MAGIC Launch Lecture 2024

Title - Instabilities in spatially-periodic, and in slightly random, media

Speaker - Professor Jonathan Healey (Keele University)

We start with a brief introduction to fluid mechanics for the uninitiated, leading into the subject of 'hydrodynamic stability theory' (HST), which can describe how waves added to a steady fluid flow travel and grow, possibly leading to turbulence. The appearance of turbulence has countless practical applications (e.g. in physiological, atmospheric, oceanographic, astrophysical, aeronautical flows), and is often called the outstanding problem of classical mechanics. It has attracted the attention of an impressive list of mathematicians and scientists, including Leonardo da Vinci, who carried out experiments, making beautiful, detailed, sketches of his observations, Newton, who explained how friction can act in fluids (viscosity), Euler, who derived the equations of motion of inviscid (frictionless) fluid flow. The Navier-Stokes equations for viscous flow were finally derived mid-19th century and one of the millenium prize problems in mathematics is concerned with the Navier-Stokes equations.

The first HST was derived by Kelvin in 1871 for a flow proposed by Helmholtz in 1868 for two parallel streams moving at different speeds, resulting in a 'mixing layer' between the two streams (google 'billow clouds' for dramatic images and videos). While modern studies of unstable flows are often based on direct numerical simulations of the Navier-Stokes equations, many results and insights continue to be obtained using theoretical approaches, complementing and informing numerical and physical experiments. In this talk we review a line of development of the Kelvin-Helmholtz instability theory focused particularly on the interplay between the propagation and instability properties of waves in mixing layers, and the unexpectedly strong effects that can be produced when the basic flow varies by a small amount in the direction of the flow. The effect of spatially-periodic variation at the boundary of a flow is considered first (described by Floquet theory), which leads, surprisingly, to solutions for small random spatial variations at a boundary (described by stochastic differential equations) that differ greatly from the usual mathematically convenient, but physically unrealistic, model of perfectly flat boundaries. This work is in collaboration with Dr Matt Turner from the University of Surrey, another member of the wonderful MAGIC consortium!