Plane shearing waves of arbitrary form: exact solutions of the Navier-Stokes equations

TL;DR

This paper derives exact solutions for incompressible Navier-Stokes equations describing plane shearing waves in a linear shear flow, expanding understanding of wave behavior in fluid dynamics.

Contribution

It introduces a method to construct explicit Kelvin mode solutions and superpositions for general plane shearing waves in shear flows.

Findings

Explicit formulas for Kelvin modes in shear flow.

Superposition method for general shearing waves.

Solutions applicable to various boundary conditions.

Abstract

We present exact solutions of the incompressible Navier-Stokes equations in a background linear shear flow. The method of construction is based on Kelvin's investigations into linearized disturbances in an unbounded Couette flow. We obtain explicit formulae for all three components of a Kelvin mode in terms of elementary functions. We then prove that Kelvin modes with parallel (though time-dependent) wave vectors can be superposed to construct the most general plane transverse shearing wave. An explicit solution is given, with any specified initial orientation, profile and polarization structure, with either unbounded or shear-periodic boundary conditions.