On Howard's conjecture in heterogeneous shear flow problem

TL;DR

This paper rigorously proves Howard's conjecture for certain heterogeneous shear flows, showing that the growth rate of unstable waves diminishes as wavelength decreases, under specific conditions of negligible buoyancy.

Contribution

It provides a rigorous mathematical proof of Howard's conjecture for plane parallel heterogeneous shear flows with negligible buoyancy force.

Findings

Growth rate approaches zero as wavelength decreases.

Howard's conjecture holds for flows with negligible buoyancy.

Mathematically establishes the conjecture in a specific flow regime.

Abstract

Howard's conjecture, which states that in the linear instability problem of inviscid heterogeneous parallel shear flow growth rate of an arbitrary unstable wave must approach zero as the wave length decreases to zero, is established in a mathematically rigorous fashion for plane parallel heterogeneous shear flows with negligible buoyancy force $g\beta \ll 1$ (Miles J W, {\it J. Fluid Mech.} {\bf 10} (1961) 496--508), where $\beta$ is the basic heterogeneity distribution function).