Hydromagnetic Stability of Two Rivlin-Ericksen Elastico-Viscous Superposed Conducting Fluids R. C. Sharma and P. Kumar Department of Mathematics, Himachal Pradesh University, Summer Hill, Shimla-171 005, India Z. Naturforsch. 52a, 528-532 (1997); received April 18, 1996 The stability of the plane interface separating two Rivlin-Ericksen elastico-viscous superposed fluids of uniform densities when the whole system is immersed in a uniform horizontal magnetic field has been studied. The stability analysis has been carried out, for mathematical simplicity, for two highly viscous fluids of equal kinematic viscosities and equal kinematic viscoelasticities. It is found that the stability criterion is independent of the effects of viscosity and viscoelasticity and is dependent on the orientation and magnitude of the magnetic field. The magnetic field is found to stabilize a certain wave-number range of the unstable configuration. The behaviour of growth rates with respect to kinematic viscosity and kinematic viscoelasticity parameters are examined numeri- cally. 1. Introduction The instability of the plane interface separating two Newtonian fluids when one is accelerated towards the other or when one is superposed over the other has been studied by several authors, and Chandrasekhar [1] has given a detailed account of these investigations. Roberts [2] has extended the analysis to the case of two fluids of equal kinematic viscosities in presence of a vertical magnetic field, while Gerwin [3] has studied the case of compressible streaming fluids. The influ- ence of viscosity on the stability of the plane interface separating two incompressible superposed fluids of uniform densities, when the whole system is immersed in a uniform horizontal magnetic field has been stud- ied by Bhatia [4]. He has carried out the stability analysis for two fluids of equal kinematic viscosities and different uniform densities. A good account of hydrodynamic stability problems has also been given by Drazin and Reid [5] and Joseph [6]. The fluids have been considered to be Newtonian in all the above studies. The stability of a layer of vis- coelastic (Oldroyd) fluid heated from below and sub- ject to a magnetic field has been studied by Sharma [7], In another study Sharma and Sharma [8] have studied the stability of the plane interface separating two vis- coelastic (Oldroyd) superposed fluids of uniform den- sities. Fredricksen [9] has given a good review of non- Newtonian fluids whereas Joseph [6] has also considered the stability of viscoelastic fluids. Molten Reprint requests to Prof. R. C. Sharma. plastics, petroleum oil additives and whipped cream are examples of incompressible viscoelastic fluids. There are many non-Newtonian fluids that cannot be characterized by Oldroyd's [10] constitutive relations. The Rivlin-Ericksen elastico-viscous fluid is one such fluid. It is this class of elastico-viscous fluids we are interested in particularly to study the stability of the plane interface separating two incrompressible super- posed Rivlin-Ericksen fluids of uniform densities per- vaded by a uniform horizontal magnetic field in addi- tion to a constant gravity field. This aspect forms the subject of the present paper where we have carried out the stability analysis for two fluids of equal kinematic viscosities, equal kinematic viscoelasticities and differ- ent densities. 2. Perturbation Equations Consider a static state, in which an incompressible, infinitely conducting, Rivlin-Ericksen elastico-viscous fluid of variable density pervaded by a uniform mag- netic field H(H X ,H Y , 0) is arranged in horizontal strata and the pressure p and density q are functions of the vertical coordinate z only. The character of the equilibrium of this initial static state is determined, as usual, by supposing that the system is slightly dis- turbed and then by following its further evolution. Let q(u, v, w), öq, öp and h(h x ,h y ,h : ) denote the perturbations in velocity (0, 0, 0), density g, pressure p and magnetic field H, respectively. Then the linearized hydromagnetic perturbation equations relevant to the 0932-0784 / 97 / 0600-538 $ 06.00 © - Verlag der Zeitschrift für Naturforschung, D-72027 Tübingen