Translationally invariant exact steady flows of gas and fluid

Abstract

Abstract New exact 3-D steady translationally invariant (or z-invariant) flows of isentropic gas and ideal incompressible fluid are constructed. New exact solutions for steady magnetohydrodynamics equations are derived. We show that Euler equations for steady z-invariant flows of isentropic gas are equivalent to a coupled system of a partial differential equation of second order for the streamfunction ψ(x, y) which contains an arbitrary differentiable function H(ψ) and a transcendental equation connecting gas density ρ(x, y) with function ψ(x, y) and depending on equation of state p(ρ) = cρ γ and on the function H(ψ). We prove that functions ψ(x, y) and ρ(x, y) satisfy a universal nonlinear equation that is independent of equation of state p = p(ρ) and of the function H(ψ).