Stability Analysis on Compressible Walters B′ Viscoelastic Fluid

The present paper considered the thermosolutal convection in a compressible Walters B' viscoelastic fluid layer heated and soluted from below in the presence of uniform rotation. Following the linearized stability theory and normal mode analysis, the dispersion relation is obtained. For the case of stationary convection, Walters B′ viscoelastic fluid behaves like a Newtonian fluid and compressibility, rotation and stable solute gradient have stabilizing effect on the system for G >1. Graphs have been plotted by giving numerical values to the parameters to depict the stability characteristics. The stable solute gradient, rotation and viscoelasticity introduce oscillatory modes in the system which were non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained.


Introduction
The theoretical and experimental results of the onset of thermal convection (Be'nard convection) in a fluid layer under varying assumptions of hydrodynamics has been treated in detail by Chandrasekhar [1] in his celebrated monograph. Veronis' [2] has investigated the problem of thermohaline convection in a layer of flu id heated fro m below and subjected to a stable salinity gradient. The buoyancy forces can arise not only fro m density differences due to variations in temperature but also fro m those due to variations in solute concentration. Double-diffusive convecti on problems arise in oceanography, limnology and engineering. Examples of particu lar interest are provided by ponds built to trap solar heat (Tabor and Mat z [3]) and so me Antarctic lakes (Shirtcliffe [4]).The physics is quite similar in the stellar case in that Heliu m acts like salt in raising the density and in diffusing more slowly than heat. The conditions under which convective motions are impo rtant in stellar at mospheres are usually far removed fro m considerat ion of single component fluid and rigid boundaries and therefore, it is desirable to consider a fluid acted on by solute gradient and free boundaries. The flu ids have been considered to be Newtonian in all the above studies.
With the growing importance of non-Newtonian fluids in modern technology and industries, investigations on such flu ids are desirable. Widely used theoretical models (models A and B, respectively) for certain classes of viscoelasticfluids have been proposed by Old royd [5]. Renardy [6] has studied the stability of the interface in a two-layer Couette flo w of upper convected Maxwell liquids. The thermal instability of Maxwellian viscoelastic fluid in the presence of a uniform rotation has been considered by Bhatia and Steiner [7], where rotation is found to have a destabilizing effect. This is in contrast to the thermal instability of a Newtonian flu id where rotation has a stabilizing effect. The thermal instability of an Oldroydian viscoelastic fluid acted on by a uniform rotation has been studied by Sharma [8]. An experimental demonstration by Toms and Strawbridge [9] revealed that a dilute solution of methyl methacrylate in n-butyl acetate agrees with the theoretical model of Oldroyd [5]. There are many v iscoelastic flu ids that cannot be characterized by constitutive relations of the Maxwell/Oldroyd type. One such class of viscoelastic flu ids is Walters B' v iscoelastic fluid [10], having relevance and importance in geophysical fluid dynamics, chemical technology and petroleum industry. Walters [11] reported that the mixtu re of poly methyl methacrylate and pyridine at 250C containing 30.5g of poly mer per litre with a density of 0.98 g/ l behaves very nearly as the Walters B' elastico-viscous fluid. Ku mar [12] has studied the effect of rotation on thermal instability in Walters B' elastico-viscous flu id. In another study, Ku mar and Lal [13] have studied the effect of magnetic field and rotation on thermal convection in Walters B' elastico-viscous fluid. Sunil et al. [14] have studied the thermosolutal convection in Walters B' flu id in porous mediu m in p resence of magnetic field.
Brakke [15] exp lained a double-diffusive instability that occurs when a solution of a slowly diffusing protein is layered over a denser solution of more rapidly d iffusing sucrose. Nason et al. [16] found that this instability, which is deleterious to certain biochemical separations, can be suppressed by rotation in the ult racentrifuge. The effect of rotation on double-diffusive convection in compressible Walters B' flu id is important in certain chemical engineering and biochemical situations.
Keeping in mind the importance and applications in chemical engineering and bio mechanics, the effect of unifor m rotation on thermosolutal convection in compressible Walters B' viscoelastic fluid has been considered in the present paper. The analysis of the present work begins with Section 2, which fo rmulates the problem for Walters B' viscoelastic compressible flu id in the presence of uniform rotation by using the Boussinesq approximat ion, linearized theory and the perturbation theory. In Section 3, a dispersion relation is obtained by using the normal mode technique. The effects of stable solute gradient and rotation for the case of stationary convection are discussed analytically and graphically in Section 4. In Sect ion 5, the existence of oscillatory modes is discussed. Sufficient conditions for non-existence of overstability are obtained in Section 6. Section 7, the conclusion section, summarizes the results obtained in the preceding sections.

Formulation of the Problem and Perturbation Equations
Consider an infin ite compressible layer of Walters B' viscoelastic fluid, confined between the planes z = 0 and z = d, acted on by a gravity force   The equations governing the system become qu ite complicated when the flu ids are co mpressible. To simp lify them, Boussinesq try to justify the appro ximat ion for nearly incompressible flu ids when the density variations arise principally fro m thermal effects by noting that atmospheric pressure fluctuations are much too small to produce the observed density changes. Spiegel and Veronis' [17] have simp lified the set of equations governing the flow of compressible fluids under the follo wing assumptions; (1) The vertical d imension of the flu id is much less than any scale height, as defined by them, if only motions of infinitesimal amp litudes are considered and (2) The motioninduced fluctuations in density and pressure do not exceed in order of their total static variations.
Under the above approximat ions, Spiegel and Veronis' [17] have shown that the equations governing convection in a perfect gas are formally equivalent to those for an incompressible fluid if the static temperature gradient is replaced by its excess over the adiabatic and v c is replaced where suffix zero refers to values at the reference level z = 0, α is the coefficient of thermal expansion and α′ is the analogous solvent expansion.
The basic motionless solution is The linearized perturbation equations are The change in density  caused by the perturbations  and  in temperature and solute concentration is given by  .

The Dispersion Relation
Analyzing the d isturbances into normal modes, we assume that perturbation quantities are of the form Using (17), equations (12)-(15) in non-dimensional form become  x ip x iq

The Stationary Convection
For the stationary convection 0   , equation (24) reduces to   where cc R and R denote, respectively, the critical Rayleigh nu mbers in the presence and absence of compressibility. Since crit ical nu mber is positive and finite, so G > 1 and we obtain a stabilizing effect of co mpressibility as its result is to postpone the onset of double-diffusive convection. The cases G < 1 and G =1 correspond to negative and infin ite values of critical Ray leigh nu mbers in the presence of compressibility, that are not relevant in the present study.
To investigate the effect of stable solute gradient and rotation, we examine the behaviour of It is evident fro m equation (25) that The stable solute gradient and rotation, therefore, have stabilizing effects on the system for G >1.
We now give some realistic values to various parameters in equation (25) to demonstrate the above results through graphs. Figure 1 Figure 1 that the stable solute gradient has stabilizing effect on the system. Figure 2

Stability of the System and Oscillatory Modes
To examine the possibility of oscillatory modes if any due to the presence of kinemat ic viscoelasticity, stable solute gradient and rotation, we mult iply ing equation (18)

The Case of Overstablity
Here we d iscuss the possibility of whether instability may occur as an overstability. Since we wish to determine the critical Rayleigh number for overstability, it suffices to find conditions for which equation (24) will ad mit of solution with 1  real. Separating real and imaginary parts of equation, we have by eliminating 1

Conclusions
The thermosolutal convection in a co mpressible Walters B' viscoelastic fluid layer heated and soluted from below in the presence of uniform rotation is considered in the present paper. The investigation of thermosolutal convection is motivated by its interesting complexit ies as a double diffusion phenomena as well as its direct relevance to geophysics and astrophysics. The main conclusions from the analysis of this paper are as follows:  For the case of stationary convection, the compressible Walters B' viscoelastic fluid behaves like an ord inary compressible Newtonian fluid.
 It is observed for the case of stationary convection that the stable solute gradient and rotation have stabilizing effects on the system for G>1.
 It is also observed graphically fro m Figures 1 and 2 that the stable solute gradient and rotation postpone the onset of convection.
 It is observed that the presence of rotation and solute gradient introduce oscillatory modes in the system, which were non-existent in their absence.
 The conditions 1