Explicit superposed and forced plane wave generalized Beltrami flows

Abstract

Abstract We revisit and present new linear spaces of explicit solutions to incompressible Euler and Navier–Stokes equations on ${{{\mathbb{R}}}}^n$, as well as the rotating Boussinesq equations on ${{{\mathbb{R}}}}^3$. We cast these solutions are superpositions of certain linear plane waves of arbitrary amplitudes that also solve the nonlinear equations by constraints on wave vectors and flow directions. For $n\leqslant 3$, these are explicit examples for generalized Beltrami flows. We show that forcing terms of corresponding plane wave type yield explicit solutions by linear variation of constants. We work in Eulerian coordinates and distinguish the two situations of vanishing and of gradient nonlinear terms, where the nonlinear terms modify the pressure. The methods that we introduce to find explicit solutions in nonlinear fluid models can also be used in other equations with material derivative. Our approach offers another view on known explicit solutions of different fluid models from a plane wave perspective and provides transparent nonlinear interactions between different flow components.