# A Linear Stability Analysis of Thermal Convection in a  Fluid Layer with Simultaneous Rotation and Magnetic Field Acting in Different Directions

Authors: Ruben Avila; Ares Cabello
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The onset of thermal convection of a Boussinesq fluid located in an unbounded layer heated from below and subject simultaneously to rotation and magnetic field, whose vectors act in different directions, is presented. To the knowledge of the authors, the convective thermal instability analysis for this complex problem has not been previously reported. In this paper, we use the Tau Chebyshev spectral method to calculate the value of the critical parameters (wave number and Rayleigh number at the onset of convection) as a function of (i) different kinds of boundaries, (ii) angle between the three vectors, and (iii) different values of the Taylor numberT (rate of rotation) and magnetic parameter Q (strength of the magnetic force). For the classical problems previously reported in the literature, we compare our calculations with Chandrasekhar’s variational method results and show that the present method is applicable.


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## Body

## 1. Introduction

The thermal instability of a horizontal layer of fluid heated from below (Rayleigh-Bénard problem) is an important and classical fluid dynamics problem in science and engineering. The linear stability analysis of this problem, in terms of normal modes, has been carried out and reported in several investigations.The rotating Rayleigh-Bénard problem has also been widely studied and it has been found that the effect of the Coriolis force due to rotation of the infinite layer of fluid, when the rotation vectorΩ acts about the vertical direction, that is, parallel to the gravity force vector g, inhibits the onset of convection and thus induces a stabilizing effect [1–3]. The amount of the stabilizing effect depends on the rate of rotation (the Taylor number T). Results for the problem in which the vectors Ω and g act in different directions, with an angle ϑR between them, were not explicitly reported by [1, 4]; instead an analogy with the closely related problem of thermal convection subject to the action of a magnetic field was presented by [1].Moreover, it has also been reported that, in the magnetic Rayleigh-Bénard problem, a stabilizing effect is obtained when the layer of an electrically conducting fluid is immersed in a uniform magnetic field vectorH that is parallel to the gravity force vector g [5]. The extent of the inhibition depends on the value of the nondimensional magnetic parameter Q. It has been found that when H and g are parallel, convection at marginal stability is characterized by a cellular pattern; hence, longitudinal and transverse rolls appear simultaneously. Results for the problem in which the vectors g and H act in different directions, with an angle ϑH between them, and when instability sets in as stationary convection have been previously obtained by considering two cases: (i) solution of the perturbation equations which are independent of x1 and (ii) solutions of the perturbation equations that conduct to a more general patterns of motion. In the former case, the onset of instability is characterized by rolls in the x1 direction, whereas in the latter case, by using a variational procedure, [1] shows that as the parameter CH, defined as CH=axtanϑH (where ax is the wave number in the x1 direction), increases, keeping the Chandrasekhar number Q (defined by Q=μ2(HcosϑH)2D^2σ^/ρoν, where μ, H, D^, σ^, ρo, and ν are the magnetic permeability, the magnitude of the magnetic field, the distance between the two plates, the coefficient of electrical conductivity, the density of reference, and the kinematic viscosity, resp.) as a constant value, both the critical Rayleigh number R and the wave number a=ax2+ay2 (where ay is the wave number in the x2 direction) at the onset of stationary convection shift to higher values. It has been shown that when H is very slightly inclined to the direction of gravity, the extent to which transverse rolls are suppressed at marginal stability is also very slight. According to [1], when longitudinal rolls and transverse rolls appear simultaneously, a cellular pattern of convection emerges. Therefore if H and g (as well as Ω and g) are not parallel, convection at marginal stability occurs as longitudinal rolls. On the other hand, when H is impressed with an inclination angle ϑH=π/2 to the direction of the vertical and H lies in the x1x3-plane, the vertical component of the magnetic field (HcosϑH) is equal to zero. The authors in [6] found that a magnetic field, impressed in a horizontal direction (ϑH=π/2), did not inhibit convection, even though the magnetic field was five times stronger than that one needed to suppress convection when acting in the vertical direction (ϑH=0); hence, it was experimentally confirmed that the vertical component of the magnetic field (HcosϑH) is the critical parameter to inhibit the onset of convection as it is included in the Chandrasekhar number Q. According to the experimental and theoretical evidence, it is well known that rotation and magnetic field have a stabilizing effect on the Rayleigh-Bénard problem when they act separately [2, 4, 5, 7].When rotation and magnetic field act simultaneously and when both of them are parallel to the gravity vector, the stabilizing effects have conflicting tendencies; that is, the results reveal some very unexpected features showing the complex behaviour of the flow; thus, it has been found that the total effect is not always stabilizing with respect to both fields [1, 4, 8, 9]. The conflictual behaviour depends on the values of the parameters of the system: rotation rate (Taylor number T) and magnetic parameter (Chandrasekhar number Q). Reference [4] only shows explicit results on the dependence of the critical Rayleigh number R on Q and T, for the problem in which the conducting liquid is located between two free boundaries, and when the medium adjoining the fluid is nonconducting, because in this situation the system of equations can be solved explicitly.To the knowledge of the authors, the solution of the problem, in which the three implicated vectors (g, H, and Ω) are not parallel, has not been previously reported. In this paper we show, for the rotating-magnetic Rayleigh-Bénard problem with rigid and free surfaces and nonconducting boundaries, the influence on the critical Rayleigh number R of the cooperative work (or competition), between the strength of the magnetic field and the Coriolis force, when they act simultaneously, but in different directions relatively with the gravity vector. In the literature, very few investigations have been carried out aimed to study the cooperative work of perturbations in the flow to inhibit the thermal convection of an infinite layer of fluid; for example, [7, 10, 11] have found that when the layer of fluid is simultaneously subject to rotation and to a salt concentration field, the stabilizing effects are cumulative; that is, rotation and salt concentration (salted from below) show a cooperative behaviour.By using the Tau Chebyshev method, we find that depending on the value of the parametersCR=axtanϑR and CH=axtanϑH, a critical situation may appear, at which the competition between H and Ω conduct to an unstable system; however as the values of CR or CH are modified, a cooperative work between H and Ω may appear which leads to a system in which the thermal convection of the layer of fluid is inhibited.The paper is organized as follows. In Section2 we briefly describe the physical model of the problem under study. In Section 3 we formulate the general perturbation equations, from which we get the solution of the problems 1 to 6 described in Table 1. Note that the classical linear problems 1 to 5 of this table have been previously solved by [1], while the linear problem 6 of Table 1 is the subject of the present paper that is, thermal instability analysis of a fluid layer subject to gravity g, rotation Ω, and magnetic field H vectors, with angles ϑR and ϑH different from zero. In Section 4 we describe the Tau Chebyshev spectral method that is used to solve the set of perturbation equations. Section 5 shows the solution of the six formulated linear problems by using the Tau Chebyshev method, together with a comparison of our calculations with the solution of the classical problems previously published in the literature; see problems 1 to 5 of Table 1. Finally, in Section 6 we present our concluding remarks.Table 1
Linear problems under study.ϑR is the angle between the rotation vector Ω and the gravity vector g. If ϑR≠0, Ω lies in the x1x3-plane. ϑH is the angle between the magnetic field vector H and the gravity vector g. If ϑH≠0, H lies in the x1x3-plane. g acts along the x3 direction; see Figure 1.
Problem
Gravity,g
Rotation,Ω
Magnetic field,H
ϑ
R
ϑ
H
1 (Section5.1)
Yes
Yes
No
=0
—
2 (Section5.2)
Yes
Yes
No
≠
0
—
3 (Section5.3)
Yes
No
Yes
—
=0
4 (Section5.4)
Yes
No
Yes
—
≠
0
5 (Section5.5)
Yes
Yes
Yes
=0
=0
6 (Section5.6)
Yes
Yes
Yes
≠
0
≠
0


## 2. Physical Situation

Figure1 shows a schematic representation of the geometry under study. A layer of fluid of constant thickness D^=2d^ parallel to the x1x2-plane with no lateral boundaries is considered. The upper and lower boundaries are located at x3=±d^. The temperatures Th and Tc (where Th>Tc) are applied at the lower and upper boundaries, respectively. The whole layer can rotate with an angular velocity vector Ω, which can be inclined at an angle ϑR to the direction of the vertical x3-axis, in such a way that Ω lies in the x1x3-plane. Gravity acts in the negative x3-direction g=-gi3. The fluid layer is assumed to be located in a magnetic field represented by the vector H, which can also be inclined at an angle ϑH to the direction of the vertical x3-axis, in such a way that H also lies in the x1x3-plane. The surfaces are electrically nonconducting; thus, no currents can cross the boundary, and hence we must require J3=0, where J3 is the third component of the current density vector J.Figure 1
Thermal instability analysis in the presence of a rotation vectorΩ and a magnetic field H. The gravity vector g acts in the negative x3-direction. The temperatures Th and Tc (where Th>Tc) are applied at the lower and upper boundaries, respectively. The distance between the two surfaces is D^=2d^. The angle between g and Ω is ϑR, while the angle between g and H is ϑH. Note that the vectors Ω and H lay in the x1x3-plane. The surfaces are electrically nonconducting; thus, no currents can cross the boundary.


## 3. Thermal Instability Analysis in the Presence of a Rotation Vector and a Magnetic Field

In this section we analyse the linear thermal instability of a rotating infinite layer of an incompressible, electrically conducting fluid upon which a uniform magnetic field is impressed. The gravity vectorg, the rotation vector Ω, and the magnetic field vector H may act in different directions.


### 3.1. Governing Equations

Let us consider an infinite fluid layer of an electrically conducting fluid upon which a uniform magnetic fieldH is impressed. The fluid is also subjected to rotation about a fixed axis. For an incompressible conducting fluid, the linear convective thermal stability of a fluid layer (initially at rest and with a steady adverse temperature gradient) is analyzed by solving the following perturbation equations in the Boussinesq approximation and in a rotating frame of [1, page 199]:
(1)∂ui∂t=-∂∂xi(δϖ)+βθgλi+ν∇2ui+2ϵijkujΩk+μ4πρoHj∂hi∂xj,(2)∂θ∂t=β^λjuj+α∇2θ,(3)∂ui∂xi=0,(4)∂hi∂t=Hj∂ui∂xj+η∇2hi,(5)∂hi∂xi=0,
where t is the time, ui are the components of the velocity vector, ν denotes the kinematic viscosity, Ωk are the components of the rotation vector, β is the coefficient of volume expansion, α is the thermal diffusivity, θ is the perturbation in the temperature, g is the acceleration of gravity, μ is the magnetic permeability, η denotes the resistivity, H is the impressed magnetic field, h is the perturbation of the magnetic field, λ=(0,0,1) is the unit vector in the direction of the vertical, and β^ is the imposed adverse temperature gradient defined as
(6)β^=Th-TcΔx3,
where Th and Tc are the temperature of the lower hot surface and the temperature of the upper cold surface, respectively. The last two terms on the right hand side of (1) are the Coriolis force and the Lorentz force, respectively, and the term δϖ is defined as
(7)δϖ=δpρo-12|Ω×r|2+μH·h4πρo,
where
(8)12|Ω×r|2
is the centrifugal force, δp is the perturbation in the pressure, and ρo is the density at a mean temperature To.Taking the curl of (1) we may write
(9)∂ωi∂t=gβϵijk∂θ∂xjλk+ν∇2ωi+2Ωj∂ui∂xj+μ4πρoHj∂vi∂xj,
where ωi are the components of the vorticity vector and the vector v=viii=∇×h is the current density induced by the perturbation. Taking the curl of (9) once again, we obtain
(10)∂∇2ui∂t=gβ(λi∇2θ-λj∂2θ∂xj∂xi)+ν∇4ui-2Ωj∂ωi∂xj+μ4πρoHj∂∂xj∇2hi.
Taking the curl of (4) we have
(11)∂vi∂t=Hj∂ωi∂xj+η∇2vi,
Multiplying (2), (4), (9), (10), and (11) by the components λi we get
(12)∂θ∂t=β^u3+α∇2θ,(13)∂h3∂t=Hj∂u3∂xj+η∇2h3,(14)∂v3∂t=Hj∂ω3∂xj+η∇2v3,(15)∂ω3∂t=ν∇2ω3+2Ωj∂u3∂xj+μ4πρoHj∂v3∂xj,(16)∂∇2u3∂t=gβ(∂2θ∂x12+∂2θ∂x22)+ν∇4u3-2Ωj∂ω3∂xj+μ4πρoHj∂∂xj∇2h3,
where h3, v3, ω3, and u3 are the x3 components of the perturbation of the magnetic field, the current density, the vorticity, and the velocity, respectively. In this investigation the set of equations to be solved are (12)–(16). Let the rotation vector Ω be inclined at an angle ϑR to the direction of the vertical x3-axis. Additionally let the magnetic field vector H be also inclined at an angle ϑH to the direction of the vertical x3-axis. The x1-axis of the Cartesian coordinate system is defined such as the vectors Ω and H lie in the x1x3-plane; see Figure 1. Thus the unit vector in the direction of the vertical, the rotation vector, and the magnetic field vector are written as
(17)λ=(0,0,1),Ω=Ω(sinϑR,0,cosϑR),H=H(sinϑH,0,cosϑH),
where Ω and H are the magnitude of the rotation vector Ω and the magnitude of the magnetic field vector H, respectively. Using (17) into (13)–(16) we obtain
(18)∂h3∂t=H(cosϑH∂∂x3+sinϑH∂∂x1)u3+η∇2h3,∂v3∂t=H(cosϑH∂∂x3+sinϑH∂∂x1)ω3+η∇2v3,∂ω3∂t=ν∇2ω3+μ4πρoH(cosϑH∂∂x3+sinϑH∂∂x1)v3+2Ω(cosϑR∂∂x3+sinϑR∂∂x1)u3,∂∇2u3∂t=gβ(∂2θ∂x12+∂2θ∂x22)+ν∇4u3+μ4πρoH(cosϑH∂∂x3+sinϑH∂∂x1)∇2h3-2Ω(cosϑR∂∂x3+sinϑR∂∂x1)ω3.


#### 3.1.1. The Analysis into Normal Modes

We analyse the perturbationsθ, h3, v3, ω3, and u3 into two dimensional periodic waves characterized by a particular wave number k. We assume the forms of the perturbations as
(19)θ=Θ(x3)exp[i(kxx1+kyx2)+pt],h3=K(x3)exp[i(kxx1+kyx2)+pt],v3=X(x3)exp[i(kxx1+kyx2)+pt],ω3=Z(x3)exp[i(kxx1+kyx2)+pt],u3=W(x3)exp[i(kxx1+kyx2)+pt].
If we define the wave number of the disturbance k as k=(kx2+ky2)1/2 and p as a constant, (12) and (18) can be written as
(20)Θp=β^W+α[d2dx32-k2]Θ,Kp=HcosϑHdWdx3+HsinϑHWkxi+η[d2dx32-k2]K,Xp=HcosϑHdZdx3+HsinϑHZkxi+η[d2dx32-k2]X,Zp=ν[d2dx32-k2]Z+μ4πρoHcosϑHdXdx3+μ4πρoHsinϑHXkxi+2ΩcosϑRdWdx3+2ΩsinϑRWkxi,[d2dx32-k2]Wp=-gβk2Θ+ν[d2dx32-k2]2W+μ4πρoHcosϑHddx3[d2dx32-k2]K+μ4πρoHsinϑH[d2dx32-k2]Kkxi-2ΩcosϑRdZdx3-2ΩsinϑRZkxi.
As it is shown in Figure 1, the origin of the coordinate system is located at the middle between the two surfaces; hence, the distance between the two surfaces is D^=2d^. If we define the following nondimensional variables
(21)a=kD^,σ=pD^2ν,x3*=2x3D^,Pr=να,Pr2=νη,
(20) is written as
(22)[4D2-a2-σPr]Θ=-(D^2β^α)W,(23)[4D2-a2-σPr2]K=-2D^ηHcosϑHDW-D^ηHsinϑHaxiW,(24)[4D2-a2-σPr2]X=-2D^ηHcosϑHDZ-D^ηHsinϑHaxiZ,(25)[4D2-a2-σ]Z=-4D^νΩcosϑRDW-2D^νΩsinϑRaxiW-μD^2πρoνHcosϑHDX-μD^4πρoνHsinϑHaxiX,(26)[4D2-a2][4D2-a2-σ]W+μD^2πρoνHcosϑHD[4D2-a2]K+μD^4πρoνHsinϑH[4D2-a2]axiK-4D^3νΩcosϑRDZ-2D^3νΩsinϑRaxiZ=D^2νgβa2Θ,
where ax=D^kx. If instability sets in as stationary convection, σ=0, we can operate (25) by [4D2-a2] and in the resulting equation we use (24) to get (let us assume that D^=2)
(27)[(4D2-a2)2-Q^D2-Q^CHiD+14Q^CH2]Z=-T1/2D[4D2-a2]W-T1/22CRi[4D2-a2]W,
where
(28)T=4Ω2D^4cos2ϑRν2
is the Taylor number,
(29)CH=axtanϑH,CR=axtanϑR,(30)4D^νΩcosϑR=2D^T1/2,2D^νΩcosϑR=T1/2D^,(31)Q^=μD^2(HcosϑH)2πρoνη.
If we define the resistivity η as
(32)η=14πμσ^,
it is possible to write (31) as
(33)Q^=4(μ2(HcosϑH)2D^2σ^ρoν)=4Q,
where the variable Q defined by Chandrasekhar [1] is written as
(34)Q=μ2(HcosϑH)2D^2σ^ρoν.We can eliminateK from (26) by using (23), then we obtain
(35)[(4D2-a2)2-Q^D2-Q^CHiD+14Q^CH2]W-4D^3νΩcosϑRDZ-2D^3νΩcosϑRCRiZ=(D^2gβν)a2Θ.
We operate (35) by (4D2-a2), and in the resulting equation we use (22); hence, we obtain
(36)(4D2-a2){24D^3ν[(4D2-a2)2-Q^D2-Q^CHiD+14Q^CH2]W-4D^3νΩcosϑRDZ-2D^3νΩcosϑRCRiZ}=-Ra2W,
where the Rayleigh number R is defined as
(37)R=D^4gββ^να.
From (36) we can define the variable F as
(38)F=[(4D2-a2)2-Q^D2-Q^CHiD+14Q^CH2]W-4D^3νΩcosϑRDZ-2D^3νΩcosϑRCRiZ;
hence, (36) is written as
(39)(4D2-a2)F=-Ra2W.
By using the following relationships
(40)4D^3νΩcosϑR=2D^T1/2,2D^3νΩcosϑR=D^T1/2,
into (38) we can formulate the relevant equations to be solved as
(41)16D4W-(8a2+4Q)D2W-4QCHiDW+(a4+QCH2)W-4T1/2DZ-2T1/2CRiZ-F=0,(42)4D2F-a2F=-Ra2W,(43)16D4Z-(8a2+4Q)D2Z-4QCHiDZ+(a4+QCH2)Z+4T1/2D3W-T1/2a2DW+2T1/2CRiD2W-12T1/2CRia2W=0.
Notice that when the magnetic field is negligible H=0, that is, pure rotation, and the relevant equations to be solved are the following:
(44)16D4W-8a2D2W+a4W-4T1/2DZ-2T1/2CRiZ-F=0,(45)4D2F-a2F=-Ra2W,(46)16D4Z-8a2D2Z+a4Z+4T1/2D3W-T1/2a2DW+2T1/2CRiD2W-12T1/2CRia2W=0
together with the following boundary conditions.(a)
For rigid surfaces atx3*=-1  and atx3*=1:
(47)F=0,Z=0,W=0,DW=0,atx3*=-1,atx3*=1.(b)
For free surfaces atx3*=-1  and atx3*=1:
(48)F=0,DZ=0,W=0,D2W=0,atx3*=-1,atx3*=1.(It must be pointed out that due to the process we follow to derive the general relevant equations, the vorticity equation (46) has both different form and higher order than the one we obtained by following Chandrasekhar’s procedure; see Chandrasekhar [1], Chapter III, page 90. The equation for the vorticity that we obtained by considering only rotation without magnetic field is 4D2Z-a2Z+T1/2DW+(1/2)T1/2CRiW=0, which is the same as the one obtained by Chandrasekhar [1].)On the other hand when the rotation vectorΩ is equal to zero, the relevant equations are
(49)16D4W-(8a2+4Q)D2W-4QiCHDW+(a4+QCH2)W-F=0,4D2F-a2F=-Ra2W,
together with the following boundary conditions.(a)
For rigid surfaces atx3*=-1  and atx3*=1:
(50)F=0,W=0,DW=0,atx3*=-1,atx3*=1.(b)
For free surfaces atx3*=-1  and atx3*=1:
(51)F=0,W=0,D2W=0,atx3*=-1,atx3*=1.(For this particular case in which Ω=0, the x3 component of vorticity Z vanishes, see Chandrasekhar [1], Chapter IV, page 166; hence the general equation (43) is not taken into account in the analysis.)Therefore the boundary conditions needed to solve the general equations (41)–(43) are defined by (47) and (48).


## 3.1. Governing Equations

Let us consider an infinite fluid layer of an electrically conducting fluid upon which a uniform magnetic fieldH is impressed. The fluid is also subjected to rotation about a fixed axis. For an incompressible conducting fluid, the linear convective thermal stability of a fluid layer (initially at rest and with a steady adverse temperature gradient) is analyzed by solving the following perturbation equations in the Boussinesq approximation and in a rotating frame of [1, page 199]:
(1)∂ui∂t=-∂∂xi(δϖ)+βθgλi+ν∇2ui+2ϵijkujΩk+μ4πρoHj∂hi∂xj,(2)∂θ∂t=β^λjuj+α∇2θ,(3)∂ui∂xi=0,(4)∂hi∂t=Hj∂ui∂xj+η∇2hi,(5)∂hi∂xi=0,
where t is the time, ui are the components of the velocity vector, ν denotes the kinematic viscosity, Ωk are the components of the rotation vector, β is the coefficient of volume expansion, α is the thermal diffusivity, θ is the perturbation in the temperature, g is the acceleration of gravity, μ is the magnetic permeability, η denotes the resistivity, H is the impressed magnetic field, h is the perturbation of the magnetic field, λ=(0,0,1) is the unit vector in the direction of the vertical, and β^ is the imposed adverse temperature gradient defined as
(6)β^=Th-TcΔx3,
where Th and Tc are the temperature of the lower hot surface and the temperature of the upper cold surface, respectively. The last two terms on the right hand side of (1) are the Coriolis force and the Lorentz force, respectively, and the term δϖ is defined as
(7)δϖ=δpρo-12|Ω×r|2+μH·h4πρo,
where
(8)12|Ω×r|2
is the centrifugal force, δp is the perturbation in the pressure, and ρo is the density at a mean temperature To.Taking the curl of (1) we may write
(9)∂ωi∂t=gβϵijk∂θ∂xjλk+ν∇2ωi+2Ωj∂ui∂xj+μ4πρoHj∂vi∂xj,
where ωi are the components of the vorticity vector and the vector v=viii=∇×h is the current density induced by the perturbation. Taking the curl of (9) once again, we obtain
(10)∂∇2ui∂t=gβ(λi∇2θ-λj∂2θ∂xj∂xi)+ν∇4ui-2Ωj∂ωi∂xj+μ4πρoHj∂∂xj∇2hi.
Taking the curl of (4) we have
(11)∂vi∂t=Hj∂ωi∂xj+η∇2vi,
Multiplying (2), (4), (9), (10), and (11) by the components λi we get
(12)∂θ∂t=β^u3+α∇2θ,(13)∂h3∂t=Hj∂u3∂xj+η∇2h3,(14)∂v3∂t=Hj∂ω3∂xj+η∇2v3,(15)∂ω3∂t=ν∇2ω3+2Ωj∂u3∂xj+μ4πρoHj∂v3∂xj,(16)∂∇2u3∂t=gβ(∂2θ∂x12+∂2θ∂x22)+ν∇4u3-2Ωj∂ω3∂xj+μ4πρoHj∂∂xj∇2h3,
where h3, v3, ω3, and u3 are the x3 components of the perturbation of the magnetic field, the current density, the vorticity, and the velocity, respectively. In this investigation the set of equations to be solved are (12)–(16). Let the rotation vector Ω be inclined at an angle ϑR to the direction of the vertical x3-axis. Additionally let the magnetic field vector H be also inclined at an angle ϑH to the direction of the vertical x3-axis. The x1-axis of the Cartesian coordinate system is defined such as the vectors Ω and H lie in the x1x3-plane; see Figure 1. Thus the unit vector in the direction of the vertical, the rotation vector, and the magnetic field vector are written as
(17)λ=(0,0,1),Ω=Ω(sinϑR,0,cosϑR),H=H(sinϑH,0,cosϑH),
where Ω and H are the magnitude of the rotation vector Ω and the magnitude of the magnetic field vector H, respectively. Using (17) into (13)–(16) we obtain
(18)∂h3∂t=H(cosϑH∂∂x3+sinϑH∂∂x1)u3+η∇2h3,∂v3∂t=H(cosϑH∂∂x3+sinϑH∂∂x1)ω3+η∇2v3,∂ω3∂t=ν∇2ω3+μ4πρoH(cosϑH∂∂x3+sinϑH∂∂x1)v3+2Ω(cosϑR∂∂x3+sinϑR∂∂x1)u3,∂∇2u3∂t=gβ(∂2θ∂x12+∂2θ∂x22)+ν∇4u3+μ4πρoH(cosϑH∂∂x3+sinϑH∂∂x1)∇2h3-2Ω(cosϑR∂∂x3+sinϑR∂∂x1)ω3.


### 3.1.1. The Analysis into Normal Modes

We analyse the perturbationsθ, h3, v3, ω3, and u3 into two dimensional periodic waves characterized by a particular wave number k. We assume the forms of the perturbations as
(19)θ=Θ(x3)exp[i(kxx1+kyx2)+pt],h3=K(x3)exp[i(kxx1+kyx2)+pt],v3=X(x3)exp[i(kxx1+kyx2)+pt],ω3=Z(x3)exp[i(kxx1+kyx2)+pt],u3=W(x3)exp[i(kxx1+kyx2)+pt].
If we define the wave number of the disturbance k as k=(kx2+ky2)1/2 and p as a constant, (12) and (18) can be written as
(20)Θp=β^W+α[d2dx32-k2]Θ,Kp=HcosϑHdWdx3+HsinϑHWkxi+η[d2dx32-k2]K,Xp=HcosϑHdZdx3+HsinϑHZkxi+η[d2dx32-k2]X,Zp=ν[d2dx32-k2]Z+μ4πρoHcosϑHdXdx3+μ4πρoHsinϑHXkxi+2ΩcosϑRdWdx3+2ΩsinϑRWkxi,[d2dx32-k2]Wp=-gβk2Θ+ν[d2dx32-k2]2W+μ4πρoHcosϑHddx3[d2dx32-k2]K+μ4πρoHsinϑH[d2dx32-k2]Kkxi-2ΩcosϑRdZdx3-2ΩsinϑRZkxi.
As it is shown in Figure 1, the origin of the coordinate system is located at the middle between the two surfaces; hence, the distance between the two surfaces is D^=2d^. If we define the following nondimensional variables
(21)a=kD^,σ=pD^2ν,x3*=2x3D^,Pr=να,Pr2=νη,
(20) is written as
(22)[4D2-a2-σPr]Θ=-(D^2β^α)W,(23)[4D2-a2-σPr2]K=-2D^ηHcosϑHDW-D^ηHsinϑHaxiW,(24)[4D2-a2-σPr2]X=-2D^ηHcosϑHDZ-D^ηHsinϑHaxiZ,(25)[4D2-a2-σ]Z=-4D^νΩcosϑRDW-2D^νΩsinϑRaxiW-μD^2πρoνHcosϑHDX-μD^4πρoνHsinϑHaxiX,(26)[4D2-a2][4D2-a2-σ]W+μD^2πρoνHcosϑHD[4D2-a2]K+μD^4πρoνHsinϑH[4D2-a2]axiK-4D^3νΩcosϑRDZ-2D^3νΩsinϑRaxiZ=D^2νgβa2Θ,
where ax=D^kx. If instability sets in as stationary convection, σ=0, we can operate (25) by [4D2-a2] and in the resulting equation we use (24) to get (let us assume that D^=2)
(27)[(4D2-a2)2-Q^D2-Q^CHiD+14Q^CH2]Z=-T1/2D[4D2-a2]W-T1/22CRi[4D2-a2]W,
where
(28)T=4Ω2D^4cos2ϑRν2
is the Taylor number,
(29)CH=axtanϑH,CR=axtanϑR,(30)4D^νΩcosϑR=2D^T1/2,2D^νΩcosϑR=T1/2D^,(31)Q^=μD^2(HcosϑH)2πρoνη.
If we define the resistivity η as
(32)η=14πμσ^,
it is possible to write (31) as
(33)Q^=4(μ2(HcosϑH)2D^2σ^ρoν)=4Q,
where the variable Q defined by Chandrasekhar [1] is written as
(34)Q=μ2(HcosϑH)2D^2σ^ρoν.We can eliminateK from (26) by using (23), then we obtain
(35)[(4D2-a2)2-Q^D2-Q^CHiD+14Q^CH2]W-4D^3νΩcosϑRDZ-2D^3νΩcosϑRCRiZ=(D^2gβν)a2Θ.
We operate (35) by (4D2-a2), and in the resulting equation we use (22); hence, we obtain
(36)(4D2-a2){24D^3ν[(4D2-a2)2-Q^D2-Q^CHiD+14Q^CH2]W-4D^3νΩcosϑRDZ-2D^3νΩcosϑRCRiZ}=-Ra2W,
where the Rayleigh number R is defined as
(37)R=D^4gββ^να.
From (36) we can define the variable F as
(38)F=[(4D2-a2)2-Q^D2-Q^CHiD+14Q^CH2]W-4D^3νΩcosϑRDZ-2D^3νΩcosϑRCRiZ;
hence, (36) is written as
(39)(4D2-a2)F=-Ra2W.
By using the following relationships
(40)4D^3νΩcosϑR=2D^T1/2,2D^3νΩcosϑR=D^T1/2,
into (38) we can formulate the relevant equations to be solved as
(41)16D4W-(8a2+4Q)D2W-4QCHiDW+(a4+QCH2)W-4T1/2DZ-2T1/2CRiZ-F=0,(42)4D2F-a2F=-Ra2W,(43)16D4Z-(8a2+4Q)D2Z-4QCHiDZ+(a4+QCH2)Z+4T1/2D3W-T1/2a2DW+2T1/2CRiD2W-12T1/2CRia2W=0.
Notice that when the magnetic field is negligible H=0, that is, pure rotation, and the relevant equations to be solved are the following:
(44)16D4W-8a2D2W+a4W-4T1/2DZ-2T1/2CRiZ-F=0,(45)4D2F-a2F=-Ra2W,(46)16D4Z-8a2D2Z+a4Z+4T1/2D3W-T1/2a2DW+2T1/2CRiD2W-12T1/2CRia2W=0
together with the following boundary conditions.(a)
For rigid surfaces atx3*=-1  and atx3*=1:
(47)F=0,Z=0,W=0,DW=0,atx3*=-1,atx3*=1.(b)
For free surfaces atx3*=-1  and atx3*=1:
(48)F=0,DZ=0,W=0,D2W=0,atx3*=-1,atx3*=1.(It must be pointed out that due to the process we follow to derive the general relevant equations, the vorticity equation (46) has both different form and higher order than the one we obtained by following Chandrasekhar’s procedure; see Chandrasekhar [1], Chapter III, page 90. The equation for the vorticity that we obtained by considering only rotation without magnetic field is 4D2Z-a2Z+T1/2DW+(1/2)T1/2CRiW=0, which is the same as the one obtained by Chandrasekhar [1].)On the other hand when the rotation vectorΩ is equal to zero, the relevant equations are
(49)16D4W-(8a2+4Q)D2W-4QiCHDW+(a4+QCH2)W-F=0,4D2F-a2F=-Ra2W,
together with the following boundary conditions.(a)
For rigid surfaces atx3*=-1  and atx3*=1:
(50)F=0,W=0,DW=0,atx3*=-1,atx3*=1.(b)
For free surfaces atx3*=-1  and atx3*=1:
(51)F=0,W=0,D2W=0,atx3*=-1,atx3*=1.(For this particular case in which Ω=0, the x3 component of vorticity Z vanishes, see Chandrasekhar [1], Chapter IV, page 166; hence the general equation (43) is not taken into account in the analysis.)Therefore the boundary conditions needed to solve the general equations (41)–(43) are defined by (47) and (48).


## 3.1.1. The Analysis into Normal Modes

We analyse the perturbationsθ, h3, v3, ω3, and u3 into two dimensional periodic waves characterized by a particular wave number k. We assume the forms of the perturbations as
(19)θ=Θ(x3)exp[i(kxx1+kyx2)+pt],h3=K(x3)exp[i(kxx1+kyx2)+pt],v3=X(x3)exp[i(kxx1+kyx2)+pt],ω3=Z(x3)exp[i(kxx1+kyx2)+pt],u3=W(x3)exp[i(kxx1+kyx2)+pt].
If we define the wave number of the disturbance k as k=(kx2+ky2)1/2 and p as a constant, (12) and (18) can be written as
(20)Θp=β^W+α[d2dx32-k2]Θ,Kp=HcosϑHdWdx3+HsinϑHWkxi+η[d2dx32-k2]K,Xp=HcosϑHdZdx3+HsinϑHZkxi+η[d2dx32-k2]X,Zp=ν[d2dx32-k2]Z+μ4πρoHcosϑHdXdx3+μ4πρoHsinϑHXkxi+2ΩcosϑRdWdx3+2ΩsinϑRWkxi,[d2dx32-k2]Wp=-gβk2Θ+ν[d2dx32-k2]2W+μ4πρoHcosϑHddx3[d2dx32-k2]K+μ4πρoHsinϑH[d2dx32-k2]Kkxi-2ΩcosϑRdZdx3-2ΩsinϑRZkxi.
As it is shown in Figure 1, the origin of the coordinate system is located at the middle between the two surfaces; hence, the distance between the two surfaces is D^=2d^. If we define the following nondimensional variables
(21)a=kD^,σ=pD^2ν,x3*=2x3D^,Pr=να,Pr2=νη,
(20) is written as
(22)[4D2-a2-σPr]Θ=-(D^2β^α)W,(23)[4D2-a2-σPr2]K=-2D^ηHcosϑHDW-D^ηHsinϑHaxiW,(24)[4D2-a2-σPr2]X=-2D^ηHcosϑHDZ-D^ηHsinϑHaxiZ,(25)[4D2-a2-σ]Z=-4D^νΩcosϑRDW-2D^νΩsinϑRaxiW-μD^2πρoνHcosϑHDX-μD^4πρoνHsinϑHaxiX,(26)[4D2-a2][4D2-a2-σ]W+μD^2πρoνHcosϑHD[4D2-a2]K+μD^4πρoνHsinϑH[4D2-a2]axiK-4D^3νΩcosϑRDZ-2D^3νΩsinϑRaxiZ=D^2νgβa2Θ,
where ax=D^kx. If instability sets in as stationary convection, σ=0, we can operate (25) by [4D2-a2] and in the resulting equation we use (24) to get (let us assume that D^=2)
(27)[(4D2-a2)2-Q^D2-Q^CHiD+14Q^CH2]Z=-T1/2D[4D2-a2]W-T1/22CRi[4D2-a2]W,
where
(28)T=4Ω2D^4cos2ϑRν2
is the Taylor number,
(29)CH=axtanϑH,CR=axtanϑR,(30)4D^νΩcosϑR=2D^T1/2,2D^νΩcosϑR=T1/2D^,(31)Q^=μD^2(HcosϑH)2πρoνη.
If we define the resistivity η as
(32)η=14πμσ^,
it is possible to write (31) as
(33)Q^=4(μ2(HcosϑH)2D^2σ^ρoν)=4Q,
where the variable Q defined by Chandrasekhar [1] is written as
(34)Q=μ2(HcosϑH)2D^2σ^ρoν.We can eliminateK from (26) by using (23), then we obtain
(35)[(4D2-a2)2-Q^D2-Q^CHiD+14Q^CH2]W-4D^3νΩcosϑRDZ-2D^3νΩcosϑRCRiZ=(D^2gβν)a2Θ.
We operate (35) by (4D2-a2), and in the resulting equation we use (22); hence, we obtain
(36)(4D2-a2){24D^3ν[(4D2-a2)2-Q^D2-Q^CHiD+14Q^CH2]W-4D^3νΩcosϑRDZ-2D^3νΩcosϑRCRiZ}=-Ra2W,
where the Rayleigh number R is defined as
(37)R=D^4gββ^να.
From (36) we can define the variable F as
(38)F=[(4D2-a2)2-Q^D2-Q^CHiD+14Q^CH2]W-4D^3νΩcosϑRDZ-2D^3νΩcosϑRCRiZ;
hence, (36) is written as
(39)(4D2-a2)F=-Ra2W.
By using the following relationships
(40)4D^3νΩcosϑR=2D^T1/2,2D^3νΩcosϑR=D^T1/2,
into (38) we can formulate the relevant equations to be solved as
(41)16D4W-(8a2+4Q)D2W-4QCHiDW+(a4+QCH2)W-4T1/2DZ-2T1/2CRiZ-F=0,(42)4D2F-a2F=-Ra2W,(43)16D4Z-(8a2+4Q)D2Z-4QCHiDZ+(a4+QCH2)Z+4T1/2D3W-T1/2a2DW+2T1/2CRiD2W-12T1/2CRia2W=0.
Notice that when the magnetic field is negligible H=0, that is, pure rotation, and the relevant equations to be solved are the following:
(44)16D4W-8a2D2W+a4W-4T1/2DZ-2T1/2CRiZ-F=0,(45)4D2F-a2F=-Ra2W,(46)16D4Z-8a2D2Z+a4Z+4T1/2D3W-T1/2a2DW+2T1/2CRiD2W-12T1/2CRia2W=0
together with the following boundary conditions.(a)
For rigid surfaces atx3*=-1  and atx3*=1:
(47)F=0,Z=0,W=0,DW=0,atx3*=-1,atx3*=1.(b)
For free surfaces atx3*=-1  and atx3*=1:
(48)F=0,DZ=0,W=0,D2W=0,atx3*=-1,atx3*=1.(It must be pointed out that due to the process we follow to derive the general relevant equations, the vorticity equation (46) has both different form and higher order than the one we obtained by following Chandrasekhar’s procedure; see Chandrasekhar [1], Chapter III, page 90. The equation for the vorticity that we obtained by considering only rotation without magnetic field is 4D2Z-a2Z+T1/2DW+(1/2)T1/2CRiW=0, which is the same as the one obtained by Chandrasekhar [1].)On the other hand when the rotation vectorΩ is equal to zero, the relevant equations are
(49)16D4W-(8a2+4Q)D2W-4QiCHDW+(a4+QCH2)W-F=0,4D2F-a2F=-Ra2W,
together with the following boundary conditions.(a)
For rigid surfaces atx3*=-1  and atx3*=1:
(50)F=0,W=0,DW=0,atx3*=-1,atx3*=1.(b)
For free surfaces atx3*=-1  and atx3*=1:
(51)F=0,W=0,D2W=0,atx3*=-1,atx3*=1.(For this particular case in which Ω=0, the x3 component of vorticity Z vanishes, see Chandrasekhar [1], Chapter IV, page 166; hence the general equation (43) is not taken into account in the analysis.)Therefore the boundary conditions needed to solve the general equations (41)–(43) are defined by (47) and (48).


## 4. The Tau Chebyshev Numerical Method

In order to find an approximate solution to the set of perturbations equations formulated in the previous section, together with the boundary conditions, we use expansions in Chebyshev polynomials. Therefore we expand the variablesW(x3*), F(x3*), and Z(x3*) as well as their derivatives D=d/dx3*, D2=d2/dx3*2, D3=d3/dx3*3, and D4=d4/dx3*4, in the interval -1≤x3*≤1 as follows:
(52)W(x3*)=∑n=0N‍aWnTn(x3*),(53)F(x3*)=∑n=0N‍aFnTn(x3*),(54)Z(x3*)=∑n=0N‍aZnTn(x3*),(55)dqWdx3*q=∑n=0N‍aWndqTn(x3*)dx3*q=∑n=0N‍bWnTn(x3*),(56)dqFdx3*q=∑n=0N‍aFndqTn(x3*)dx3*q=∑n=0N‍bFnTn(x3*),(57)dqZdx3*q=∑n=0N‍aZndqTn(x3*)dx3*q=∑n=0N‍bZnTn(x3*),
where q is an integer number defining the order of the derivative of the functions W, F, and Z, and Tn(x3*) are the nth-degree Chebyshev polynomials of the first kind (with N as the maximum degree used for the polynomials). The Chebyshev polynomials satisfy the orthogonality relation [12–14]:
(58)∫-11‍Tn(x3*)Tm(x3*)(1-x3*2)-1/2dx3*=π2cnδnm,
where c0=2, cn=1 for n>0. Equations for the expansion coefficients aWn, aFn, aZn, bWn (in terms of aWn), bFn (in terms of aFn), and bZn (in terms of aZn) are generated by substituting (52)–(57) into the set of perturbation equations shown in the previous section and applying the orthogonality property (58). In the literature it is possible to find explicit formulae for the expansion coefficients bWn in terms of aWn, bFn in terms of aFn, and bZn in terms of aZn [12, 15].The method that we have selected to obtain the equations for the coefficientsaWn, aFn, and aZn is the Tau Chebyshev method. The Tau Chebyshev method is an efficient numerical procedure for finding eigenvalues and eigenfunctions of systems in fluid and solid mechanics theories. One advantage of the algorithm is that it is able to calculate as many eigenvalues as required, not just the dominant one [13, 14]. Using this numerical methodology we generate, by substituting (52), (53), (55), and (56) into the couple of the general perturbation equations for the case when Ω=0; see (49), one equation for each n=0,1,2,…,2N+2-6. The remaining six equations, those for 2N+2-6+1≤n≤2N+2, are generated by using the six boundary conditions. Hence for rigid surfaces, the set of boundary conditions are written as
(59)W(1)=∑n=0N‍aWn=0,dW(1)dx3*=∑n=0N‍aWnn2=0,W(-1)=∑n=0N‍aWn(-1)n=0,dW(-1)dx3*=∑n=0N‍aWn(-1)n-1n2=0,F(1)=∑n=0N‍aFn=0,F(-1)=∑n=0N‍aFn(-1)n=0,
whereas for free surfaces the set of boundary conditions are written as
(60)W(1)=∑n=0N‍aWn=0,d2W(1)dx3*2=∑n=0N‍aWnn2(n2-13)=0,W(-1)=∑n=0N‍aWn(-1)n=0,d2W(-1)dx3*2=∑n=0N‍aWn(-1)n+2n2(n2-13)=0,F(1)=∑n=0N‍aFn=0,F(-1)=∑n=0N‍aFn(-1)n=0.
Using the Tau Chebyshev method in the cases with rotation (Ω≠0) we generate, by substituting (52)–(57) into the three perturbation equations, one equation for each n=0,1,2,…,3N+3-8. The remaining eight equations, those for 3N+3-8+1≤n≤3N+3, are generated by using the eight boundary conditions. Hence for rigid surfaces, the set of boundary conditions are written as
(61)W(1)=∑n=0N‍aWn=0,dW(1)dx3*=∑n=0N‍aWnn2=0,W(-1)=∑n=0N‍aWn(-1)n=0,dW(-1)dx3*=∑n=0N‍aWn(-1)n-1n2=0,F(1)=∑n=0N‍aFn=0,F(-1)=∑n=0N‍aFn(-1)n=0,Z(1)=∑n=0N‍aZn=0,Z(-1)=∑n=0N‍aZn(-1)n=0,
whereas for free surfaces the set of boundary conditions are written as
(62)W(1)=∑n=0N‍aWn=0,d2W(1)dx3*2=∑n=0N‍aWnn2(n2-13)=0,W(-1)=∑n=0N‍aWn(-1)n=0,d2W(-1)dx3*2=∑n=0N‍aWn(-1)n+2n2(n2-13)=0,F(1)=∑n=0N‍aFn=0,F(-1)=∑n=0N‍aFn(-1)n=0,dZ(1)dx3*=∑n=0N‍aZnn2=0,dZ(-1)dx3*=∑n=0N‍aZn(-1)n-1n2=0.
Notice that the high frequency behaviour (i.e., high n) of the solution is not governed by the perturbation equations but by the boundary conditions [12]. The Tau equations can be written as a generalized eigenvalue problem (Ax)=R(Bx), where the vector x includes the coefficients aWn, aFn (and aZn), R is the Rayleigh number, and A and B are (2N+2)×(2N+2) matrices (or (3N+3)×(3N+3) matrices) whose first 2N+2-6 (or 3N+3-8) rows are defined by the two (or three) perturbation equations. The last six (or eight) rows of A are given by the boundary conditions, and the last six (or eight) rows of B vanish [16]. The system of equations has been solved numerically by using the routine F02GJF from the NAG library. The routine F02GJF calculates all the eigenvalues of the eigenproblem Ax=λ^Bx, by using the QZ algorithm (see http://www.nag.co.uk/numeric/Fl/manual/xhtml/F02/f02gjf.xml).


## 5. Results

### 5.1. Thermal Instability Analysis in the Presence of a Rotation Vector,ϑR=0, H=0

Tables2 and 3 show the critical Rayleigh number R and wave number a of the unstable modes at marginal stability for the onset of stationary convection when both bounding surfaces are rigid and when both bounding surfaces are free, respectively. It is possible to observe that the results obtained by the Tau Chebyshev method are in full agreement with the variational formulation; see [1]. It is also observed the inhibition effect of the Coriolis force as the Taylor number T increases. As T becomes greater, the value of the critical wave number also increases.Table 2
Critical Rayleigh numbers and wave numbers at marginal stability for the onset of stationary convection when both bounding surfaces are rigid. Gravity vectorg parallel to the rotation vector Ω. T is the Taylor number. R is the Rayleigh number obtained by the Tau Chebyshev method. RCh is the Rayleigh number obtained by Chandrasekhar [1].
T
a
R
R
Ch
10
3.1
1.7127
E
3
1.713
E
3
1
0
2
3.15
1.756
E
3
1.7566
E
3
1
0
3
3.50
2.1514
E
3
2.1517
E
3
1
0
5
7.20
1.6720
E
4
1.6721
E
4
1
0
10
55.5
4.476
E
7
3.4574
E
7Table 3
Critical Rayleigh numbers and wave numbers at marginal stability for the onset of stationary convection when both bounding surfaces are free. Gravity vectorg parallel to the rotation vector Ω. T is the Taylor number. R is the Rayleigh number obtained by the Tau Chebyshev method. RCh is the Rayleigh number obtained by Chandrasekhar [1].
T
a
R
R
Ch
T
a
R
R
Ch
1
0
2
2.594
8.262
E
2
8.263
E
2
1
0
6
12.86
9.222
E
4
9.222
E
4
1
0
3
3.710
1.676
E
3
1.676
E
3
1
0
8
28.02
1.897
E
6
1.897
E
6
1
0
4
5.698
5.377
E
3
5.377
E
3
1
0
11
88.87
1.875
E
8
1.876
E
8
1
0
5
8.626
2.130
E
4
2.131
E
4
1
0
13
191.51
4.037
E
9
4.037
E
9


### 5.2. Thermal Instability Analysis in the Presence of a Rotation Vector,ϑR≠0, H=0

Tables4 and 5 show the results obtained by the Tau Chebyshev method when both bounding surfaces are rigid and when both bounding surfaces are free, respectively. It is observed that when the Taylor number is T=100, as the value of the CR parameter increases, the critical parameters (Rayleigh number Rc and wave number ac) become greater. Note in Table 4 that when CR=1, ac=3.15 and Rc=1766.28; however, when CR=4, ac=3.35 and Rc=1905.43. Also note in Table 5 that when CR=1, ac=2.66 and Rc=855.0; however, when CR=4, ac=3.2 and Rc=1145.92. Hence for a fixed Taylor number, the net effect of increasing CR is to stabilize the system.Table 4
Thermal instability analysis in the presence of a rotation vector when both bounding surfaces are rigid: gravity vector and rotation vector acting in different directions. Rayleigh numbersR for the Taylor number T = 100 and for various values of a and CR.
C
R
a
R
C
R
a
R
0.0
3.15
1756.37
—
—
—
0.5
3.0
1765.91
2.0
3.0
1806.85
3.15
1758.85
3.15
1796.01
3.35
1767.31
3.35
1800.20
1.0
3.0
1774.10
4.0
3.0
1937.84
3.15
1766.28
3.35
1905.43
3.35
1773.89
3.45
1908.84Table 5
Thermal instability analysis in the presence of a rotation vector when both bounding surfaces are free: Gravity vector and rotation vector acting in different directions. Rayleigh numbersR for the Taylor number T = 100 and for various values of a and CR.
C
R
a
R
C
R
a
R
0.0
2.594
826.28
—
—
—
0.5
2.5
836.54
2.0
2.62
943.16
2.6
833.77
2.74
932.00
2.64
833.84
3.00
933.42
1.0
2.5
861.92
4.0
2.76
1228.56
2.66
855.00
3.2
1145.92
2.72
855.35
3.4
1151.32


### 5.3. Thermal Instability Analysis in the Presence of a Magnetic Field,ϑH=0, Ω=0

Tables6 and 7 show the critical Rayleigh number R and wave number a of the unstable modes at marginal stability for the onset of stationary convection when both bounding surfaces are rigid and when both bounding surfaces are free, respectively. It is possible to observe that the results obtained by the Tau Chebyshev method are in full agreement with the variational formulation; see [1]. It is also observed the inhibition effect of the magnetic field, as the magnetic number Q increases. As Q becomes greater, the value of the critical wave number also increases.Table 6
Critical Rayleigh numbers and wave numbers at marginal stability for the onset of stationary convection when both bounding surfaces are rigid. Gravity vectorg parallel to the magnetic field vector H. Q is the dimensionless magnetic parameter. R is the Rayleigh number obtained by the Tau Chebyshev method. RCh is the Rayleigh number obtained by Chandrasekhar [1].
Q
a
R
R
Ch
10
3.25
1945.8
1945.8
100
4.00
3757.27
3757.3
1000
5.80
17102.94
17103
4000
7.40
54696.93
54700
10000
8.66
124508.47
124509Table 7
Critical Rayleigh numbers and wave numbers at marginal stability for the onset of stationary convection when both bounding surfaces are free. Gravity vectorg parallel to the magnetic field vector H. Q is the dimensionless magnetic parameter. R is the Rayleigh number obtained by the Tau Chebyshev method. RCh is the Rayleigh number obtained by Chandrasekhar [1].
Q
a
R
R
Ch
Q
a
R
R
Ch
1
0
2.590
923.069
923.070
7500
8.159
91705.75
91705.7
100
3.702
2653.708
2653.71
9000
8.429
108622.9
108623
1000
5.684
15207.01
15207.0
15000
9.227
175279.33
175279
2000
6.453
27699.95
27699.9
17000
9.433
197248.97
197249
5000
7.585
63135.98
63135.9
35000
10.70
392013.13
392013


### 5.4. Thermal Instability Analysis in the Presence of a Magnetic Field,ϑH≠0, Ω=0

Tables8 and 9 show the results obtained by the Tau Chebyshev method when both bounding surfaces are rigid and when both bounding surfaces are free, respectively. It is observed that when the Chandrasekhar number Q is fixed to Q=100, as the value of the CH parameter increases, the critical Rayleigh number becomes greater. In Table 8 we present a comparison between the results obtained by [1] and the results of the Tau Chebyshev method. We observe that if the data obtained by Chandrasekhar are the exact values, the maximum percentage error (% error), when CH≠0, of the critical R number calculated by the Tau Chebyshev method occurs when CH=4.0 and a=4.1, with an error of 3%, while the minimum percentage error occurs when CH=0.5 and a=3.9, with an error of 0.3%. Thus we may conclude that our calculations are in good agreement with Chandrasekhar’s results. Note in Table 8 that when CH=1, the critical wave number a=4; however, when CH=4, the critical wave number a=4.1 Also note in Table 9 that when CH=1, the critical wave number a=3.7; however, when CH=4, the critical wave number a=4.1. Hence for a fixed Chandrasekhar number Q, the net effect of increasing CH is to stabilize the system.Table 8
Thermal instability analysis in the presence of a magnetic field when both bounding surfaces are rigid: gravity vector and magnetic field vector acting in different directions. Rayleigh numbers for the dimensionless parameterQ = 100 and for various values of a and CH. R is the Rayleigh number obtained by the Tau Chebyshev method. RCh is the Rayleigh number obtained by Chandrasekhar [1].
C
H
a
R
R
Ch
C
H
a
R
R
Ch
0.0
4.0
3757.27
3757.3
—
—
—
—
0.5
3.9
3785.12
3797.0
2.0
3.9
4146.18
4190.0
4.0
3780.70
3793.0
4.0
4136.55
4181.0
4.1
3782.82
3795.0
4.1
4133.94
4178.0
1.0
3.9
3856.64
3875.0
4.0
3.9
5356.98
5523.0
4.0
3851.20
3878.0
4.0
5328.39
5494.0
4.1
3852.41
3871.0
4.1
5308.46
5474.0Table 9
Thermal instability analysis in the presence of a magnetic field when both bounding surfaces are free: gravity vector and magnetic field vector acting in different directions. Rayleigh numbers for the dimensionless parameterQ = 100, and for various values of a and CH. R is the Rayleigh number obtained by the Tau Chebyshev method. RCh is the Rayleigh number obtained by Chandrasekhar [1].
C
H
a
R
C
H
a
R
0.0
3.702
2653.70
—
—
—
0.5
3.6
2679.49
2.0
3.6
3028.72
3.7
2676.17
3.8
3017.01
3.8
2678.55
3.9
3020.24
1.0
3.6
2748.31
4.0
3.6
4226.99
3.7
2743.9
4.1
4151.21
3.8
2745.3
4.2
4157.48


### 5.5. Thermal Instability Analysis in the Presence of a Rotation Vector and a Magnetic Field,ϑR=0, ϑH=0

Table10 shows a comparison between the results obtained by the Tau Chebyshev method and those of Chandrasekhar’s variational formulation when both bounding surfaces are free. The results have been obtained for several values of the modified magnetic number Q1=Q/π2 and for a fixed value of the modified Taylor number T1=T/π4=105. Notice that the agreement is encouraging. Also note that the two minima of the Rayleigh number are successfully reproduced by the Tau Chebyshev method. It is interesting to mention that by using the Tau Chebyshev method we found two additional critical Rayleigh numbers that were not reported by Chandrasekhar (see [1], Table XVIII, page 205, for T1=105). We found (not shown in Table 10) that when Q1=30, a=3.93 and R=1.125×106, and when Q1=200, a=17.1 and R=3.855×105. Figure 2 shows that if T1=105, when both bounding surfaces are free (see row (a)), and the magnetic parameter Q1 is either Q1=40 (curve labelled a), Q1=100 (curve labelled b), or Q1=200 (curve labelled c), see left panel, the two minima of the Rayleigh number curves reported by Chandrasekhar are recovered (see [1], Figure 49, page 208). Row (b), left panel of Figure 2, shows that when the bounding surfaces are rigid, the two minima only slightly appear when Q1=100 (curve labelled b) and Q1=200 (curve labelled c); however, when Q1=40 (curve labelled a), only one minimum value is present. Figure 2 shows on the left panel of both rows the case with pure rotation T1=105, that is, without magnetic field (see the curves with circles). It is pointed out that when the surfaces are free (row (a), left panel), the effect of the magnetic field is to destabilize the system, particularly when Q1=200 (curve labelled c). This effect is clearly observed for low values of the modified wave number x=a2/π2. It is also observed that the system with Q1=200 is more unstable than the pure rotation case. However when the surfaces are rigid (see row (b), left panel), it is possible to observe that as the magnetic field becomes greater, it firstly induces a stabilizing effect, with respect to the pure rotation case (see curve labelled a for Q1=40 and the curve with circles), but for Q1=100 and 200, it initiates a destabilizing effect; however, the pure rotation case (see the curve with circles) is more unstable. On the other hand Figure 2 shows, on the right panel of both rows, the influence of greater magnetic numbers (Q1=400 (see continuous lines), Q1=1500 (see dashed lines), Q1=2000 (see dotted lines), and Q1=4000 (see dash-dotted lines)). Also shown on the right panel of both rows is the case with pure magnetic field Q1=4000, that is, without rotation (see the curves with asterisks). It is pointed out that when the surfaces are free or rigid, when Q1=400 (see the continuous lines), the system reaches its critical situation; that is, the minimum value of the Reynolds number is attained. However as Q1 increases even more, the system becomes more stable. Now when the surfaces are free (row (a)), and when the curve with asterisks (Q1=4000, without rotation) is compared with the dash-dotted line (Q1=4000, T1=105), it is possible to observe that the Coriolis force induces a marginal stabilizing effect. However when the surfaces are rigid (row (b)) and when the curve with asterisks (Q1=4000, without rotation) is compared with the dash-dotted line (Q1=4000, T1=105), we observe that rotation has a significant stabilizing effect. According to the previous discussion we confirm the fact that when H and Ω act simultaneously and when both are parallel to the gravity vector, the stabilizing effects have conflicting tendencies. Figure 3 shows the influence of the Coriolis force when the magnetic number is fixed to Q1=2000 and when the surfaces are free (left panel) and rigid (right panel). Notice that when the surfaces are free, the Coriolis force when T1=1000 (curve labelled a) does not produce any effect on the system as compared to that when there is no rotation (see the asterisks curve for T1=0). However when the surfaces are rigid (right panel), the Coriolis force when T1=1000 induces a stabilizing effect as compared with the case without rotation (see the asterisks curve for T1=0). It is clearly observed that as the modified Taylor number T1 acquires the values 5×105 and 106, the system becomes more stable.Table 10
Thermal instability analysis in the presence of a rotation and a magnetic field when both bounding surfaces are free: gravity vectorg parallel to both rotation vector Ω and magnetic field vector H. Critical Rayleigh numbers and wave numbers at marginal stability for the onset of stationary convection. Taylor number T1 = T/π4=105 for various values of the dimensionless magnetic parameter Q1 = Q/π2. R and a are the Rayleigh number and the wave number, respectively, obtained by the Tau Chebyshev method. RCh and aCh are the Rayleigh number and the wave number, respectively, obtained by Chandrasekhar [1]. The superscripts 1 and 2 mean the two minima of the Rayleigh number.
Q
1
a
1
R
1
a
Ch
1
R
Ch
1
a
2
R
2
a
Ch
2
R
Ch
2
10
18.87
4.066
E
5
18.9
4.067
E
5
—
—
—
—
50
18.58
4.028
E
5
18.6
4.028
E
5
3.49
7.261
E
5
3.48
7.261
E
5
100
18.2
3.976
E
5
18.2
3.977
E
5
3.365
3.932
E
5
3.37
3.933
E
5
1000
—
—
—
—
5.675
1.828
E
5
5.67
1.829
E
5
10000
—
—
—
—
12.29
1.081
E
6
12.3
1.082
E
6
100000
—
—
—
—
18.9
1.015
E
7
18.9
1.015
E
7Thermal instability analysis in the presence of a rotation and a magnetic field. Gravity vector parallel to both rotation vector and magnetic field vector. The dependence of the Rayleigh numberR (R*=R/100) on the wave number a (x=a2/π2) when T1=105. Left panels: Q1=40 (curves labelled a), Q1=100 (curves labelled b), Q1=200 (curves labelled c), circles: pure rotation T1=105 without magnetic field. Right panels: Q1=400 (continuous lines), Q1=1500 (dashed lines), Q1=2000 (dotted lines), Q1=4000 (dash-dotted lines), asterisks: pure magnetic field Q1=2000, without rotation. Row (a): free boundaries. Row (b): rigid boundaries.
(a)
(b)Thermal instability analysis in the presence of a rotation and a magnetic field. Gravity vector parallel to both rotation vector and magnetic field vector. The dependence of the Rayleigh numberR (R*=R/100) on the wave number a (x=a2/π2) when Q1=2000. The effect of the Coriolis force: T1=0 (without rotation, asterisks curves), T1=1000 (curves labelled a), T1=5×105 (curves labelled b), and T1=106 (curves labelled c). Left panel: free boundaries. Right panel: rigid boundaries.
(a)
(b)


### 5.6. Thermal Instability Analysis in the Presence of a Rotation Vector and a Magnetic Field,ϑR≠0, ϑH≠0

Figure4 shows, for free surfaces, the dependence of the critical Rayleigh number R*=R/100 and the critical wave number x=a2/π2, on the parameter CR, when the Taylor number T1=T/π4=105, the magnetic parameter Q1=Q/π2=40, and the gravity vector g is parallel to the magnetic field vector H (CH=0). On the left panel the competition between the stabilizing effect of the Coriolis force and the strong destabilizing properties of the CR parameter when it is in the range 0.5≤CR≤4 is observed. When CR=0 and CH=0, the critical Rayleigh number is R*=8829.8; see left panel of Figure 2, while left panel of Figure 4 shows that when CR=0.5, the critical Rayleigh number has been lowered to R*=3009.2. Note, on the left panel, the presence of the two minima in the curves of the Rayleigh number R*. On the right panel of Figure 4 it is shown that when the CR parameter is in the range 6≤CR≤10, the fluid layer becomes more stable as CR increases. Therefore, we have found that there is a critical value of the CR parameter at which the destabilizing behaviour is changed to a stabilizing performance, and the critical value occurs around CR=6 (see right panel). It is also possible to observe that the curves corresponding to CR=8 (curve labelled b) and CR=10 (curve labelled c) only show one minimum value. In general it is observed that as the CR parameter increases, the wave number x=a2/π2 also increases and a tendency to get only one minimum is apparent.Thermal instability analysis in the presence of a rotation and a magnetic field. Gravity vector parallel to the magnetic field vector (CH=0). Rotation vector at different values of CR≠0. The dependence of the Rayleigh number R (R*=R/100) for the onset of instability on the wave number a (x=a2/π2) when T1=105 and Q1=40 (for free surfaces). Q1=Q/π2 and T1=T/π4. Left panel: CR=0.5 (continuous line), CR=1 (dashed line), CR=2 (dotted line), and CR=4 (dotted-dashed line). Right panel: CR=6 (curve labelled a), CR=8 (curve labelled b), and CR=10 (curve labelled c).
(a)
(b)Figure5 shows, for free surfaces, the dependence of the critical Rayleigh number R* and the critical wave number x on the parameter CH, when the Taylor number T1=105, the magnetic parameter Q1=40, and the gravity vector g is parallel to the rotation vector Ω (CR=0). On the left panel it is shown that the influence of the parameter CH, when its value is 0.5 (curve labelled a), is almost negligible; hence, curve for CH=0.5 resembles curve labelled a, of Figure 2 left panel, when CR=0 and CH=0. On the left panel of Figure 5, it is possible to observe the competition between the stabilizing effect of the Coriolis force and the destabilizing influence of the CH parameter when its values are in the range 0.5≤CH≤10; that is, as the CH parameter increases, the fluid layer gets more unstable. Right panel of Figure 5 shows the cooperative behaviour between the Coriolis force and the CH parameter; that is, as the CH parameter increases even more in the range 15≤CH≤35, the fluid layer becomes more stable. Hence we have found again that a critical value of the CH parameter exists at which the destabilizing performance of CH is transformed to a stabilizing behaviour. The critical CH is around CH=15. We can also conclude that as the CH parameter increases, the wave number x=a2/π2 also increases and the tendency to get only one minimum is clear.Thermal instability analysis in the presence of a rotation and a magnetic field. Gravity vector parallel to the rotation vector (CR=0). Magnetic field vector at different values of CH≠0. The dependence of the Rayleigh number R* for the onset of instability on the wave number x when T1=105 and Q1=40 (for free surfaces). Left panel: CH=0.5 (curve labelled a), CH=1 (curve labelled b), CH=2 (curve labelled c), CH=4 (curve labelled d), CH=6 (curve labelled e), CH=8 (curve labelled f), and CH=10 (curve labelled g). Right panel: CH=15 (curve labelled a), CH=20 (curve labelled b), CH=25 (curve labelled c), CH=30 (curve labelled d), and CH=35 (curve labelled e).
(a)
(b)Finally left panel of Figure6 shows for free surfaces, the dependence of the Rayleigh number R*=R/100 on the wave number x=a2/π2 when CR is a fixed value equal to CR=6 and CH has the values CH=2, 10, 20, and 40. Right panel of Figure 6 also shows for free surfaces the dependence of the Rayleigh number R* on the wave number x when CH is a fixed value equal to CH=15 and CR has the values CR=0.5, 2, 10, and 15. In both panels of Figure 6, the modified Taylor and magnetic numbers are T1=105 and Q1=40, respectively. When CR is fixed to CR=6 (see left panel), we observe that the curve CH=2 (continuous line) resembles the curve a (for CH=0) shown in right panel of Figure 4. However the case CH=2 is a bit more unstable than the case CH=0. When CH increases to CH=10 (+ curve), the system becomes even more unstable; however, it is clearly observed that there exists a critical value of CH (around 10), at which the system shifts its unstable behaviour to a more stable performance, see the curves for CH=20 (* curve) and CH=40 (∘ curve). When CH is fixed to CH=15 (see right panel), we observe that the curve CR=0.5 (continuous line) is very similar to the curve a (for CR=0) shown in Figure 5 right panel. When CR increases to CR=2 (+ curve), the system becomes more unstable, however it is clearly observed that there also exists a critical value of CR (around 2), at which the system shifts its unstable behaviour to a more stable performance; see the curves for CR=10 (* curve) and CR=15 (∘ curve). In both panels of Figure 6 it is observed that before the critical situation there exists a competition between CH and CR parameters, and the result is that the fluid layer is transformed into a more unstable system. However after the critical situation, there is a cooperative work of CH and CR to inhibit the thermal convection of the infinite layer of fluid.Thermal instability analysis in the presence of a rotation and a magnetic field. The dependence of the Rayleigh numberR* for the onset of instability on the wave number x when T1=105 and Q1=40 (for free surfaces). Left panel: rotation vector at CR=6, for different values of CH, CH=2 (continuous line), CH=10 (+ curve), CH=20 (* curve), and CH=40 (∘ curve). Right panel: magnetic field vector at CH=15, for different values of CR, CR=0.5 (continuous line), CR=2 (+ curve), CR=10 (* curve), and CR=15 (∘ curve).
(a)
(b)


## 5.1. Thermal Instability Analysis in the Presence of a Rotation Vector,ϑR=0, H=0

Tables2 and 3 show the critical Rayleigh number R and wave number a of the unstable modes at marginal stability for the onset of stationary convection when both bounding surfaces are rigid and when both bounding surfaces are free, respectively. It is possible to observe that the results obtained by the Tau Chebyshev method are in full agreement with the variational formulation; see [1]. It is also observed the inhibition effect of the Coriolis force as the Taylor number T increases. As T becomes greater, the value of the critical wave number also increases.Table 2
Critical Rayleigh numbers and wave numbers at marginal stability for the onset of stationary convection when both bounding surfaces are rigid. Gravity vectorg parallel to the rotation vector Ω. T is the Taylor number. R is the Rayleigh number obtained by the Tau Chebyshev method. RCh is the Rayleigh number obtained by Chandrasekhar [1].
T
a
R
R
Ch
10
3.1
1.7127
E
3
1.713
E
3
1
0
2
3.15
1.756
E
3
1.7566
E
3
1
0
3
3.50
2.1514
E
3
2.1517
E
3
1
0
5
7.20
1.6720
E
4
1.6721
E
4
1
0
10
55.5
4.476
E
7
3.4574
E
7Table 3
Critical Rayleigh numbers and wave numbers at marginal stability for the onset of stationary convection when both bounding surfaces are free. Gravity vectorg parallel to the rotation vector Ω. T is the Taylor number. R is the Rayleigh number obtained by the Tau Chebyshev method. RCh is the Rayleigh number obtained by Chandrasekhar [1].
T
a
R
R
Ch
T
a
R
R
Ch
1
0
2
2.594
8.262
E
2
8.263
E
2
1
0
6
12.86
9.222
E
4
9.222
E
4
1
0
3
3.710
1.676
E
3
1.676
E
3
1
0
8
28.02
1.897
E
6
1.897
E
6
1
0
4
5.698
5.377
E
3
5.377
E
3
1
0
11
88.87
1.875
E
8
1.876
E
8
1
0
5
8.626
2.130
E
4
2.131
E
4
1
0
13
191.51
4.037
E
9
4.037
E
9


## 5.2. Thermal Instability Analysis in the Presence of a Rotation Vector,ϑR≠0, H=0

Tables4 and 5 show the results obtained by the Tau Chebyshev method when both bounding surfaces are rigid and when both bounding surfaces are free, respectively. It is observed that when the Taylor number is T=100, as the value of the CR parameter increases, the critical parameters (Rayleigh number Rc and wave number ac) become greater. Note in Table 4 that when CR=1, ac=3.15 and Rc=1766.28; however, when CR=4, ac=3.35 and Rc=1905.43. Also note in Table 5 that when CR=1, ac=2.66 and Rc=855.0; however, when CR=4, ac=3.2 and Rc=1145.92. Hence for a fixed Taylor number, the net effect of increasing CR is to stabilize the system.Table 4
Thermal instability analysis in the presence of a rotation vector when both bounding surfaces are rigid: gravity vector and rotation vector acting in different directions. Rayleigh numbersR for the Taylor number T = 100 and for various values of a and CR.
C
R
a
R
C
R
a
R
0.0
3.15
1756.37
—
—
—
0.5
3.0
1765.91
2.0
3.0
1806.85
3.15
1758.85
3.15
1796.01
3.35
1767.31
3.35
1800.20
1.0
3.0
1774.10
4.0
3.0
1937.84
3.15
1766.28
3.35
1905.43
3.35
1773.89
3.45
1908.84Table 5
Thermal instability analysis in the presence of a rotation vector when both bounding surfaces are free: Gravity vector and rotation vector acting in different directions. Rayleigh numbersR for the Taylor number T = 100 and for various values of a and CR.
C
R
a
R
C
R
a
R
0.0
2.594
826.28
—
—
—
0.5
2.5
836.54
2.0
2.62
943.16
2.6
833.77
2.74
932.00
2.64
833.84
3.00
933.42
1.0
2.5
861.92
4.0
2.76
1228.56
2.66
855.00
3.2
1145.92
2.72
855.35
3.4
1151.32


## 5.3. Thermal Instability Analysis in the Presence of a Magnetic Field,ϑH=0, Ω=0

Tables6 and 7 show the critical Rayleigh number R and wave number a of the unstable modes at marginal stability for the onset of stationary convection when both bounding surfaces are rigid and when both bounding surfaces are free, respectively. It is possible to observe that the results obtained by the Tau Chebyshev method are in full agreement with the variational formulation; see [1]. It is also observed the inhibition effect of the magnetic field, as the magnetic number Q increases. As Q becomes greater, the value of the critical wave number also increases.Table 6
Critical Rayleigh numbers and wave numbers at marginal stability for the onset of stationary convection when both bounding surfaces are rigid. Gravity vectorg parallel to the magnetic field vector H. Q is the dimensionless magnetic parameter. R is the Rayleigh number obtained by the Tau Chebyshev method. RCh is the Rayleigh number obtained by Chandrasekhar [1].
Q
a
R
R
Ch
10
3.25
1945.8
1945.8
100
4.00
3757.27
3757.3
1000
5.80
17102.94
17103
4000
7.40
54696.93
54700
10000
8.66
124508.47
124509Table 7
Critical Rayleigh numbers and wave numbers at marginal stability for the onset of stationary convection when both bounding surfaces are free. Gravity vectorg parallel to the magnetic field vector H. Q is the dimensionless magnetic parameter. R is the Rayleigh number obtained by the Tau Chebyshev method. RCh is the Rayleigh number obtained by Chandrasekhar [1].
Q
a
R
R
Ch
Q
a
R
R
Ch
1
0
2.590
923.069
923.070
7500
8.159
91705.75
91705.7
100
3.702
2653.708
2653.71
9000
8.429
108622.9
108623
1000
5.684
15207.01
15207.0
15000
9.227
175279.33
175279
2000
6.453
27699.95
27699.9
17000
9.433
197248.97
197249
5000
7.585
63135.98
63135.9
35000
10.70
392013.13
392013


## 5.4. Thermal Instability Analysis in the Presence of a Magnetic Field,ϑH≠0, Ω=0

Tables8 and 9 show the results obtained by the Tau Chebyshev method when both bounding surfaces are rigid and when both bounding surfaces are free, respectively. It is observed that when the Chandrasekhar number Q is fixed to Q=100, as the value of the CH parameter increases, the critical Rayleigh number becomes greater. In Table 8 we present a comparison between the results obtained by [1] and the results of the Tau Chebyshev method. We observe that if the data obtained by Chandrasekhar are the exact values, the maximum percentage error (% error), when CH≠0, of the critical R number calculated by the Tau Chebyshev method occurs when CH=4.0 and a=4.1, with an error of 3%, while the minimum percentage error occurs when CH=0.5 and a=3.9, with an error of 0.3%. Thus we may conclude that our calculations are in good agreement with Chandrasekhar’s results. Note in Table 8 that when CH=1, the critical wave number a=4; however, when CH=4, the critical wave number a=4.1 Also note in Table 9 that when CH=1, the critical wave number a=3.7; however, when CH=4, the critical wave number a=4.1. Hence for a fixed Chandrasekhar number Q, the net effect of increasing CH is to stabilize the system.Table 8
Thermal instability analysis in the presence of a magnetic field when both bounding surfaces are rigid: gravity vector and magnetic field vector acting in different directions. Rayleigh numbers for the dimensionless parameterQ = 100 and for various values of a and CH. R is the Rayleigh number obtained by the Tau Chebyshev method. RCh is the Rayleigh number obtained by Chandrasekhar [1].
C
H
a
R
R
Ch
C
H
a
R
R
Ch
0.0
4.0
3757.27
3757.3
—
—
—
—
0.5
3.9
3785.12
3797.0
2.0
3.9
4146.18
4190.0
4.0
3780.70
3793.0
4.0
4136.55
4181.0
4.1
3782.82
3795.0
4.1
4133.94
4178.0
1.0
3.9
3856.64
3875.0
4.0
3.9
5356.98
5523.0
4.0
3851.20
3878.0
4.0
5328.39
5494.0
4.1
3852.41
3871.0
4.1
5308.46
5474.0Table 9
Thermal instability analysis in the presence of a magnetic field when both bounding surfaces are free: gravity vector and magnetic field vector acting in different directions. Rayleigh numbers for the dimensionless parameterQ = 100, and for various values of a and CH. R is the Rayleigh number obtained by the Tau Chebyshev method. RCh is the Rayleigh number obtained by Chandrasekhar [1].
C
H
a
R
C
H
a
R
0.0
3.702
2653.70
—
—
—
0.5
3.6
2679.49
2.0
3.6
3028.72
3.7
2676.17
3.8
3017.01
3.8
2678.55
3.9
3020.24
1.0
3.6
2748.31
4.0
3.6
4226.99
3.7
2743.9
4.1
4151.21
3.8
2745.3
4.2
4157.48


## 5.5. Thermal Instability Analysis in the Presence of a Rotation Vector and a Magnetic Field,ϑR=0, ϑH=0

Table10 shows a comparison between the results obtained by the Tau Chebyshev method and those of Chandrasekhar’s variational formulation when both bounding surfaces are free. The results have been obtained for several values of the modified magnetic number Q1=Q/π2 and for a fixed value of the modified Taylor number T1=T/π4=105. Notice that the agreement is encouraging. Also note that the two minima of the Rayleigh number are successfully reproduced by the Tau Chebyshev method. It is interesting to mention that by using the Tau Chebyshev method we found two additional critical Rayleigh numbers that were not reported by Chandrasekhar (see [1], Table XVIII, page 205, for T1=105). We found (not shown in Table 10) that when Q1=30, a=3.93 and R=1.125×106, and when Q1=200, a=17.1 and R=3.855×105. Figure 2 shows that if T1=105, when both bounding surfaces are free (see row (a)), and the magnetic parameter Q1 is either Q1=40 (curve labelled a), Q1=100 (curve labelled b), or Q1=200 (curve labelled c), see left panel, the two minima of the Rayleigh number curves reported by Chandrasekhar are recovered (see [1], Figure 49, page 208). Row (b), left panel of Figure 2, shows that when the bounding surfaces are rigid, the two minima only slightly appear when Q1=100 (curve labelled b) and Q1=200 (curve labelled c); however, when Q1=40 (curve labelled a), only one minimum value is present. Figure 2 shows on the left panel of both rows the case with pure rotation T1=105, that is, without magnetic field (see the curves with circles). It is pointed out that when the surfaces are free (row (a), left panel), the effect of the magnetic field is to destabilize the system, particularly when Q1=200 (curve labelled c). This effect is clearly observed for low values of the modified wave number x=a2/π2. It is also observed that the system with Q1=200 is more unstable than the pure rotation case. However when the surfaces are rigid (see row (b), left panel), it is possible to observe that as the magnetic field becomes greater, it firstly induces a stabilizing effect, with respect to the pure rotation case (see curve labelled a for Q1=40 and the curve with circles), but for Q1=100 and 200, it initiates a destabilizing effect; however, the pure rotation case (see the curve with circles) is more unstable. On the other hand Figure 2 shows, on the right panel of both rows, the influence of greater magnetic numbers (Q1=400 (see continuous lines), Q1=1500 (see dashed lines), Q1=2000 (see dotted lines), and Q1=4000 (see dash-dotted lines)). Also shown on the right panel of both rows is the case with pure magnetic field Q1=4000, that is, without rotation (see the curves with asterisks). It is pointed out that when the surfaces are free or rigid, when Q1=400 (see the continuous lines), the system reaches its critical situation; that is, the minimum value of the Reynolds number is attained. However as Q1 increases even more, the system becomes more stable. Now when the surfaces are free (row (a)), and when the curve with asterisks (Q1=4000, without rotation) is compared with the dash-dotted line (Q1=4000, T1=105), it is possible to observe that the Coriolis force induces a marginal stabilizing effect. However when the surfaces are rigid (row (b)) and when the curve with asterisks (Q1=4000, without rotation) is compared with the dash-dotted line (Q1=4000, T1=105), we observe that rotation has a significant stabilizing effect. According to the previous discussion we confirm the fact that when H and Ω act simultaneously and when both are parallel to the gravity vector, the stabilizing effects have conflicting tendencies. Figure 3 shows the influence of the Coriolis force when the magnetic number is fixed to Q1=2000 and when the surfaces are free (left panel) and rigid (right panel). Notice that when the surfaces are free, the Coriolis force when T1=1000 (curve labelled a) does not produce any effect on the system as compared to that when there is no rotation (see the asterisks curve for T1=0). However when the surfaces are rigid (right panel), the Coriolis force when T1=1000 induces a stabilizing effect as compared with the case without rotation (see the asterisks curve for T1=0). It is clearly observed that as the modified Taylor number T1 acquires the values 5×105 and 106, the system becomes more stable.Table 10
Thermal instability analysis in the presence of a rotation and a magnetic field when both bounding surfaces are free: gravity vectorg parallel to both rotation vector Ω and magnetic field vector H. Critical Rayleigh numbers and wave numbers at marginal stability for the onset of stationary convection. Taylor number T1 = T/π4=105 for various values of the dimensionless magnetic parameter Q1 = Q/π2. R and a are the Rayleigh number and the wave number, respectively, obtained by the Tau Chebyshev method. RCh and aCh are the Rayleigh number and the wave number, respectively, obtained by Chandrasekhar [1]. The superscripts 1 and 2 mean the two minima of the Rayleigh number.
Q
1
a
1
R
1
a
Ch
1
R
Ch
1
a
2
R
2
a
Ch
2
R
Ch
2
10
18.87
4.066
E
5
18.9
4.067
E
5
—
—
—
—
50
18.58
4.028
E
5
18.6
4.028
E
5
3.49
7.261
E
5
3.48
7.261
E
5
100
18.2
3.976
E
5
18.2
3.977
E
5
3.365
3.932
E
5
3.37
3.933
E
5
1000
—
—
—
—
5.675
1.828
E
5
5.67
1.829
E
5
10000
—
—
—
—
12.29
1.081
E
6
12.3
1.082
E
6
100000
—
—
—
—
18.9
1.015
E
7
18.9
1.015
E
7Thermal instability analysis in the presence of a rotation and a magnetic field. Gravity vector parallel to both rotation vector and magnetic field vector. The dependence of the Rayleigh numberR (R*=R/100) on the wave number a (x=a2/π2) when T1=105. Left panels: Q1=40 (curves labelled a), Q1=100 (curves labelled b), Q1=200 (curves labelled c), circles: pure rotation T1=105 without magnetic field. Right panels: Q1=400 (continuous lines), Q1=1500 (dashed lines), Q1=2000 (dotted lines), Q1=4000 (dash-dotted lines), asterisks: pure magnetic field Q1=2000, without rotation. Row (a): free boundaries. Row (b): rigid boundaries.
(a)
(b)Thermal instability analysis in the presence of a rotation and a magnetic field. Gravity vector parallel to both rotation vector and magnetic field vector. The dependence of the Rayleigh numberR (R*=R/100) on the wave number a (x=a2/π2) when Q1=2000. The effect of the Coriolis force: T1=0 (without rotation, asterisks curves), T1=1000 (curves labelled a), T1=5×105 (curves labelled b), and T1=106 (curves labelled c). Left panel: free boundaries. Right panel: rigid boundaries.
(a)
(b)


## 5.6. Thermal Instability Analysis in the Presence of a Rotation Vector and a Magnetic Field,ϑR≠0, ϑH≠0

Figure4 shows, for free surfaces, the dependence of the critical Rayleigh number R*=R/100 and the critical wave number x=a2/π2, on the parameter CR, when the Taylor number T1=T/π4=105, the magnetic parameter Q1=Q/π2=40, and the gravity vector g is parallel to the magnetic field vector H (CH=0). On the left panel the competition between the stabilizing effect of the Coriolis force and the strong destabilizing properties of the CR parameter when it is in the range 0.5≤CR≤4 is observed. When CR=0 and CH=0, the critical Rayleigh number is R*=8829.8; see left panel of Figure 2, while left panel of Figure 4 shows that when CR=0.5, the critical Rayleigh number has been lowered to R*=3009.2. Note, on the left panel, the presence of the two minima in the curves of the Rayleigh number R*. On the right panel of Figure 4 it is shown that when the CR parameter is in the range 6≤CR≤10, the fluid layer becomes more stable as CR increases. Therefore, we have found that there is a critical value of the CR parameter at which the destabilizing behaviour is changed to a stabilizing performance, and the critical value occurs around CR=6 (see right panel). It is also possible to observe that the curves corresponding to CR=8 (curve labelled b) and CR=10 (curve labelled c) only show one minimum value. In general it is observed that as the CR parameter increases, the wave number x=a2/π2 also increases and a tendency to get only one minimum is apparent.Thermal instability analysis in the presence of a rotation and a magnetic field. Gravity vector parallel to the magnetic field vector (CH=0). Rotation vector at different values of CR≠0. The dependence of the Rayleigh number R (R*=R/100) for the onset of instability on the wave number a (x=a2/π2) when T1=105 and Q1=40 (for free surfaces). Q1=Q/π2 and T1=T/π4. Left panel: CR=0.5 (continuous line), CR=1 (dashed line), CR=2 (dotted line), and CR=4 (dotted-dashed line). Right panel: CR=6 (curve labelled a), CR=8 (curve labelled b), and CR=10 (curve labelled c).
(a)
(b)Figure5 shows, for free surfaces, the dependence of the critical Rayleigh number R* and the critical wave number x on the parameter CH, when the Taylor number T1=105, the magnetic parameter Q1=40, and the gravity vector g is parallel to the rotation vector Ω (CR=0). On the left panel it is shown that the influence of the parameter CH, when its value is 0.5 (curve labelled a), is almost negligible; hence, curve for CH=0.5 resembles curve labelled a, of Figure 2 left panel, when CR=0 and CH=0. On the left panel of Figure 5, it is possible to observe the competition between the stabilizing effect of the Coriolis force and the destabilizing influence of the CH parameter when its values are in the range 0.5≤CH≤10; that is, as the CH parameter increases, the fluid layer gets more unstable. Right panel of Figure 5 shows the cooperative behaviour between the Coriolis force and the CH parameter; that is, as the CH parameter increases even more in the range 15≤CH≤35, the fluid layer becomes more stable. Hence we have found again that a critical value of the CH parameter exists at which the destabilizing performance of CH is transformed to a stabilizing behaviour. The critical CH is around CH=15. We can also conclude that as the CH parameter increases, the wave number x=a2/π2 also increases and the tendency to get only one minimum is clear.Thermal instability analysis in the presence of a rotation and a magnetic field. Gravity vector parallel to the rotation vector (CR=0). Magnetic field vector at different values of CH≠0. The dependence of the Rayleigh number R* for the onset of instability on the wave number x when T1=105 and Q1=40 (for free surfaces). Left panel: CH=0.5 (curve labelled a), CH=1 (curve labelled b), CH=2 (curve labelled c), CH=4 (curve labelled d), CH=6 (curve labelled e), CH=8 (curve labelled f), and CH=10 (curve labelled g). Right panel: CH=15 (curve labelled a), CH=20 (curve labelled b), CH=25 (curve labelled c), CH=30 (curve labelled d), and CH=35 (curve labelled e).
(a)
(b)Finally left panel of Figure6 shows for free surfaces, the dependence of the Rayleigh number R*=R/100 on the wave number x=a2/π2 when CR is a fixed value equal to CR=6 and CH has the values CH=2, 10, 20, and 40. Right panel of Figure 6 also shows for free surfaces the dependence of the Rayleigh number R* on the wave number x when CH is a fixed value equal to CH=15 and CR has the values CR=0.5, 2, 10, and 15. In both panels of Figure 6, the modified Taylor and magnetic numbers are T1=105 and Q1=40, respectively. When CR is fixed to CR=6 (see left panel), we observe that the curve CH=2 (continuous line) resembles the curve a (for CH=0) shown in right panel of Figure 4. However the case CH=2 is a bit more unstable than the case CH=0. When CH increases to CH=10 (+ curve), the system becomes even more unstable; however, it is clearly observed that there exists a critical value of CH (around 10), at which the system shifts its unstable behaviour to a more stable performance, see the curves for CH=20 (* curve) and CH=40 (∘ curve). When CH is fixed to CH=15 (see right panel), we observe that the curve CR=0.5 (continuous line) is very similar to the curve a (for CR=0) shown in Figure 5 right panel. When CR increases to CR=2 (+ curve), the system becomes more unstable, however it is clearly observed that there also exists a critical value of CR (around 2), at which the system shifts its unstable behaviour to a more stable performance; see the curves for CR=10 (* curve) and CR=15 (∘ curve). In both panels of Figure 6 it is observed that before the critical situation there exists a competition between CH and CR parameters, and the result is that the fluid layer is transformed into a more unstable system. However after the critical situation, there is a cooperative work of CH and CR to inhibit the thermal convection of the infinite layer of fluid.Thermal instability analysis in the presence of a rotation and a magnetic field. The dependence of the Rayleigh numberR* for the onset of instability on the wave number x when T1=105 and Q1=40 (for free surfaces). Left panel: rotation vector at CR=6, for different values of CH, CH=2 (continuous line), CH=10 (+ curve), CH=20 (* curve), and CH=40 (∘ curve). Right panel: magnetic field vector at CH=15, for different values of CR, CR=0.5 (continuous line), CR=2 (+ curve), CR=10 (* curve), and CR=15 (∘ curve).
(a)
(b)


## 6. Conclusions

The Tau Chebyshev method has been used to solve the classical problem of thermal convection on an infinite fluid layer with rotation and an impressed magnetic field. Our methodology has been verified with results previously published in the literature, through the solution of the first five cases shown in Table1. By using the Tau Chebyshev method, we find that, for the sixth case shown in Table 1, a critical value of the CR and CH parameters exists at which the competition between H and Ω, which leads to an unstable system, can be transformed to a cooperative work that promotes a system in which the thermal convection of the layer of fluid is inhibited.



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*Source: 236901-2013-08-06.xml*