Global stability for thermal convection in a partially-ionized plasma

Global stability for thermal convection in a partially-ionized plasma

Vishal Chandel¹, Sunil¹, Poonam Sharma², Reeta Devi³

¹Department of Mathematics and Scientific Computing, National Institute of Technology,

Hamirpur-177005, Himachal Pradesh, India

²Department of Mathematics, NSCBM Govt. College Hamirpur-177005, Himachal Pradesh, India

³Department of Mathematics, Govt. College Nagrota Bagwan (Kangra)-176047, Himachal Pradesh, India

ABSTRACT:

A nonlinear (energy) stability analysis is performed for a partially-ionized plasma layer heated from below, in three different bounding cases (both free, both rigid, one free and one rigid). A nonlinear stability threshold in a partially-ionized plasma is exactly the same as the linear instability boundary. It is found that the nonlinear critical stability Rayleigh number coincide with the linear instability critical Rayleigh number and this indicates the existence of global stability. So, no sub-critical instabilities are possible. The optimal result is important because it shows that linearized instability theory has captured completely the physics of the onset of thermal convection. The collisional effect plays an important role in decaying of energy whereas it doesn’t affect the value of the Rayleigh number.

1. INTRODUCTION:

In many areas of fluid dynamics, the theory of thermal instability (or Bénard convection) is considered as a basic problem of uttermost importance. In 1900, Bénard demonstrates the onset of thermal instability in fluid¹. In most of the circumstances, when a layer of fluid is heated from below, the fluid in the lower part of the layer expands due to increase in its temperature and hence become lighter as compared to the fluid in the upper layer. This arrangement is potentially unstable and when the temperature gradient or layer depth is adequately large to conquer the effect of gravity, the fluid rises and a cellular pattern may be seen.

In this work, we are dealing with a layer of partially-ionized plasma. Partially-ionized plasma represents a state which often exists in the universe. Partially-ionized plasma are essential constituents of many astrophysical environments, including the solar atmosphere, the interstellar medium, molecular clouds, cometary tails, etc., where the ionization degree may vary from weak ionization to almost full ionization². In cosmic physics, there are many examples where interaction between neutral and ionized (hydromagnetic) gas components are important. Alfvén’s³ theory on the origin of the planetary system gives the example of existence of such situations, where a high ionization rate is suggested to appear from collision between a plasma and a neutral gas cloud and by the absorption of plasma waves due to ion-neutral collisions such as in the solar photosphere and chromosphere and in cool interstellar clouds^4,5. The dynamics of partially-ionized plasma is heavily affected by the interactions between the various charged and neutral species that compose the plasma. It has been shown that partial ionization effects influence the triggering and development of fluid instabilities as, e.g., Kelvin-Helmholtz, Rayleigh-Taylor, thermal and thermosolutal instabilities, among others.

Hans⁶ and Bhatia⁷ have shown that the collision has a stabilizing effect on the Rayleigh-Taylor instability. For Kelvin-Helmholtz configuration, Rao and Kalra⁸ and Hans⁶ have found that the collisions have destabilizing effect for a sufficiently large collision frequency. Sharma⁹ has studied the thermal linear instability of a partially-ionized plasma in free-free boundaries, whereas the thermosolutal instability of a partially-ionized plasma in porous medium has been considered in the presence of a uniform vertical magnetic field to include the effect of collisions and Hall currents by Sharma and Sunil¹⁰. Nonlinear analysis and linear analysis with different boundary conditions have not been explored in any of the aforementioned analyses.

The current work deals with the linear as well as nonlinear analysis of partially-ionized plasma in free-free, free-rigid and rigid-rigid boundaries. Linear instability is studied by normal mode analysis [Chandrasekhar¹] whereas energy method have been used for nonlinear stability analysis [Straughan¹¹]. Sunil and Mahajan¹² have also used energy method in their work of “A nonlinear stability analysis for magnetized ferrofluid heated from below”. For numerical analysis, the Galerkin method is employed. After comparing the results of both studies, we established the optimal result that, the Rayleigh number is same for both linear instability and nonlinear stability theories. Also, we have studied the comparison between all three boundaries. As far as we know, no attempts have been made to perform the aforementioned analysis.

2. MATHEMATICAL FORMULATION: