A variational treatment of hydrodynamic and magnetohydrodynamic flows

In order to describe stationary plasma flows in thrusters based on plasma propulsion, an ideal, axial symmetric, single-fluid motion is assumed. The conservation laws of conductive fluids and the Maxwell’s equations lead to a second order differential equation for the magnetic flux function ψ, i.e. the generalized Grad Shafranov (GS) equation, and to two implicit constraints relating ψ to the plasma density and the azimuthal velocity. This set of three equations, one differential and two algebraic, is then expressed using a variational approach and the solution is obtained in a straightforward manner from the extremum of the appropriate Lagrangian functional. The numerical approach is based on Ritz’s method, which has the advantage of producing analytic (though approximate) solutions. Both non-conductive fluids, where the acceleration can only be obtained exploiting the internal energy of the flow (thermodynamic process), and conductive fluids, where the electromagnetic forces play a fundamental role, are considered. In order to apply this approach to the acceleration processes in nozzle-like configurations, an open-boundary geometry is investigated and specific attention is paid to a physical definition of boundary conditions. Hydrodynamic shocks are taken into account and it is shown that the appropriate jump conditions follow implicitly from a natural extension of the Lagrangian variational principle. A comparison test with an explicit solution permits an estimate of the approximate results.