On the interaction between Tollmien-Schlichting and Rayleigh-Bénard instabilities in the presence of a longitudinal magnetic field

Abstract

The method used by Gage and Reid(10) to investigate hydrodynamic stability of thermally stratified fluid is extended to hydromagnetic stability to study the effect of aligned magnetic field on the stability of unstable thermal stratification under the assumption of small magnetic Reynolds number. The interaction between the Tollmien-Schlichting-Stuart mechanism of instability due to shear and magnetic field and Rayleigh-Bénard-Thompson mechanism of instability due to thermally unstable stratification and magnetic field is brought out in detail. It is shown that, although Squire's transformation can be used to reduce the three-dimensional problem to an equivalent two-dimensional one, Squire's theorem is not valid. This conclusion follows from the fact that in our analysis the Richardson number R_i ( < 0) will not be greater than the value -0.92 × 10^-6. In particular, it is shown that for the values of stratification parameter n ≤ 0.6 the effect of magnetic field for small values of Stuart number N is to augment instability and impose the restriction on the validity of our numerical procedure. However, for η = 0.8 a sharp transition from unstable to stable flow takes place at N = 0.3. A physical explanation for this based on eddies is given.