Introduction
Stability of a dynamical system is closest to real life, in the sense that realization of a dynamical system depends upon its stability. Right from the conceptualizations of turbulence, instability of fluid flows is being regarded at its root. The thermal instability of a fluid layer with maintained adverse temperature gradient by heating the underside plays an important role in Geophysics, interiors of the Earth, Oceanography and Atmospheric Physics, and has been investigated by several authors (e.g., Bénard   1 , Rayleigh   2 , Jeffreys   acted upon by a uniform rotation and found that rotation has destabilizing as well as stabilizing effects under certain conditions in contrast to that of a Maxwell fluid where it has a destabilizing effect. In another study Sharma   10 has studied the stability of a layer of an electrically conducting Oldroyd fluid   5 in the presence of magnetic field and has found that the magnetic field has a stabilizing influence. 
Introduction
, for all wave numbers and for any combination of perfectly conducting dynamically free and rigid boundaries. 
II. Formulation Of The Problem And Perturbation Equations
Where the suffix zero refer to the values at the reference level z = 0. Here  
II. Formulation Of The Problem And Perturbation Equations
is acceleration due to gravity and  is the coefficient of thermal expansion. In writing the equation (1), we made use of the Boussinesq approximation, which states that the density variations are ignored in all terms in the equation of motion except the external force term. The magnetic permeability e  , thermal diffusivity , and electrical resistivity , are all assumed to be constant. The initial state is one in which the velocity, density, pressure, and temperature at any point in the fluid are, respectively, given by
II. Formulation Of The Problem And Perturbation Equations
Assume small perturbations around the basic solution and let , p 
II. Formulation Of The Problem And Perturbation Equations
Then the linearized perturbation equations are
II. Formulation Of The Problem And Perturbation Equations
Within the framework of Boussinesq approximation, equations (9) - (13), become 
III. Normal Mode Analysis
Analyzing the disturbances into normal modes, we assume that the Perturbation quantities are of the form
III. Normal Mode Analysis
Where y x k k , are the wave numbers along the x-and y-directions, respectively,
III. Normal Mode Analysis
, is the resultant wave number, and n is the growth rate which is, in general, a complex constant. Using (17) , equations (14) - (16), in non-dimensional form transform to
III. Normal Mode Analysis
And
III. Normal Mode Analysis
Where we have introduced new coordinates 
III. Normal Mode Analysis
Further, for 
IV. Mathematical Analysis
We prove the following lemma: 
IV. Mathematical Analysis
which gives that
IV. Mathematical Analysis
inequality (28) 
IV. Mathematical Analysis
This completes the proof of lemma. We prove the following theorems: (18) - (20) and the boundary conditions (21), (24) and any combination of (22) and (23) 
IV. Mathematical Analysis
Taking complex conjugate on both sides of equation (19), we get
IV. Mathematical Analysis
Therefore, using (32), we get
IV. Mathematical Analysis
Also taking complex conjugate on both sides of equation (20), we get
IV. Mathematical Analysis
Therefore, using (34) and using boundary condition (21), we get
IV. Mathematical Analysis
Substituting (33) and (35) in the right hand side of equation (31), we get
IV. Mathematical Analysis
Integrating the terms on both sides of equation (36) for an appropriate number of times by making use of the appropriate boundary conditions (21) - (24), we get
IV. Mathematical Analysis
And equating the imaginary parts on both side of Eq. (37), and cancelling 
IV. Mathematical Analysis
and therefore , we must have In the context of existence of instability in 'oscillatory modes' and that of 'overstability in the present configuration, we can state the above theorem as follow:-Theorem 3: The necessary condition for the existence of instability in 'oscillatory modes' and that of 'overstability' in a Rivlin-Ericksen viscoelastic fluid heated from below, in the presence of uniform vertical magnetic field, is that the Chandrasekhar number Q , the viscoelasticity parameter F and the magnetic Prandtl 

{'paper_id': '8254012', 'title': 'A Characterization Theorem in Magneto-Thermal Convection in Rivlin-Ericksen Viscoelastic Fluid', 's2_url': 'https://api.semanticscholar.org/CorpusID:8254012', 'source': 'S2ORC'}