Some Exact Solutions of the Navier-Stokes and the Hydromagnetic Equations

Abstract

Some exact, closed-form solutions of the Navier-Stokes equations for incompressible flow and of the hydromagnetic equations for high-conductivity, incompressible flow are presented. They can be considered to be generalizations of Taylor's solutions. The solutions are two dimensional and cellular containing a single-space Fourier component; the spatial behavior is chosen in such a way that the nonlinear inertial term and the pressure term cancel one another, leaving a linear system to be solved. The time behavior of the solutions is quite general. The solutions to the hydromagnetic equations are such that the velocity and the magnetic fields are parallel and decoupled. The velocity behaves as it does in the purely mechanical case while the magnetic field simply decays in time; there is no source term for it in the present treatment.