium, J. Geophys. Eng. 1 (2) (2004) 116–127. [21] Sunil, P.K. Bharti, R.C. Sharma, Thermosolutal convection in ferromagnetic fluid, Arch. Mech. 56 (2) (2004) 117–135. [22] M.D. Gupta, A.S. Gupta, Convective instability of a layer of a ferromagnetic fluid rotating about a vertical axis, Int. J. Eng. Sci. 17 (1979) 271–277. [23] P.G. Saffman, On the stability of a laminar flow of a dusty gas, J. Fluid Mech. 13 (1962) 120–128. [24] J.W. Scanlon, L.A. Segel, Some effects of suspended particles on the onset of Be´nard convection, Phys. Fluids 16 (1973) 1573–1578. [25] R.C. Sharma, K. Prakash, S.N. Dube, Effect of suspended particles on the onset of Be´nard convection in hydromagnetics, Acta Phys. Hungarica 40 (1976) 3–10. [26] V.A. Palaniswamy, C.M. Purushotham, Stability of stratified fluids with fine dust, Phys. Fluids 24 (1981) 1224–1228. [27] R.C. Sharma, Sunil, Thermal instability of Oldroydian viscoelastic fluid with suspended particles in hydromagnetics in porous medium, Poly.-Plast. Tech. Eng. 33 (3) (1994) 323–339. [28] R.C. Sharma, Sunil, Y.D. Sharma, R.S. Chandel, On couple-stress fluid permeated with suspended particles heated from below, Arch. Mech. 54 (2002) 287–298. [29] Sunil, R.C. Sharma, R.S. Chandel, Stability of superposed couple-stress fluids in presence of suspended particles in porous medium, Int. J. Appl. Mech. Eng. 8 (4) (2003) 693–704. [30] Sunil, R.C. Sharma, R.S. Chandel, Effect of suspended particles on couple-stress fluid heated and soluted from below in porous medium, J. Porous Medium 7 (1) (2004) 9–18. [31] G. Vaidyanathan, R. Sekar, A. Ramanathan, Ferrothermohaline convection, J. Magn. Magn. Mater. 176 (1997) 321–330. [32] Sunil, Anupama, R.C. Sharma, The effect of magnetic field dependent viscosity on thermosolutal convection in ferromagnetic fluid, Appl. Math. Comput. 163 (3) (2005) 1197–1214. [33] G. Vaidyanathan, R. Sekar, A. Ramanathan, Ferrothermohaline convection in a porous medium, J. Magn. Magn. Mater. 149 (1–2) (1995) 137–142.
Effect of rotation on a ferromagnetic fluid heated and soluted from below in the presence dust particles Sunil a
a,*
, Anu Sharma a, R.G. Shandil
b
Department of Applied Sciences, National Institute of Technology (Deemed University), Hamirpur, HP 177 005, India b Department of Mathematics, Himachal Pradesh University, Summer Hill, Shimla, HP 171 005, India
Abstract This paper deals with the theoretical investigation of the effect of rotation on a layer of a ferromagnetic fluid permeated with dust particles, heated and soluted from below subjected to a transverse uniform magnetic field. For a flat fluid layer contained between two free boundaries, an exact solution is obtained using a linearized stability theory and a normal mode analysis method. For the case of stationary convection, non-buoyancy magnetization and dust particles have always a destabilizing effect, whereas stable solute gradient and rotation have a stabilizing effect on the onset of instability. The critical wave number and the critical magnetic thermal Rayleigh number for the onset of instability are also determined numerically for sufficiently large values of buoyancy magnetization M1. Graphs have been plotted by giving numerical values to the parameters, to depict the stability characteristics. It is observed that the critical magnetic thermal Rayleigh number is reduced solely because the heat capacity of clean fluid is supplemented by that of the dust particles. The principle of exchange of stabilities is found to hold true for the ferromagnetic fluid heated from below in the absence of dust particles, stable solute gradient and rotation. The oscillatory modes are introduced due to the presence of the dust particles, stable solute gradient and rotation, which were non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained. 2005 Elsevier Inc. All rights reserved. Keywords: Ferromagnetic fluid; Thermosolutal convection; Dust particles; Rotation; Magnetization
1. Introduction In the last millennium, the investigation on ferromagnetic fluid attracted researchers because of the increase of applications of magnetic fluids. The most famous application of magnetic fluids is the sealing of rotating shafts. This advantage is commonly used in various technical applications like the sealing of hard disc drives, rotating X-ray tubes and rotating vacuum feed-throughs where reliable sealing at low friction is required. The use of magnetic fluids as a heat transfer medium that may be magnetically hold in a certain position is
*
Corresponding author. E-mail addresses: [email protected], [email protected] (Sunil).
0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.09.092
Sunil et al. / Applied Mathematics and Computation 177 (2006) 614–628
615
nowadays the commercially most important branch of ferrofluid manufacturing. The major application in this field is cooling of loudspeakers, enabling a significant increase of the maximum acoustical power without any geometrical changes of the speaker system. The third important field of application of magnetic fluids is their use in biomedical applications. For examples, their use as a contrast medium in X-ray examinations [1] and for positioning tamponade for retinal detachment repair in eye surgery [2] has been reported. Rosensweig [3] has given an authoritative introduction to research on magnetic liquids in his monograph and the study of the effect of magnetization yields interesting information. This magnetization, in general, is function of the magnetic field, temperature and density of the fluid. Any variation of these quantities can induce a change of body force distribution in the fluid. This leads to convection in ferrofluids in the presence of magnetic field gradient. This mechanism is known as ferroconvection, which is similar to Be´nard convection. A detailed account of the theoretical and experimental results of the onset of thermal instability (Be´nard convection) in a fluid (non-magnetic) layer under varying assumptions of hydrodynamics and hydromagnetics has been given in the celebrated monograph by Chandrasekhar [4]. The convective instability of a ferromagnetic fluid for a fluid layer heated from below in the presence of uniform vertical magnetic field has been considered by Finlayson [5]. Thermoconvective stability of ferrofluids without considering buoyancy effects has been investigated by Lalas and Carmi [6], whereas Shliomis [7] analyzed the linearized relation for magnetized perturbed quantities at the limit of instability. The thermal convection in a magnetic fluid has been considered by Zebib [8], whereas the stability of a static magnetic fluid under the action of an external pressure drop has been studied by Polevikov [9]. Schwab et al. [10] investigated experimentally the Finlayson’s problem in the case of a strong magnetic field and detected the onset of convection by plotting the Nusselt number versus the Rayleigh number. Then, the critical Rayleigh number corresponds to a discontinuity in the slope. Later, Stiles and Kagan [11] examined the experimental problem reported by Schwab et al. [10] and generalized the Finlayson’s model assuming that under a strong magnetic field, the rotational viscosity augments the shear viscosity. The Be´nard convection in ferromagnetic fluids has been considered by many authors [12–22]. The ferromagnetic fluid has been considered to be clean in all the above studies. In many situations, the fluid is often not pure but contains suspended/dust particles. Saffman [23] has considered the stability of laminar flow of a dusty gas. Scanlon and Segel [24] have considered the effects of suspended particles on the onset of Be´nard convection, whereas Sharma et al. [25] have studied the effect of suspended particles on the onset of Be´nard convection in hydromagnetics. The suspended particles were thus found to destabilize the layer. Palaniswamy and Purushotham [26] have studied the stability of shear flow of stratified fluids with fine dust and found the effects of fine dust to increase the region of instability. On the other hand, the multiphase fluid systems are concerned with the motion of a liquid or gas containing immiscible inert identical particles. Of all multiphase fluid systems observed in nature, blood flows in arteries, flow in rocket tubes, dust in gas cooling systems to enhance the heat transfer processes, movement of inert solid particles in atmosphere, sand or other particles in sea or ocean beaches are the most common examples of multiphase fluid systems. Naturally studies of these systems are mathematically interesting and physically useful for various good reasons. The effect of dust particles on non-magnetic fluids has been investigated by many authors [27–30]. In the standard Be´nard problem the instability is driven by a density difference caused by a temperature difference between the upper and lower planes bounding the fluid. If the fluid is additionally has salt dissolved in it then there are potentially two destabilizing sources for the density difference, the temperature field and salt field. When there are two effects such as this, the phenomenon of convection which arises is called double-diffusive convection or thermohaline convection. Vaidyanathan et al. [31] studied ferrothermohaline convection in which a horizontal layer of an incompressible ferromagnetic fluid of thickness d in the presence of transverse magnetic field, heated from below and salted from above is considered. They found that the salinity of a ferromagnetic fluid enables the fluid to get destabilized more when it is salted from above. The really interesting situation from both a physical and a mathematical viewpoint arises when the layer is simultaneously heated from below and salted from below. More recently, Sunil et al. [32] have studied the effect of magnetic field dependent viscosity on thermosolutal convection in a ferromagnetic fluid. In view of the above investigations, it is attempted to discuss the effect of rotation on a ferromagnetic fluid heated and soluted from below in the presence of dust particles, subjected to a vertical magnetic field. The present study can serve as a theoretical support for experimental investigations e.g. evaluating the influence
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of impurities in a ferromagnetic fluid on thermal convection phenomena. This problem, to the best of our knowledge, has not been investigated yet. 2. Mathematical formulation of the problem Consider an infinite, horizontal, electrically non-conducting incompressible thin ferromagnetic fluid layer of thickness d, embedded in dust particles, heated and soluted from below (see Fig. 1). A uniform magnetic field H0 acts along the vertical direction which is taken as z-axis. The temperature T and solute concentration C at the bottom and top surfaces z ¼ 12 d are T0, T1 and C0, C1, respectively, and a uniform temperature gradient b ð¼ j dT jÞ and a uniform solute gradient b0 ð¼ j dC jÞ are maintained. The whole system is assumed dz dz to rotate with angular velocity X along the vertical axis, which is taken as z-axis. Both the boundaries are taken to be free and perfect conductors of heat. The gravity field g = (0, 0, g) and uniform vertical magnetic field intensity H = (0, 0, H0) pervade the system. The mathematical equations governing the motion of a ferromagnetic fluid for the above model are as follows: The continuity equation for an incompressible ferromagnetic fluid is r q ¼ 0.
ð1Þ
The momentum equation is o þ ðq rÞ q ¼ rp þ qg þ KN ðqd qÞ þ r ðHBÞ þ lr2 q þ 2q0 ðq XÞ; q0 ot
ð2Þ
where q, q0, q, l and p are the fluid density, reference density, velocity, dynamic viscosity (constant) and pressure of ferromagnetic fluid, respectively; t and B denote, respectively, the time and magnetic induction. qd(x, t) and N(x, t) denote velocity and number density of the dust particles, respectively. x = (x, y, z) and K = 6plg, g being the particle radius, is the Stokes drag coefficient. Assuming a uniform particle size, a spherical shape, and small relative velocities between the fluid and dust particles, the presence of dust particles adds an extra force term in the equations of motion (2), proportional to the velocity difference between the dust particles and 2 the fluid. The effect of rotation contributes two terms: (a) centrifugal force q20 gradjX rj and (b) coriolis z-axis
z=
Ω = (0, 0, Ω)
d 2
T1, C1 g = (0, 0, –g)
Incompressible rotating ferromagnetic fluid with dust particles
H = (0, 0, H0) z=0
z=– y-axis
d 2 T0, C0 Heated and soluted from below Fig. 1. Geometrical configuration.
x-axis
Sunil et al. / Applied Mathematics and Computation 177 (2006) 614–628
617
2
force 2q0(q · X). In Eq. (2), p ¼ p0 12 q0 jX rj is the reduced pressure, whereas p 0 stands for fluid pressure. Two additional simplifications are assumed in Eq. (2): we assume the viscosity is isotropic and independent of the magnetic field intensity. Both approximations simplify the analysis without changing the ultimate conclusion. We also use the Boussinesq approximation by allowing the density to change only in the gravitational body force term. Since density variations are mainly due to variations in temperature and solute concentration, the equation of state for the fluid is given by q ¼ q0 ½1 aðT T a Þ þ a0 ðC C a Þ;
ð3Þ
where a and a 0 thermal expansion coefficient and an analogous solvent coefficient of expansion, respectively. 1Þ Ta is the average temperature given by T a ¼ ðT 0 þT , where T0 and T1 are the constant average temperatures of 2 1Þ the lower and upper surfaces of the layer and Ca is the average concentration given by C a ¼ ðC0 þC , where C0 2 and C1 are the constant average concentrations of the lower and upper surfaces of the layer. Since the force exerted by the fluid on the particles is equal and opposite to that exerted by the particles on the fluid, there must be an extra force term, equal in magnitude but opposite in sign, in the equations of motion for the particles. The buoyancy force on the particles is neglected. Inter-particle reactions are also ignored since we assume that the distances between particles are quite large compared with their diameters. The effects due to pressure, gravity, viscous force on the particles are negligibly small, and therefore ignored. If mN is the mass of particles per unit volume, then the equations of motion and continuity of the dust particles, under the above assumptions, are o mN þ ðqd rÞ qd ¼ KN ðq qd Þ; ð4Þ ot oN þ r ðN qd Þ ¼ 0. ð5Þ ot The equations expressing the conservation of temperature and solute concentration in presence of dust particles are " # oM DT oM DH o þ l0 T þ mNC pt þ qd r T ¼ K 1 r 2 T ; . ð6Þ q0 C V ;H l0 H oT V ;H Dt oT V ;H Dt ot " # oM DC oM DH o þ l0 C þ mNC pt þ qd r C ¼ K 01 r2 C; ð7Þ q0 C V ;H l0 H oC V ;H Dt oC V ;H Dt ot where CV,H, Cpt, K1, K 01 , M and l0 are the specific heat at constant volume and magnetic field, specific heat of dust particles, thermal conductivity, solute conductivity, magnetization [defined by (10)] and magnetic permeability of free space (l0 = 4p · 107 Henry m1), respectively. The partial derivatives of M are material properties which can be evaluated once the magnetic equation of state, such as (11), is known. Since in ferrohydrodynamics it is usual to assume that the free charge and the electric displacement are absent, Maxwell’s equations are employed, r B ¼ 0;
r H ¼ 0;
ð8a; bÞ
where the magnetic induction is given by B ¼ l0 ðH þ MÞ.
ð9Þ
We assume that the magnetization is aligned with the magnetic field, but allow a dependence on the magnitude of the magnetic field, temperature and salinity, so that H ð10Þ M ¼ MðH ; T ; CÞ. H The magnetic equation of state is linearized about the magnetic field, H0, an average temperature, Ta, and the average salinity, Ca, to become M ¼ M 0 þ vðH H 0 Þ K 2 ðT T a Þ þ K 3 ðC C a Þ;
ð11Þ
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where magnetic susceptibility, pyromagnetic coefficient and salinity magnetic coefficient are defined by v ¼ ðoM Þ ; K 2 ¼ ðoM Þ and K 3 ¼ ðoM Þ , respectively. oH H 0 ;T a oT H 0 ;T a oC H 0 ;C a Here H0 is the uniform magnetic field of the fluid layer when placed in an external magnetic field ^ ^ H ¼ H ext 0 k; k is unit vector in the z-direction, H = jHj, M = jMj and M0 = M(H0, Ta, Ca). The basic state is assumed to be quiescent state and is given by q ¼ qb ¼ 0;
qd ¼ ðqd Þb ¼ 0;
C ¼ C b ðzÞ ¼ b0 z þ C a ; b ¼ K 2 bz K 3 b0 z ^ k; Mb ¼ M 0 þ 1þv 1þv
q ¼ qb ðzÞ; T1 T0 ; d
p ¼ pb ðzÞ;
T ¼ T b ðzÞ ¼ bz þ T a ; C1 C0 K 2 bz K 3 b0 z ^ þ b0 ¼ ; Hb ¼ H 0 k; 1þv 1þv d
N ¼ Nb ¼ N0
ð12Þ
and H 0 þ M 0 ¼ H ext 0 .
Only the spatially varying parts of H0 and M0 contribute to the analysis, so that the direction of the external magnetic field is unimportant and the convective phenomenon is the same whether the external magnetic field is parallel or antiparallel to the gravitational force. 3. The perturbation equations and normal mode analysis method Assume small perturbations around the basic state, and let q 0 = (u, v, w), q01 ¼ ð‘; r; sÞ, p 0 , q 0 , h, c, H 0 , M 0 and N 0 denote, respectively, the perturbations in ferromagnetic fluid velocity, particle velocity, pressure, density, temperature, concentration, magnetic field intensity, magnetization and suspended particles number density N0. The change in density q 0 , caused mainly by the perturbations h and c in temperature and concentration, is given by q0 ¼ q0 ðah a0 cÞ.
ð13Þ
Then the linearized perturbation equations of the ferromagnetic fluid become ou op0 oH 01 ou 2 þ l0 ðM 0 þ H 0 Þ þ 2q0 Xv þ lr u mN 0 ; L0 q0 ¼ L0 ot ot ox oz ov op0 oH 0 ov L0 q0 ¼ L0 þ l0 ðM 0 þ H 0 Þ 2 2q0 Xu þ lr2 v mN 0 ; ot ot oy oz ow op0 oH 0 l K 2b 0 ¼ L0 H 3 ð1 þ vÞ K 2 h þ l0 ðM 0 þ H 0 Þ 3 þ lr2 w 0 L0 q0 ot 1þv oz oz þ
l0 K 3 b0 l K 2K 3 0 ow ; fH 03 ð1 þ vÞ þ K 3 cg 0 ðb h þ bcÞ þ gq0 ðah a0 cÞ mN 0 ot 1þv ð1 þ vÞ
ð14Þ
ð15Þ
ð16Þ
ou ov ow þ þ ¼ 0; ð17Þ ox oy oz 0 oh o oU1 l T 0 K 22 b L0 fqC 1 þ mN 0 C pt g l0 T 0 K 2 w þ mN 0 bC pt w; ð18Þ ¼ L0 K 1 r2 h þ qC 1 b 0 ot ot oz ð1 þ vÞ 0 oc o oU2 l0 C 0 K 23 b0 0 2 0 0 0 L0 fqC 1 þ mN 0 C pt g l0 C 0 K 3 ¼ L0 K 1 r c þ qC 1 b w þ mN 0 b0 C pt w; ð19Þ ot ot oz ð1 þ vÞ where qC 1 ¼ q0 C V ;H þ l0 K 2 H 0 ;
qC 01
¼ q0 C V ;H l0 K 3 H 0 ;
L0 ¼
m o þ1 . K ot
ð20Þ
Sunil et al. / Applied Mathematics and Computation 177 (2006) 614–628
Eqs. (10) and (11) yield H 03 þ M 03 ¼ ð1 þ vÞH 03 K 2 h;
9 > > =
H 03 þ M 03 ¼ ð1 þ vÞH 03 þ K 3 c;
> > 0 H 0i þ M 0i ¼ 1 þ M H 0i ði ¼ 1; 2Þ; ; H0
619
ð21Þ
where we have assumed K2bd (1 + v)H0 and K3b 0 d (1 + v)H0 as the analysis is restricted to physical situation in which the magnetization induced by temperature and concentration variations is small compared to that induced by the external magnetic field. Eq. (8b) means we can write H0 ¼ rðU01 U02 Þ, where U01 is the perturbed magnetic potential and U02 is the perturbed magnetic potential analogous to solute. Eliminating u, v, p 0 between Eqs. (14)–(16) and using Eq. (17), we obtain o o 2 L0 q0 lr þ mN 0 r2 w ot ot
l0 K 2 b 2 o 0 l0 K 3 b0 2 o 0 0 0 r ð1 þ vÞ U U2 K 2 h þ U U2 þ K 3 c r ð1 þ vÞ ¼ L0 1þv 1 oz 1 oz 1 1þv 1 l K 2K 3 2 0 of þq0 gr21 ðah a0 cÞ 0 r ðb h þ bcÞ 2q0 X . ð22Þ oz ð1 þ vÞ 1 The vertical component of the vorticity equation is of ow 2 þ lr f ; ½L0 q0 þ mN 0 ¼ L0 2q0 X ot oz where f ¼ ov ou stands for the z-component of the vorticity. ox oy From (21), we have o2 U01 M0 oh ð1 þ vÞ 2 þ 1 þ r21 U01 K 2 ¼ 0; oz oz H0 o2 U02 M0 oc ð1 þ vÞ 2 þ 1 þ r21 U02 K 3 ¼ 0. oz oz H0
ð23Þ
ð24Þ ð25Þ
Now we analyze the perturbations w, h, c, f, U01 , and U02 into two-dimensional periodic waves; and considering disturbances characterized by a particular wave number k. Thus we assume to all quantities describing the perturbation a dependence on x, y, and t of the form
ð26Þ w; h; c; f; U01 ; U02 ¼ ½W ðz; tÞ; Hðz; tÞ; Cðz; tÞ; Zðz; tÞ; U1 ðz; tÞ; U2 ðz; tÞ exp iðk x x þ k y yÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where kx, ky are the wave numbers along the x- and y-directions, respectively, and k ¼ ðk 2x þ k 2y Þ is the resultant wave number. Following the normal mode analysis, the linearized perturbation dimensionless equations become o o 2 2 L0 ðD a Þ þ f ðD2 a2 ÞW ot ot ¼ aR1=2 L0 ½ðM 1 M 4 ÞDU1 ð1 þ M 1 M 4 ÞT
1=2 þ aS 1=2 L0 M 01 M 04 DU2 þ 1 M 01 þ M 04 C L0 T A DZ ; o o 1=2 2 2 L0 ðD a Þ þ f Z ¼ L0 T A DW ; ot ot oT o L0 P r ð1 þ hÞ M 2 DU1 ¼ L0 ðD2 a2 ÞT þ aR1=2 L0 ð1 M 2 Þ þ h W ; ot ot
oC o L0 P s ð1 þ h0 Þ M 02 DU2 ¼ L0 ðD2 a2 ÞC þ aS 1=2 L0 1 M 02 þ h0 W ; ot ot
ð27Þ ð28Þ ð29Þ ð30Þ
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Sunil et al. / Applied Mathematics and Computation 177 (2006) 614–628
D2 U1 a2 M 3 U1 DT ¼ 0; D
2
U2
a
2
M 3 U2
ð31Þ
DC ¼ 0;
ð32Þ
where the following non-dimensional parameters are introduced: t ¼
mt ; d2
R¼
gabd 4 qC 1 ; mK 1
d W ¼ W; m S¼
U1 ¼
ð1 þ vÞK 1 aR1=2 U1 ; K 2 qC 1 bmd 2
ga0 b0 d 4 qC 01 ; mK 01
T ¼
U2 ¼
K 1 aR1=2 H; qC 1 bmd
ð1 þ vÞK 01 aS 1=2 U2 ; K 3 qC 01 b0 md 2 C ¼
K 01 aS 1=2 C; qC 01 b0 md
a ¼ kd;
z z ¼ ; d
l0 K 22 b l0 K 23 b0 ; M 01 ¼ ; M1 ¼ ð1 þ vÞaq0 g ð1 þ vÞa0 q0 g
0 2 1þM H l0 T 0 K 22 l C K l0 K 2 K 3 b0 0 0 0 3 M2 ¼ ; M ; ; M 02 ¼ ; M ¼ ¼ 3 4 ð1 þ vÞ ð1 þ vÞqC 1 ð1 þ vÞqC 01 ð1 þ vÞaq0 g l0 K 2 K 3 b M 4 M 01 K 3 b0 mm o mN 0 ; M ; s ¼ M 04 ¼ ¼ ¼ ¼ ; L ¼ s þ 1 ; f ¼ ; 5 0 0 2 0 ð1 þ vÞa q0 g ot M 1 M 4 K 2b q0 Kd 2 mN 0 C pt mN 0 C pt 2Xd 2 d2 0 Z. h¼ ; h ¼ ; T ¼ and Z ¼ A qC 1 qC 01 m m
D¼
o ; oz
Pr ¼
m qC 1 ; K1
Ps ¼
m qC 0 ; K 01 1
4. Exact solution for free boundaries Here we consider the case where both boundaries are free as well as perfect conductors of heat. The case of two free boundaries is of little physical interest, but it is mathematically important because one can derive an exact solution, whose properties guide our analysis. Here we consider the case of an infinite magnetic susceptibility v and we neglect the deformability of the horizontal surfaces. Thus the exact solution of the system (27)–(32) subject to the boundary conditions W ¼ D2 W ¼ T ¼ C ¼ DU1 ¼ DU2 ¼ 0
at z ¼
1 2
ð33Þ
is written in the form
W ¼ A1 ert cos pz ; T ¼ B1 ert cos pz ; C ¼ F 1 ert cos pz ; DU1 ¼ C 1 ert cos pz ; C 1 rt E1 rt rt U1 ¼ e sin pz ; DU2 ¼ E1 e cos pz ; U2 ¼ e sin pz ; p p
ð34Þ
where A1, B1, C1, E1, F1 are constants and r is the growth rate which is, in general, a complex constant. Substituting Eq. (34) in Eqs. (27)–(32) and dropping asterisks for convenience, we get the following equations: h i fðr þ p2 þ a2 Þð1 þ srÞ þ f rg2 ðp2 þ a2 Þ þ p2 ð1 þ srÞ2 T A A1 aR1=2 fðr þ p2 þ a2 Þð1 þ srÞ þ f rg½ð1 þ srÞð1 þ M 1 M 4 ÞB1 þ aR1=2 fðr þ p2 þ a2 Þð1 þ srÞ þ f rg½ðM 1 M 4 Þð1 þ srÞC 1 þ aS 1=2 fðr þ p2 þ a2 Þð1 þ srÞ þ f rg½ð1 þ srÞð1 M 01 þ M 04 ÞF 1
þ aS 1=2 fðr þ p2 þ a2 Þð1 þ srÞ þ f rg M 01 M 04 ð1 þ srÞ E1 ¼ 0; 1=2
2
ð35Þ
2
½aR fh þ ð1 M 2 Þð1 þ srÞgA1 ½fp þ a þ P r ð1 þ hÞrgð1 þ srÞB1 þ ½P r M 2 rð1 þ srÞC 1 ¼ 0; 1=2 0
aS h þ 1 M 02 ð1 þ srÞ A1 p2 þ a2 þ P s ð1 þ h0 Þr ð1 þ srÞ F 1 þ P s M 02 rð1 þ srÞ E1 ¼ 0;
ð36Þ ð37Þ
Sunil et al. / Applied Mathematics and Computation 177 (2006) 614–628
p2 B1 þ ðp2 þ a2 M 3 ÞC 1 ¼ 0; 2
2
2
p F 1 þ ðp þ a M 3 ÞE1 ¼ 0.
621
ð38Þ ð39Þ
For existence of non-trivial solutions of the above equations, the determinant of the coefficients of A1, B1, C1, E1 and F1 in Eqs. (35)–(39) must vanish. This determinant on simplification yields T 6 r61 þ iT 5 r51 þ T 4 r41 iT 3 r31 T 2 r21 þ iT 1 r1 þ T 0 ¼ 0.
ð40Þ
T 6 ¼ s21 bL2 L3 ;
ð41Þ
T 5 ¼ s21 b2 L1 ðL3 þ L2 Þ þ 2s1 bL2 L3 L6 ; T 4 ¼ s21 L21 b3 þ 2s1 L1 L6 ðL2 þ L3 Þb2 þ ðL26 þ 2s1 bÞL2 L3 b þ x1 s21 S 1 L2 L5 L8 x1 s21 R1 L3 L4 L7 þ s21 T A1 L2 L3 ;
ð42Þ
Here
T3 ¼
2L21 s1 L6 b3
þ b 2L2 L3 L6 þ L1 ðL26 þ 2s1 bÞðL2 þ L3 Þ
ð43Þ
2
þ T A1 f2s1 L2 L3 þ bs21 L1 ðL2 þ L3 Þg þ x1 S 1 s1 L5 ½fh0 þ L8 gL2 þ L6 L8 L2 þ s1 bL1 L8 x1 R1 s1 L4 ½ðh þ L7 ÞL3 þ L6 L7 L3 þ s1 bL1 L7 ;
T 2 ¼ L2 L3 þ L21 L26 þ 2s1 b þ 2L1 L6 ðL2 þ L3 Þ b3 x1 R1 L4 ½s1 bL7 ðL3 þ L1 L6 Þ þ ðh þ L7 Þðs1 bL1 þ L3 L6 Þ þ x1 S 1 L5 ½s1 bL8 ðL2 þ L1 L6 Þ þ ðh0 þ L8 Þðs1 bL1 þ L2 L6 Þ þ T A1 s21 L21 b2 þ 2s1 L1 ðL2 þ L3 Þb þ L2 L3 ;
ð44Þ
ð45Þ
4
T 1 ¼ L1 ½2L1 L6 þ L2 þ L3 b þ L1 T A1 ½2s1 bL1 þ L2 þ L3 b x1 R1 L4 ½bðh þ L7 ÞðL3 þ L1 L6 Þ þ s1 L1 L7 b2 þ x1 S 1 L5 bðh0 þ L8 ÞðL2 þ L1 L6 Þ þ s1 L1 L8 b2 ; T 0 ¼ b2 L1 b3 L1 þ L1 T A1 x1 R1 ðh þ L7 ÞL4 þ x1 S 1 ðh0 þ L8 ÞL5 ;
ð46Þ ð47Þ
a2
where R1 ¼ pR4 ; S 1 ¼ pS4 ; x1 ¼ p2 ; ir1 ¼ pr2 ; s1 ¼ sp2 ; T A1 ¼ TpA2 ; b ¼ ð1 þ x1 Þ, L1 ¼ 1 þ x1 M 3 ; L2 ¼ P r ½fð1 M 2 Þ þ x1 M 3 g þ L1 h;
L3 ¼ P s 1 M 02 þ x1 M 3 þ L1 h0 ; L4 ¼ f1 þ x1 M 3 ð1 þ M 1 M 4 Þg; L5 ¼ f1 þ x1 M 3 ð1 M 01 þ M 04 Þg; L6 ¼ ð1 þ s1 b þ f Þ; L7 ¼ 1 M 2 and L8 ¼ 1 M 02 . 5. The case of stationary convection Here we first consider the case when the instability sets in as stationary convection (and M 2 ffi 0; M 02 ffi 0Þ, the marginal state will be characterized by r1 = 0, then the Rayleigh number is given by n
o n o ð1 þ x1 Þ3 þ T A1 ð1 þ x1 M 3 Þ S 1 h01 ð1 þ x1 M 3 Þ þ x1 M 3 M 01 M15 1 R1 ¼ þ ; ð48Þ x1 h1 fð1 þ x1 M 3 Þ þ x1 M 3 M 1 ð1 M 5 Þg h1 fð1 þ x1 M 3 Þ þ x1 M 3 M 1 ð1 M 5 Þg which expresses the modified Rayleigh number R1 as a function of the dimensionless wave number x1, buoyancy magnetization parameter M1, the non-buoyancy magnetization parameter M3, Taylor number T A1 , solute gradient parameter S1, dust particles parameter h1, dust particles parameter analogous to solute h01 and the ratio of the salinity effect on magnetic field and pyromagnetic coefficient M5. Here we put h1 = (1 + h) and h01 ¼ ð1 þ h0 Þ. In the absence of dust particles, the values of h1 and h01 is one. Since the marginal state dividing stability from instability is stationary, this means that at the onset of instability there is no relative velocity between particles and fluid and hence no particle drag on the fluid. Therefore, the critical Rayleigh number is reduced solely because the heat capacity of the clean fluid is supplemented by that of the suspended (dust) particles.
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To investigate the effects of non-buoyancy magnetization, dust particles, stable solute gradient and rotadR1 dR1 dR1 tion, we examine the behaviour of dM ; ; and dTdRA1 , analytically. 3 dh1 dS 1 1 Eq. (48) yields h
i M 01 3 0 ð1 M Þ M fð1 þ x Þ þ T g þ S h x M 5 1 1 A 1 1 1 1 1 M5 dR1 ¼ ; ð49Þ 2 dM 3 h1 f1 þ x1 M 3 þ x1 M 3 M 1 ð1 M 5 Þg n
o ð1 þ x1 M 3 Þfð1 þ x1 Þ3 þ T A1 g þ x1 S 1 h01 ð1 þ x1 M 3 Þ þ x1 M 3 M 01 M15 1 dR1 ¼ ; ð50Þ dh1 x1 h21 f1 þ x1 M 3 þ x1 M 3 M 1 ð1 M 5 Þg n
o 0 0 1 h 1 þ x M þ x M M 1 1 3 1 3 1 M5 1 dR1 ; ð51Þ ¼ h1 f1 þ x1 M 3 þ x1 M 3 M 1 ð1 M 5 Þg dS 1 dR1 ð1 þ x1 M 3 Þ . ¼ dT A1 x1 h1 f1 þ x1 M 3 þ x1 M 3 M 1 ð1 M 5 Þg
ð52Þ
This shows that, for a stationary convection, the non-buoyancy magnetization and dust particles are found to have a destabilizing effect, whereas stable solute gradient and rotation have a stabilizing effect on the system. For M1 sufficiently large, we obtain the results for the magnetic mechanism n
o 0 0 1 3 S h ð1 þ x M Þ þ x M M 1 1 1 3 1 3 1 M5 1 fð1 þ x1 Þ þ T A1 gð1 þ x1 M 3 Þ þ ; ð53Þ N ¼ R1 M 1 ¼ x1 h1 M 3 ð1 M 5 Þ x21 h1 M 3 ð1 M 5 Þ where N is the magnetic thermal Rayleigh number. As a function of x1, N given by Eq. (53) attains its minimum when 2M 3 x41 þ ð3M 3 þ 1Þx31 fM 3 þ T A1 M 3 þ S 1 h01 þ 3gx1 2ð1 þ T A1 Þ ¼ 0.
ð54Þ
The values of critical wave number for the onset of instability are determined numerically using Newton– dN Raphson method by the condition dx ¼ 0. With x1 determined as a solution of Eq. (54), Eq. (53) will give 1 the required critical magnetic thermal Rayleigh number Nc. The critical magnetic thermal Rayleigh number (Nc), depends on the non-buoyancy magnetization parameter M3, Taylor number T A1 , stable solute gradient S1, dust particles parameter h1 and dust particles parameter analogous to solute h01 . Values of Nc determined in this fashion for various values of M3, T A1 , S1, h1 and h01 are given in Tables 1 and 2 and the results are further illustrated in Figs. 2–5. Figs. 2–5 represent the plots of critical magnetic thermal Rayleigh number Nc versus non-buoyancy magnetization parameter M3 [for various values of T A1 ], dust particles parameter h1 [for various values of S1], stable solute gradient S1 [for various values of M3] and Taylor number T A1 [for various values of h1], respectively. Figs. 2 and 3 illustrate that as non-buoyancy magnetization parameter M3 and dust particle parameter h1 increase, the critical magnetic Rayleigh number (Nc) decreases. Therefore, lower values of Nc are needed for onset of convection with an increase in M3 and h1, hence justifying the destabilizing effect of non-buoyancy magnetization and dust particles. It is evident from Figs. 4 and 5 that stable solute gradient and rotation have always a stabilizing effect on the system. It is also observed from Tables 1 and 2 that in the absence of dust particles ðh1 ¼ 1; h01 ¼ 1Þ, the critical magnetic thermal Rayleigh number is very high however in the presence of dust particles ðh1 > 1; h01 > 1Þ, critical magnetic thermal Rayleigh number is reduced solely because the heat capacity of clean fluid is supplemented by that of the dust particles. 6. The case of oscillatory modes Here we examine the possibility of oscillatory modes, if any, on stability problem due to the presence of stable solute gradient, rotation, dust particles and magnetization. Equating the imaginary parts of Eq. (40), we obtain
Sunil et al. / Applied Mathematics and Computation 177 (2006) 614–628
623
Table 1 Critical magnetic thermal Rayleigh numbers and wave numbers of the unstable modes at marginal stability for the onset of stationary convection for various values of Taylor number, non-buoyancy magnetization and dust parameters T A1
M3
M 01 ¼ 0:1, M5 = 0.1, S1 = 100 h1 = 1, h01 ¼ 1
h1 = 3, h01 ¼ 3
h1 = 5, h01 ¼ 5
h1 = 7, h01 ¼ 7
xc
xc
xc
xc
Nc
Nc
Nc
Nc
0
1 2 3 4 5
3.18 2.50 2.16 1.94 1.79
279.60 256.20 246.99 241.86 238.51
4.75 3.73 3.23 2.91 2.68
252.45 237.92 232.15 228.92 226.80
5.72 4.50 3.90 3.51 3.24
244.39 232.59 227.88 225.25 223.52
6.45 5.08 4.40 3.97 3.67
240.09 229.76 225.64 223.33 221.81
102
1 2 3 4 5
4.41 3.95 3.76 3.65 3.58
316.16 295.30 287.83 283.95 281.57
5.44 4.63 4.28 4.07 3.94
261.12 247.79 242.82 240.18 238.54
6.22 5.18 4.71 4.43 4.25
248.73 237.64 233.43 231.17 229.74
6.86 5.64 5.08 4.75 4.52
242.83 232.99 229.22 227.16 225.86
103
1 2 3 4 5
8.15 7.84 7.73 7.67 7.63
495.06 473.11 465.60 461.81 459.52
8.54 8.06 7.88 7.79 7.73
314.63 303.10 299.10 297.07 295.84
8.90 8.28 8.03 7.90 7.82
278.32 269.02 265.77 264.10 263.09
9.24 8.48 8.18 8.02 7.92
262.62 254.37 251.46 249.96 249.05
104
1 2 3 4 5
17.15 16.90 16.80 16.76 16.73
1313.20 1279.55 1268.21 1262.53 1259.11
17.25 16.95 16.84 16.79 16.75
582.78 569.44 564.95 562.69 561.33
17.35 17.00 16.88 16.81 16.77
436.68 427.42 424.29 422.72 421.77
17.45 17.05 16.91 16.84 16.80
374.06 366.55 364.01 362.73 361.96
105
1 2 3 4 5
36.85 36.60 36.52 36.48 36.45
4990.55 4925.73 4904.03 4893.16 4886.64
36.88 36.62 36.53 36.48 36.46
1806.27 1783.66 1776.10 1772.30 1770.03
36.90 36.63 36.54 36.49 36.46
1169.41 1155.25 1150.51 1148.13 1146.70
36.92 36.64 36.54 36.50 36.47
896.47 885.93 882.40 880.63 879.57
106
1 2 3 4 5
79.37 79.13 79.04 79.00 78.97
21747.05 21612.18 21567.13 21544.58 21531.05
79.38 79.13 79.04 79.00 78.98
7390.69 7345.27 7330.10 7322.50 7317.95
79.38 79.13 79.05 79.00 78.98
4519.42 4491.89 4482.69 4478.09 4475.32
79.39 79.13 79.05 79.00 78.98
3288.88 3269.01 3262.37 3259.05 3257.06
r1 r41 b s21 L1 ðL2 þ L3 Þb þ 2s1 L2 L3 L6 r21 2s1 L21 L6 b3 þ 2L2 L3 L6 þ L1 L26 þ 2s1 b ðL2 þ L3 Þ b2 þT A1 s1 f2L2 L3 þ bs1 L1 ðL2 þ L3 Þg x1 R1 s1 L4 ½fh þ L7 gL3 þ ðL3 L6 þ s1 L1 bÞL7 þ x1 S 1 s1 L5 ½fh0 þ L8 gL2 þðL2 L6 þ s1 L1 bÞL8 i þ L1 ð2L1 L6 þ L2 þ L3 Þb4 þ T A1 L1 ð2s1 L1 b þ L2 þ L3 Þb x1 R1 L4 bfh þ L7 gðL3 þ L1 L6 Þ þ s1 L1 L7 b2 þ x1 S 1 L5 bfh0 þ L8 gðL2 þ L1 L6 Þ þ s1 L1 L8 b2 ¼ 0.
ð55Þ
It is evident from Eq. (55) that r1 may be either zero or non-zero, meaning that the modes may be either nonoscillatory or oscillatory. In the absence of stable solute gradient, rotation and dust particles, we obtain the result (after simplification) as r1 ½P r fL7 þ x1 M 3 g þ 2ð1 þ x1 M 3 Þ ¼ 0. ð56Þ 6 Here the quantity inside the brackets is positive definite because the typical values of M2 are +10 [5]. Hence r1 ¼ 0; ð57Þ which implies that the oscillatory modes are not allowed and the principle of exchange of stabilities is satisfied, in the absence of stable solute gradient, rotation and dust particles. Thus from Eq. (55), we conclude that the
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Table 2 Critical magnetic thermal Rayleigh numbers and wave numbers of the unstable modes at marginal stability for the onset of stationary convection for various values of stable solute gradient, non-buoyancy magnetization and dust parameters S1
M3
M 01 ¼ 0:1, M5 = 0.1, T A1 ¼ 100 h1 = 1, h01 ¼ 1
h1 = 3, h01 ¼ 3
h1 = 5, h01 ¼ 5
h1 = 7, h01 ¼ 7
xc
xc
xc
xc
Nc
Nc
Nc
Nc
0
1 2 3 4 5
3.72 3.53 3.45 3.40 3.38
77.75 69.34 66.44 64.97 64.08
3.72 3.53 3.45 3.40 3.38
25.92 23.11 22.15 21.66 21.36
3.72 3.53 3.45 3.40 3.38
15.55 13.87 13.29 12.99 12.82
3.72 3.53 3.45 3.40 3.38
11.11 9.91 9.49 9.28 9.15
100
1 2 3 4 5
4.41 3.95 3.76 3.65 3.58
316.16 295.30 287.83 283.95 281.57
5.44 4.63 4.28 4.07 3.94
261.12 247.79 242.82 240.18 238.54
6.22 5.18 4.71 4.43 4.25
248.73 237.64 233.43 231.17 229.74
6.86 5.64 5.08 4.75 4.52
242.83 232.99 229.22 227.16 225.86
200
1 2 3 4 5
4.97 4.32 4.03 3.87 3.77
550.93 519.85 508.44 502.45 498.73
6.55 5.42 4.90 4.60 4.39
490.72 469.94 461.99 457.70 454.98
7.65 6.23 5.56 5.16 4.89
475.81 458.47 451.74 448.06 445.71
8.52 6.88 6.11 5.63 5.31
468.35 452.96 446.93 443.62 441.49
300
1 2 3 4 5
5.44 4.63 4.28 4.07 3.94
783.37 743.37 728.47 720.55 715.61
7.40 6.04 5.41 5.03 4.77
717.73 690.73 680.28 674.58 670.94
8.71 7.03 6.23 5.74 5.40
700.49 677.96 669.13 664.26 661.13
9.74 7.81 6.89 6.32 5.93
691.62 671.62 663.74 659.37 656.55
400
1 2 3 4 5
5.85 4.92 4.50 4.26 4.10
1014.14 966.11 948.01 938.33 932.25
8.11 6.57 5.85 5.41 5.11
943.16 910.65 897.97 891.01 886.55
9.58 7.69 6.79 6.23 5.84
923.74 896.62 885.93 880.01 876.19
10.72 8.57 7.53 6.89 6.44
913.58 889.52 879.98 874.68 871.25
500
1 2 3 4 5
6.22 5.18 4.71 4.43 4.25
1243.65 1188.22 1167.16 1155.83 1148.69
8.71 7.03 6.23 5.74 5.40
1167.48 1129.93 1115.21 1107.10 1101.89
10.32 8.26 7.27 6.66 6.23
1146.01 1114.69 1102.31 1095.43 1090.99
11.56 9.22 8.09 7.39 6.89
1134.67 1106.87 1095.83 1089.69 1085.70
10000
1000
Nc 100
Curve 1 Curve 2 Curve 3
10
Curve 4 Curve 5
1 1
2
3
4
5
M3
Fig. 2. The variation of critical magnetic Rayleigh number (Nc) with magnetization parameter (M3) for S1 = 100, M 01 ¼ 0:1, M5 = 0.1, h1 = 3; T A1 ¼ 0 for curve 1, T A1 ¼ 100 for curve 2, T A1 ¼ 1000 for curve 3, T A1 ¼ 10000 for curve 4 and T A1 ¼ 100000 for curve 5.
oscillatory modes are introduced due to the presence of the stable solute gradient, rotation and dust particles, which were non-existent in their absence.
Sunil et al. / Applied Mathematics and Computation 177 (2006) 614–628
625
10000
1000
N c 100 Curve Curve Curve Curve Curve
10
1 1
1 2 3 4 5
2
3
4
5
6
7
h1
Fig. 3. The variation of critical magnetic Rayleigh number (Nc) with dust parameter (h1) for M 01 ¼ 0:1, M5 = 0.1, T A1 ¼ 100, M3 = 1; S1 = 0 for curve 1, S1 = 100 for curve 2, S1 = 200 for curve 3, S1 = 300 for curve 4 and S1 = 400 for curve 5.
1400 Curve 1
1200
Curve 2 Curve 3
1000
Nc
800 600 400 200 0 0
50
100
150
200
250
300
350
400
450
500
S1
Fig. 4. The variation of critical magnetic Rayleigh number (Nc) with stable solute gradient (S1) for M 01 ¼ 0:1, M5 = 0.1, h1 = 3, T A1 ¼ 100; M3 = 1 for curve 1, M3 = 2 for curve 2 and M3 = 5 for curve 3.
20000
15000
Curve Curve Curve Curve
1 2 3 4
Nc
10000
5000
0 100
1000
10000
TA
100000
1000000
1
Fig. 5. The variation of critical magnetic Rayleigh number (Nc) with Taylor number ðT A1 Þ for M 01 ¼ 0:1, M5 = 0.1, M3 = 1, S1 = 100; h1 = 1 for curve 1, h1 = 3 for curve 2, h1 = 5 for curve 3 and h1 = 7 for curve 4.
7. The case of overstability The present section is devoted to find the possibility as to whether instability may occur as overstability. Since we wish to determine the Rayleigh number for the onset of instability via a state of pure oscillations, it suffices to find conditions for which (40) will admit of solutions with r1 real.
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Equating real and imaginary parts of (40) and eliminating R1 between them, we obtain A4 c41 þ A3 c31 þ A2 c21 þ A1 c1 þ A0 ¼ 0;
ð58Þ
where c1 ¼ r21 ; A3 ¼ s41 L21 L7 ðL2 þ L1 Þ þ s41 L23 L7 ðL2 þ L1 Þ b4
þ s31 L2 ffL7 hg L23 þ L21 þ 2s31 L23 L7 ðL2 þ L1 Þ þ s31 L1 L23 h b3 h i 2 þ 2s21 L2 L23 f ffL7 hg þ s21 L23 L7 ðf þ 1Þ ðL2 þ L1 Þ þ s21 L23 fh þ L7 gfL2 þ L1 ð1 þ f Þg b2 h i 2 þ s1 L2 L23 ðf þ 1Þ ffL7 hg þ x1 s41 S 1 L1 L5 L7 L8 ðL2 L3 Þ þ s41 T A1 L23 L7 ðL2 L1 Þ b þ s31 L2 L23 T A1 fh þ L7 f g þ x1 s31 S 1 L2 L3 L5 fhL8 h0 L7 g ; A4 ¼ b2 s41 L23 L7 ðL1 þ L2 Þ þ bs31 L2 L23 ffL7 hg;
ð59Þ ð60Þ
and the coefficients A0, A1, A2, being quite lengthy and not needed in the discussion of overstability, have not been written here. Since r1 is real for overstability, the four values of c1 ð¼ r21 Þ are positive. The sum of roots of (58) is AA34 , and if this is to be negative, then A3 > 0 and A4 > 0. Eqs. (59) and (60) shows that this is clearly impossible if f >
h h0 > ; L7 L8
h ; L7
L2 > L3
and
ð61Þ
L2 > L1 ;
0
h 0 1 i.e. if f > 1M 0 , Pr > Ps, Pr > Ps + PrM2 and hPr > h Ps and P r > h, which implies that 2
ðq0 C V ;H l0 K 3 H 0 Þð1 M 02 Þ > q0 C pt ;
P r > P s þ P rM 2;
hP r > h0 P s
and
1 Pr > ; h
ð62Þ
and the other inequality Pr > Ps being automatically satisfied in view of (62). Thus, for ðq0 C V ;H l0 K 3 H 0 Þð1 M 02 Þ > q0 C pt , Pr > Ps + PrM2, hP r > h0 P s and P r > 1h, overstability cannot occur and the principle of the exchange of stabilities is valid. Hence the above conditions are the sufficient conditions for the non-existence of overstability, the violation of which does not necessarily imply the occurrence of overstability. 8. Discussion of results and conclusions The effect of rotation on a ferromagnetic fluid permeated with dust particles, heated and soluted from below in the presence of uniform vertical magnetic field has been studied. We have investigated the effects of non-buoyancy magnetization, rotation, stable solute gradient and dust particles on the onset of convection. The principal conclusions from the analysis of this paper are as under: (i) For the case of stationary convection for buoyancy magnetization parameter M1 sufficiently small, the non-buoyancy magnetization and dust particles have always a destabilizing effect whereas rotation and stable solute gradient have always a stabilizing effect on the system. (ii) The critical wave numbers and critical magnetic thermal Rayleigh numbers for the onset of instability are also determined numerically for sufficiently large values of buoyancy magnetization parameter M1 and the results are depicted graphically. The effects of governing parameters on the stability of the system are discussed below. • Figs. 2 and 3, display the destabilizing effect of non-buoyancy magnetization and dust particles, respectively. Therefore, lower values of Nc are needed for onset of convection with an increase in M3 and h1. The destabilizing effect of dust particles on non-magnetic fluid is accounted by many authors [24–30] and is found to be valid for a ferromagnetic fluid also. • We have also looked into the effect of stable solute gradient S1. Fig. 4 demonstrates the effect of S1. It is observed that a stable solute gradient delays the onset of convection. This is in contrast to the case
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of ‘‘soluted from above’’ where a unstable solute gradient have a destabilizing effect on the system [31,33]. • Fig. 5 depicts the effect of T A1 . It is observed that rotation T A1 delays the onset of convection. In conclusion, we see that the critical stability parameter, Nc is reduced in the presence of dust particles because the heat capacity of the clean fluid is supplemented by that of the suspended (dust) particles. (iii) The principle of exchange of stabilities is found to hold true for the ferromagnetic fluid heated from below in the absence of dust particles, stable solute gradient and rotation. The oscillatory modes are introduced due to the presence of the dust particles, stable solute gradient and rotation, which were non-existent in their absence. (iv) The conditions ðq0 C V ;H l0 K 3 H 0 Þð1 M 02 Þ > q0 C pt , Pr > Ps + PrM2, hP r > h0 P s and P r > 1h are the sufficient for the non-existence of overstability. Some special cases are as under: • In the absence of magnetic field (and hence in the absence of magnetic parameters i.e. non-magnetic fluid) the above conditions, as expected, reduces to CV > Cpt i.e. the specific heat of fluid at constant volume is greater than the specific heat of dust particles, K 1 < K 01 i.e. the thermal conductivity is less than the solute conductivity and P r > 1h, which is in good agreement with the analytical and previous results obtained earlier [25–27]. • In the absence of magnetic field, stable solute gradient and dust particles, the above conditions reduces to Pr > 1. Thus, in the absence of dust particles, rotation contributes Pr > 1 (Chandrasekhar [4]) as a condition for non-existence of overstability, but in the presence of dust particles, rotation C contributes a modified condition P r > 1h, where h ¼ f Cptv .
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