An intrinsic study on a certain isoenergetic flow of a compressible fluid, with extension to some cases in magneto-plasma dynamics

Abstract

This work studies and clarifies some local phenomena in fluid mechanics as well as in magneto-fluid dynamics. A model of a certain isoenergetic flow of an inviscid fluid is introduced, in order to establish a simpler form for the general PDE of the velocity potential. It consists mainly in using a new system of three orthogonal curvilinear coordinates (one of them being tied to the local specific entropy value). The choice of this system (with two coordinate curves lying on the “isentropic” surfaces) enables the treatment of any 3-D flow (rotational, steady and unsteady) as a potential 2-D one, introducing a 2-D velocity “quasi-potential”, specific to any isentropic surface. The dependence of the specific entropy on this velocity “quasi-potential” was also established. On the above surfaces the streamlines are orthogonal paths of a family of lines of equal velocity “quasi-potential”. The model can be extended to some special (but usual) cases in magneto-plasma dynamics (taking into account the flow vorticity effects, as well as those of the Joule–Lenz heat losses), considering a non-isentropic flow of a barotropic inviscid electroconducting fluid in an external magnetic field. There are always some space curves along which the equation of motion admits a first integral, making evident a new physical quantity (a true magneto-hydrodynamic one)—Selescu’s $ vector. For a fluid having an infinite electric conductivity, these curves are the flow isentropic lines, also enabling the treatment of any 3-D flow as a “quasi-potential” 2-D one. The case of the unsteady flow (and electric field and charge, and magnetic field as well) of an inviscid electroconducting liquid (incompressible fluid) was studied, being of general interest (the MHD generator with liquid), and giving an exact first integral for the motion equation. The model was extended to the viscous fluid flows. In all cases treated the newly found first integrals are similar to the D. Bernoulli and D. Bernoulli–Lagrange ones. The PDE of the isentropic surfaces and those of Selescu’s (roto-viscous and MHD) vector lines and zero-work surfaces are also given.