Cellular patterns with boundary forcing

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Abstract
In this paper we investigate some effects of a boundary forcing on 2-dimensional cellular patterns in instabilities above threshold. Boundary forcing is modelled as an inhomogeneous boundary condition on the slowly varying amplitude A, i.e. on boundaries. The relevant range is λ = O(ε½), where ε is the relative distance to the linear-instability threshold. A wavenumber-selection mechanism then occurs, resulting in a band of selected wavenumbers of width proportional to λ. For large values of λε−½ it is shown that no stationary solution exists outside the band of Eckhaus-stable wavenumbers (Eckhaus 1965). For finite geometries of size L, a nonlinear analogue of ‘quantization’ of modes is investigated. The amplitude equation (equivalent to a space-dependent Ginzburg–Landau model) is analysed by an expansion in powers of exp(-L/ζ), where ζ is the coherence length. The range λ = O(ε) is also investigated. A correction to previous theories of wavenumber selection through boundaries (Cross, Daniels, Hohenberg & Siggia 1983a; Pomeau & Zaleski 1981) is calculated. The latter results are general and assume only the existence of a higher-order stationary amplitude equation, which is recast in a form consistent with its boundary conditions.

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