Helical flows of an inviscid inhomogeneous incompressible fluid

The balance equation of helicity, defined in this case by the dot product of the momentum vector (the fluid density times the velocity vector) and its curl for an inviscid, incompressible, non-homogeneous fluid, is derived using the equation for the time evolution of the curl of the momentum vector, which, to our knowledge, has not been previously described in the literature. From this equation and using the degree of freedom associated with the density invariance in each fluid particle, a conservation law for the generalised helicity integral is derived, provided that the Ertel potential vorticity in the fluid flow is identically zero. Arnold's method is used to solve a variational problem that consists of finding the necessary conditions for the total energy extremum of the flow under the condition of constant generalised helicity and the existence of the Casimir invariant in the problem. These conditions are met for a helical Beltrami flow that generalises known solutions for helical Beltrami flow in a homogeneous, incompressible fluid. Examples of generalised helical Beltrami flows are considered for an incompressible fluid with non-uniform density in an external gravitational field, as well as in the presence of self-gravity within the fluid and, conversely, in a zero gravity situation. In particular, the possibility of using a solution obtained to construct an idealised fluid dynamic model of a massive self-gravitating cylindrical structure is explored. Also included is a comment on the extension of the results obtained to the case of ideal magnetohydrodynamics.