On the relation between viscoelastic and magnetohydrodynamic flows and their instabilities

We demonstrate a close analogy between a viscoelastic medium and an electrically conducting fluid containing a magnetic field. Specifically, the dynamics of the Oldroyd-B fluid in the limit of large Deborah number corresponds to that of a magnetohydrodynamic (MHD) fluid in the limit of large magnetic Reynolds number. As a definite example of this analogy, we compare the stability properties of differentially rotating viscoelastic and MHD flows. We show that there is an instability of the Oldroyd-B fluid that is physically distinct from both the inertial and elastic instabilities described previously in the literature, but is directly equivalent to the magnetorotational instability in MHD. It occurs even when the specific angular momentum increases outwards, provided that the angular velocity decreases outwards; it derives from the kinetic energy of the shear flow and does not depend on the curvature of the streamlines. However, we argue that the elastic instability of viscoelastic Couette flow has no direct equivalent in MHD.Comment: 21 pages, 3 figures, to be published in J. Fluid Mec

Singular diffusionless limits of double-diffusive instabilities in magnetohydrodynamics

We study local instabilities of a differentially rotating viscous flow of electrically conducting incompressible fluid subject to an external azimuthal magnetic field. In the presence of the magnetic field the hydrodynamically stable flow can demonstrate non - axisymmetric azimuthal magnetorotational instability (AMRI) both in the diffusionless case and in the double-diffusive case with viscous and ohmic dissipation. Performing stability analysis of amplitude transport equations of short-wavelength approximation, we find that the threshold of the diffusionless AMRI via the Hamilton-Hopf bifurcation is a singular limit of the thresholds of the viscous and resistive AMRI corresponding to the dissipative Hopf bifurcation and manifests itself as the Whitney umbrella singular point. A smooth transition between the two types of instabilities is possible only if the magnetic Prandtl number is equal to unity, $\rm Pm=1$. At a fixed $\rm Pm\ne 1$ the threshold of the double-diffusive AMRI is displaced by finite distance in the parameter space with respect to the diffusionless case even in the zero dissipation limit. The complete neutral stability surface contains three Whitney umbrella singular points and two mutually orthogonal intervals of self-intersection. At these singularities the double-diffusive system reduces to a marginally stable system which is either Hamiltonian or parity-time (PT) symmetric.Comment: 34 pages, 8 figures, typos corrected, refs adde

Absolute and convective instabilities of an inviscid compressible mixing layer: Theory and applications

This study aims to examine the effect of compressibility on unbounded and parallel shear flow linear instabilities. This analysis is of interest for industrial, geophysical, and astrophysical flows. We focus on the stability of a wavepacket as opposed to previous single-mode stability studies. We consider the notions of absolute and convective instabilities first used to describe plasma instabilities. The compressible-flow modal theory predicts instability whatever the Mach number. Spatial and temporal growth rates and Reynolds stresses nevertheless become strongly reduced at high Mach numbers. The evolution of disturbances is driven by Kelvin -Helmholtz instability that weakens in supersonic flows. We wish to examine the occurrence of absolute instability, necessary for the appearance of turbulent motions in an inviscid and compressible two-dimensional mixing layer at an arbitrary Mach number subject to a three-dimensional disturbance. The mixing layer is defined by a parametric family of mean-velocity and temperature profiles. The eigenvalue problem is solved with the help of a spectral method. We ascertain the effects of the distribution of temperature and velocity in the mixing layer on the transition between convective and absolute instabilities. It appears that, in most cases, absolute instability is always possible at high Mach numbers provided that the ratio of slow-stream temperature over fast-stream temperature may be less than a critical maximal value but the temporal growth rate present in the absolutely unstable zone remains small and tends to zero at high Mach numbers. The transition toward a supersonic turbulent regime is therefore unlikely to be possible in the linear theory. Absolute instability can be also present among low-Mach-number coflowing mixing layers provided that this same temperature ratio may be small, but nevertheless, higher than a critical minimal value. Temperature distribution within the mixing layer also has an effect on the growth rate, this diminishes when the slow stream is heated. These results are applied to the dynamics of mixing layers in the interstellar medium and to the dynamics of the heliopause, frontier between the interstellar medium, and the solar wind. (C) 2009 American Institute of Physics