An electroconvection pattern in a thin liquid-crystal cell is modelled

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

An electroconvection pattern in a thin liquid-crystal cell is modelled, in dimensionless units, by the Swift-Hohenberg equation

\partial_tu = ru - (1+\partial_x^2)^2u - u^3

with control parameter

r = 0.09

set by the applied voltage above its onset value. ((a)) Find the unstable band of wavenumbers and the critical wavenumber

q_{\rm c}

. ((b)) Find the saturated pattern amplitude

A^*

. ((c)) The cell has length

L = 200

in these dimensionless units and holds

N = L/\Lambda_{\rm c}

peaks, where

\Lambda_{\rm c} = 2\pi/q_{\rm c}

. In the experiment, noise makes the measured height of each peak scatter independently from peak to peak, with a standard deviation of

20\%

of the mean height. Using the block-averaging law, estimate the relative uncertainty in the mean peak height measured over the whole cell, and find the minimum cell length needed to bring this relative uncertainty down to

2\%

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