Evolution of Wavelike Disturbances in Shear Flows: A Class of Exact Solutions of the NavierStokes Equations
Abstract
New classes of exact solutions of the incompressible NavierStokes equations are presented. The method of solution has its origins in that first used by Kelvin (Phil. Mag. 24 (5), 188196 (1887)) to solve the linearized equations governing small disturbances in unbounded plane Couette flow. The new solutions found describe arbitrarily large, spatially periodic disturbances within certain two and threedimensional 'basic' shear flows of unbounded extent. The admissible classes of basic flow possess spatially uniform strain rates; they include two and threedimensional stagnation point flows and twodimensional flows with uniform vorticity. The disturbances, though spatially periodic, have timedependent wavenumber and velocity components. It is found that solutions for the disturbance do not always decay to zero; but in some instances grow continuously in spite of viscous dissipation. This behaviour is explained in terms of vorticity dynamics.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 July 1986
 DOI:
 10.1098/rspa.1986.0061
 Bibcode:
 1986RSPSA.406...13C