A gel-phase reaction-diffusion system has an activator with diffusion coefficient and…

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 gel-phase reaction-diffusion system has an activator with diffusion coefficient Du=5×10−10 m2 s−1D_u = 5\times10^{-10}\,\mathrm{m^2\,s^{-1}} and an inhibitor with Dv=5×10−9 m2 s−1D_v = 5\times10^{-9}\,\mathrm{m^2\,s^{-1}}. The kinetics turn over on a characteristic reaction time τ=10 s\tau = 10\,\mathrm{s}, so that the size of the Jacobian's determinant can be estimated dimensionally as det⁡J∼τ−2\det\mathbf{J} \sim \tau^{-2}. ((a)) Use the general Turing critical wavenumber, qc2=det⁡J/(DuDv)q_{\rm c}^2 = \sqrt{\det\mathbf{J}/(D_uD_v)}, together with this estimate of det⁡J\det\mathbf{J}, to estimate qcq_{\rm c}. ((b)) Hence estimate the pattern wavelength Λc=2π/qc\Lambda_{\rm c} = 2\pi/q_{\rm c}, in millimetres.

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