INTRODUCTION
A comprehensive account of thermal instability (Bénard convection) in
a fluid layer under varying assumptions of hydrodynamics has been summarized
in the celebrated monograph by Chandrasekhar (1981). AbdulBari
and AlRubai (2008) have studied the influence of Rayleighnumber in turbulent
and laminar region in parallelplate vertical channels. The influence of radiation
on the unsteady free convection flow of a viscous incompressible fluid past
a moving vertical plate with Newtonian heating has been investigated theoretically
by Narahari and Ishak (2011). Admon
et al. (2011) have considered the unsteady free convection flow near
the stagnation point of a threedimensional body. The theory of couplestress
fluid has been formulated by Stokes (1966). One of the
applications of couplestress fluid is its use to the study of the mechanisms
of lubrications of synovial joints, which has become the object of scientific
research. A human joint is a dynamically loaded bearing which has articular
cartilage as the bearing and synovial fluid as the lubricant. When a fluid is
generated, squeezefilm action is capable of providing considerable protection
to the cartilage surface. The shoulder, ankle, knee and hip joints are the loadedbearing
synovial joints of the human body and these joints have a low friction coefficient
and negligible wear. Normal synovial fluid is a viscous, nonNewtonian fluid
and is generally clear or yellowish. According to the theory of Stokes, couplestresses
appear in noticeable magnitudes in fluids with very large molecules.
Many of the flow problems in fluids with couplestresses, discussed by Stokes,
indicate some possible experiments, which could be used for determining the
material constants and the results are found to differ from those of Newtonian
fluid. Couplestresses are found to appear in noticeable magnitudes in polymer
solutions for force and couplestresses. This theory is developed in an effort
to examine the simplest generalization of the classical theory, which would
allow polar effects. The constitutive equations proposed by Stokes
(1966) are:
and:
Where:
and:
Here, T_{ij}, T_{(ij)}, T_{[ij]}, M_{ij}, D_{ij},
,
G_{s}, ,
V, ρ and λ, μ, η, η', are stress tensor, symmetric
part of T_{ij} antisymmetric part of T_{ij} the couplestress
tensor, deformation tensor, the vorticity tensor, the vorticity vector, body
couple, the alternating unit tensor, velocity field, the density and material
constants, respectively. The dimensions of λ and μ are those of viscosity
whereas the dimensions of η and η' are those of momentum.
Since the long chain hyaluronic acid molecules are found as additives in synovial
fluids, Walicki and Walicka (1999) modeled the synovial
fluid as a couplestress fluid. The synovial fluid is the natural lubricant
of joints of the vertebrates. The detailed description of the joint lubrication
has very important practical implications. Practically all diseases of joints
are caused by or connected with a malfunction of the lubrication. The efficiency
of the physiological joint lubrication is caused by sefveral mechanisms. The
synovial fluid is, due to its content of the hyaluronic acid, a fluid of high
viscosity, near to a gel. Goel et al. (1999)
have studied the hydromagnetic stability of an unbounded couplestress binary
fluid mixture under rotation with vertical temperature and concentration gradients.
Sharma et al. (2002) have considered a couplestress
fluid with suspended particles heated from below. They have found that for stationary
convection, couplestress has a stabilizing effect whereas suspended particles
have a destabilizing effect. The peristaltic flow of blood through a planar
channel of uniform thickness has been investigated by Sobh
and Oda (2008). A layer of such fluid heated from below under the action
of magnetic field and rotation may find applications in physiological processes
e.g., MHD finds applications in physiological processes such as magnetic therapy;
heating, rotation may find applications in physiotherapy.
Keeping in mind the importance of couplestress fluid, convection in fluid
layer heated from below, magnetic field and rotation; the present paper attempts
to study the effect of uniform vertical magnetic field on the couplestress
fluid heated from below in the presence of a uniform rotation.
FORMULATION OF THE PROBLEM AND PERTURBATION EQUATIONS
Consider an infinite, horizontal, incompressible, electrically conducting couplestress
fluid layer of thickness, heated from below so that, the temperatures and densities
at the bottom surface z = 0 are T_{0} and ρ_{0} and at
the upper surface z = d are T_{d} and ρ_{d}, respectively
and that a uniform temperature gradient β (=dT/dz) is maintained.
The gravity field ,
a uniform vertical magnetic field
and a uniform vertical rotation
act on the system.
Let
and
denote the fluid velocity, pressure, density, temperature, kinematic viscosity
and couplestress viscosity, respectively. Then the momentum balance, mass balance
and energy balance equations of couplestress fluid (STOKES [7]; CHANDRASEKHAR
[1]) in the presence of magnetic field and rotation are:
The equation of state is:
where, ρ_{0}, T_{0} are, respectively, the density and
temperature of the fluid at the reference level z = 0 and a is the coefficient
of thermal expansion. In writing Eq. (1), use has been made
of the Boussinesq approximation, which states that the density variations are
ignored in all terms in the equations of motion except the external force term.
The magnetic permeability μ_{e}, thermal diffusivity χ and
electrical resistivity η are all assumed to be constants.
The steady solution is:
Let
denote, respectively the perturbations in velocity
(initially zero), pressure p, density ρ, temperature T and the magnetic
field .
The change in density δρ, caused by the perturbation θ in temperature,
is given by:
i.e.:
Then the linearized perturbation equations for the couplestress fluid are:
Within the framework of Boussinesq approximation, Eq. 913,
becomes:
Where:
stand for the zcomponents of vorticity and current density, respectively.
DISPERSION RELATION
Decompose the disturbances into normal modes and assume that the perturbed quantities are of the form:
where, k_{x}, k_{y} are the wave numbers along x and y directions,
respectively,
is the resultant wave number and n is the growth rate which is, in general,
a complex constant.
Using expression (19), Eq. 1418 in nondimensional
form transform to:
where, we have introduced new coordinates:
in new units of length d and D = d/dz'. For convenience, the dashes are dropped hereafter. Also we have put a = kd:
is the Prandtl number and:
is the magnetic Prandtl number.
Here, we consider the case where both the boundaries are free as well as perfect
conductors of heat, while the adjoining medium is also perfectly conducting.
The case of two free boundaries is slightly artificial, except in stellar atmospheres
(Spiegel, 1965) and in certain geophysical situations
where it is most appropriate. However, the case of two free boundaries allows
us to obtain analytical solution without affecting the essential features of
the problem. The appropriate boundary conditions, with respect to which Eq.
2024 must be solved are:
DX = 0, K = 0 on a perfectly conducting boundary.
Using the above boundary conditions, it can be shown that all the even order derivatives of W must vanish for z = 0 and z = 1 and hence the proper solution of W characterizing the lowest mode is:
where W_{0} is a constant.
Eliminating Θ, K, Z and X between Eq. 2024
and substituting Eq. 26 in the resultant equation, we obtain
the dispersion relation:
Where:
stand for the Rayleighnumber, the Chandrasekhar number, the Taylor number, respectively and we have also put:
and .
THE STATIONARY CONVECTION
When the instability sets in as stationary convection, the marginal state will be characterized by σ = 0. Putting σ = 0, the dispersion relation Eq. 27 reduces to:
In the absence of couplestress parameter, rotation and magnetic field, Eq. 28 reduces to:
a result given by Chandrasekhar (1981), similarly, in
the absence of couplestress parameter and magnetic field, Eq.
28 reduces to:
a result given by Chandrasekhar (1981), in the absence
of couplestress parameter and rotation, Eq. 28 reduces to:
a result given by Chandrasekhar (1981).
To study the effects of magnetic field, couplestress parameter and rotation, we examine the natures of:
analytically.
Equation 28 yields:
It is evident from Eq. 29 and 30 that,
for a stationary convection:
may be positive as well as negative, thus, the magnetic field and the couplestress
parameter have both stabilizing and destabilizing effects on the system. It
is also clear from Eq. 31 that, for a stationary convection,
is always positive, thus, the rotation has a stabilizing effect on the system.
The dispersion relation (28) is analysed numerically. In Fig.
1, R_{1} is plotted against Q_{1} for fixed values of T_{1}
= 100, F_{1} = 10 and wave numbers x = 0.5, 1.0. It depicts both the
stabilizing and d e stabilizing effect o f the magnetic field on the system.
Figure 2, shows the variation of R_{1} with respect
to F_{1}, for fixed values of Q_{1} = 100, T_{1} = 100
and wave numbers x = 0.5, 1.0. It clearly depicts both the stabilizing and destabilizing
effect of the couplestress parameter on the system. Figure 3,
shows the variation of R_{1} with respect to T_{1}, for fixed
values of F_{1}= 10, Q_{1} = 100 and wave numbers x = 0.2, 1.0.

Fig. 1: 
The variation of Rayleigh number R_{1} with Q_{1}
for T_{1}=100, F_{1} = 10, x = 0.5 and 1.0 

Fig. 2: 
The variation of Rayleigh number R_{1} with F_{1}
for Q_{1}=100, T_{1} = 100, x = 0.5 and 1.0 

Fig. 3: 
The variation of Rayleigh number R_{1} with T_{1}
for F_{1}=10, Q_{1} = 100, x = 0.2 and 1.0 
The Rayleigh number R_{1} increases with increase in rotation parameter
T_{1} showing its stabilizing effect on the system.
STABILITY OF THE SYSTEM AND OSCILLATORY MODES
Here we examine the possibility of oscillatory modes, on a stability problem
due to the presence of rotation and magnetic field. Multiplying Eq.
20 by W*, the complex conjugate of W and using Eq. 2124
together with the boundary conditions (25), we obtain:
Where:
and σ* is the complex conjugate of σ. The integrals I_{1},……I_{12} are all positive definite. Putting σ = σ_{r}+ iσ_{i}, where σ_{r}, σ_{i} are real and equating the real and imaginary parts of Eq. 32, we obtain:
It is evident from Eq. 34 that σ_{r} is either positive or negative. The system is, therefore, either stable or unstable. It is clear from Eq. 35 that σ_{i} may be either zero or nonzero, meaning that the modes may be either nonoscillatory or oscillatory. In the absence of rotation and magnetic field, Eq. 35 reduces to:
and the terms in brackets are positive definite. Thus, σ_{i} = 0, which means that oscillatory modes are not allowed and the principle of exchange of stabilities is satisfied for the couplestress fluid heated from below. Thus, the magnetic field and rotation introduces oscillatory modes (as σ_{i} may not be zero) in the system which were nonexistent in their absence.
CONCLUSION
Couplestress fluid is an important and useful nonNewtonian fluid. Keeping in mind the importance of nonNewtonian fluids, the present paper considered the thermal convection in couplestress fluid in the presence of uniform vertical magnetic field and uniform rotation. For stationary convection, rotation is found to have a stabilizing effect whereas the magnetic field and couplestress have both stabilizing and destabilizing effects. The rotation and magnetic field brings oscillatory modes in the system, which were nonexistent in their absence.