The stability of a two-dimensional viscous flow between two rotating porous cylinders is studied. The basic steady flow is the most general rotationally-invariant solution of the Navier-Stokes equations in which the velocity has both radial and azimuthal components, and the azimuthal velocity profile depends on the Reynolds number. It is shown that for a wide range of the parameters of the problem, the basic flow is unstable to small twodimensional perturbations. Neutral curves in the space of parameters of the problem are computed. Calculations show that the stability properties of this flow are determined by the azimuthal velocity at the inner cylinder when the direction of the radial flow is from the inner cylinder to the outer one and by the azimuthal velocity at the outer cylinder when the direction of the radial flow is reversed. This work is a continuation of our previous study of an inviscid instability in flows between rotating porous cylinders (see Ilin & Morgulis (2013)).

The techniques developed in Part 1 of the present series are here applied to two-dimensional solutions of the equations governing the magnetohydrodynamics of ideal incompressible fluids. We first demonstrate an isomorphism between such flows and the flow of a stratified fluid subjected to a field of force that we describe as ‘pseudo-gravitational’. We then construct a general Casimir as an integral of an arbitrary function of two conserved fields, namely the vector potential of the magnetic field, and the analogous potential of the ‘modified vorticity field’, the additional frozen field introduced in Part 1. Using this Casimir, a linear stability criterion is obtained by standard techniques. In §4, the (Arnold) techniques of nonlinear stability are developed, and bounds are placed on the second variation of the sum of the energy and the Casimir of the problem. This leads to criteria for nonlinear (Lyapunov) stability of the MHD flows considered. The appropriate norm is a sum of the magnetic and kinetic energies and the mean-square vector potential of the magnetic field.

The stability of steady magnetohydrodynamic flows of an ideal incompressible fluid to small three-dimensional perturbations is studied. Two new conditions sufficient for linear stability of steady magnetohydrodynamic flows are obtained by the energy method.

The general theory developed in Part I of the present series is here applied to axisymmetric solutions of the equations governing the magnetohydrodynamics of ideal incompressible fluids. We first show a helpful analogy between axisymmetric MHD flows and flows of a stratified fluid in the Boussinesq approximation. We then construct a general Casimir as an integral of an arbitrary function of two conserved fields, namely the vector potential of the magnetic field and the scalar field associated with the ‘modified vorticity field’, the additional frozen-in field introduced in Part I. Using this Casimir, sufficient conditions for linear stability to axisymmetric perturbations are obtained by standard Arnold techniques. We exploit Arnold's method to obtain sufficient conditions for nonlinear (Lyapunov) stability of the MHD flows considered. The appropriate norm is a sum of the magnetic and kinetic energies and the mean square vector potential of the magnetic field.