A researcher studying a stirred layer argues: "diffusion always smooths out…
PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems
Many Parts, One Behaviour, Ch Pattern formation: instabilities that choose a wavelength
Problem
A researcher studying a stirred layer argues: "diffusion always smooths out concentration differences, so the growth rate of a disturbance must fall as q rises; the fastest-growing mode is therefore always q=0, and this system cannot make a pattern." They apply this reasoning to a layer whose growth rate is σ(q)=0.05−(1−q2)2.
((a)) Evaluate the claim by finding σ(0) and the wavenumber that actually maximises σ(q).
((b)) Give the unstable band of wavenumbers and the resulting pattern wavelength.