Stability Analysis of a Magneto Micropolar Fluid Layer by Variational Method

Abstract This Paper deals with the Stability of a micropolar fluid layer heated from below in the presence of uniform magnetic field. The basic hydrodynamic equations of magneto-microploar fluid layer heated from below are modified by using Boussinesq approximation and then the linearized perturbation equations are converted into a characteristic value problem with the help of Normal mode analysis. The expressions for Rayleigh number are obtained by using the variational principle. The effect of magnetic field and micropolar parameter on the Rayleigh numbers are discussed andupper bounds of critical Rayleigh number for all type of boundaries are obtained by using variational method.


Introduction
Thermal convection occurs in nature in so many forms and over such a wide range of scales that it could be claimed with some justification that convection represents the most common fluid flow in the universe.
The experiments by Bénard [1] particular have attracted great attention and are today considered as classical in fluid mechanics. The Bénard stability problem was first formulated and mathematically solved by Loard Rayleigh [2] for the case of free boundaries with a linear temperature gradient. A comprehensive account of the linearized stability theory of Rayleigh-Bénard convection problem in the presence of uniform rotation has been given in chanderasekhar [3].
Some years ago, Eringen [4] developed the theory of micropolar fluids in which local effects arising from micro-Structure and intrinsic motions of the fluid elements are taken into account. Ahmadi [5] studies the stability of a layer of micropolar fluid heated from below using linear theory as well as energy method the derived lower bounds for critical Rayleigh number. The profiles on Benard convection in micropoler fluids is also discussed by Narasimha, MY [6]. Dattav and Sastry [7] also discussed the theory of micropolar fluid layer heated from below and obtained exact solution of the eigen value problem. Joginder Singh Dhiman, Praveenkumar Sharma and Gurdeep Singh.
[8] studied the stability of micropolar fluid layer heated from below by variational principle.
The stability of magneto-micropolar fluid motion was studied by G. Ahmadi and M. Shahinpoor [9]. The stability method employed by G. Ahmadi and M. Shahinpoor was an energy technique due to James Serrin [10].
Our aim in this paper is to verify the stability of magneto-micropolar fluid motion by using the variational method and we have established this by finding the upper bounds for critical Rayleigh numbers for all combinations of different type of boundaries.

Physical Problem
A viscous finitely heat and electrically conducting, micropolar fluid is statically confined between two horizontal boundaries Z = 0 and Z = d of infinite extension and finite vertical depth which are maintained at uniform temperatures T 0 and T 1 (T 0 > T 1 ) respectively in the presence of uniform magnetic field acting antiparallel to the force field of gravity.
The boundary conditions are (When both the boundaries are rigid).
(When both boundaries are free) and (When both boundaries are either rigid or free, but boundaries are conducting). In the foregoing equations; k is dynamic micro rotation viscosity, µ is the dynamic Newtonain viscosity, p is hydrostatic pressure, J is microinertia, γis constant stands for coefficient of angular viscosity. d D dz is the differentiation with respect to z, z in the real independent variable, a 2 is the square of wave number, σ is the thermal Prandtl number, is the heat diffusivity is the Rayleigh number, g is the gravitational Acceleration, α is the coefficient of tharmal gradient, d is the depth of layer, k 1 is the thermometric conductivity, v is the kinematic µ is viscosity, w, θ and G are the pertubations in vertical velocity, temperature and microrotation respectively and The system of equations (1) -(4) together with boundary condition (5) -(7) constitutes an eigen value problem for R for given values of other parameters σ, c 0 , n 1 , a 2 K and Q.

Principle of Exchange of Stabilities (PES)
Firstly in the present problem, we shall see that whether the PES is valid or not for magneto micropolar fluid heated from below. So multiplying equation (1) by w * (the complex conjugate of w) and integrating over the range 0 ≤z≤ 1, a suitable number of times by using relevant boundary conditions (5)- (7) Taking the complex conjugate of equation (3) and multiplying the resulting equation by G on both sides and integrating it over the range of z, a suitable number of times by using boundary conditions (5) -(7), we get ( ) By using equation (9) is equation (8) and integrating by parts the resulting equation, we get Now equating real and imaginary parts of both sides of equation (10) and canceling p i (≠ 0) (supposition) throughout from imaginary part, we get; Now, multiplying the equation (4) by h z * (the complex conjugate of h z ) and integrating the resulting equation over the vertical range of z, a suitable number of times and making use of either boundary conditions (5) -(7), we have.
( ) Equating real parts from both sides of above equation, we have: Since p r ≥ 0 from above inequality, we have ( ) Further, since w(0) = 0 = w(1), we have the Reyleigh -Ritz inequality namely; Using inequalities (14) and (15) is equation (11), we get It clearly follows from above inequality that Which is a contradiction to our suppositon that p i ≠ 0. Hence, we must have p i = 0, In particular p r = 0 ⇒ p i = 0, hence PES is Valid. Therefore, we have the result that PES is not valid or have over stability when Qσ 1 > 2 , Π which can also be seen in classical magneto hydrodynamic Bénard convection.

Variational Principle
When instability sets is as stationary convection, the marginal state will be characterized by p =0 and basic equations (1) -(4) reduce to From equation (17), letting Operating the above equation lay (D 2 -a 2 ) and using equation (18), we get: Multiplying the above equation by F and integrating the resulting equation over the range of z, we have ( ) Since F(0) = 0 = F(1)because of equation (22), therefore, we have Now integrating the equation (24) by using F from equation (22) and integrating a suitable number of times by using relevant boundary conditions, we get:

Stationary Property
The stationary property is R given by equation (28) is checkedjust by giving a small variation δW to w, δG to G and δh z to h z , which are again compatible with boundary conditions i.e.

Minimum Property
Now, we shall show that the lowest characteristic value of R is indeed, a true minimum.
Let R i be a characteristic value and let corresponding characteristic function be distinguished by a subscript i. Then from equation (23) Integrating the integrants on right hand side of above equation by parts, a suitable number of times with the help of boundary conditions(30) -(31), We have Similarly using the expanded forms W i terms of basic set functions for G, F, and h z , the values of integrals I 1 and I 2 and equation (28) having the following values, also, I 2 = 1 1 In follows from equation (34) that This shows that the quantity on R.H.S has a true minimum.

Upper Bounds for Critical Raylight Number for All Combination of Boundary Conditions
Let Which obviously satisfies the boundary conditions The general solution of above differential equation is given by letting k → ∞ in the above inequality, we have : Which is the upper bound for R c for free-free boundaries,

Case II One Rigid -One Free Boundary:
Let us take lower boundary as rigid and upper dynamically free, so using these boundary conditions in the equations (39) -(41) we get:  Which is upper bound for R, for one rigid-one free boundaries.

Case III. For both rigid-rigid boundaries:
Now by using equations (39), (40) and (42), with boundary conditions in w, Dw and θ (proceeding same as in the previous two cases), We have: