Exact solutions for symmetric magnetohydrodynamic equilibria with mass flow

In this paper the problem of ideal magnetohydrodynamic equilibria with mass flow is treated. Under the assumption of general symmetry (i.e. one ignorable spatial coordinate) we derive a generalized Grad-Shafranov equation in an unspecified curvilinear coordinate system. If incompressibility is assumed an elliptical equation is derived and a new, totally analytical method of solution is proposed. This is based on a particular selfsimilar separation of the variables in the unknown flux function and leads to an ordinary, non-linear differential equation for the profile of the magnetic and flow surfaces. Three novel classes of solutions are derived in different geometries, all being flexible (they contain a minimum of three free functions) and regular, which makes them suitable for astrophysical applications. These are flows in magnetic flux tubes with non-circular section, flows in magnetic arcades above the solar surface and collimated, axisymmetric outflows.