A class of approximation.s {SN,M} to a periodic function f which uses the ideas of Pad6, or rational function, approximations based on the Fourier series representation of f, rather than on the Taylor series representation of f, is introduced and studied. Each approximation SNM is the quotient of a trigonometric polynomial of degree N and a trigonometric polynomial of degree M. The coefficients in these polynomials are determined by requiring that an appropriate number of the Fourier coefficients of SN,M agree with those of f. Explicit expressions are derived for these coefficients in terms of the Fourier coefficients of f. It is proven that these "Fourier-Pad6" approximations converge point-wise to (f(x+) + f(x-))/2 more rapidly (in some cases by a factor of 1/k2M) than the Fourier series partial sums on which they are based.
In the free surface flow of dense suspensions in confined domains, spontaneous particle enrichment occurs in the region adjacent to the free surface with an attendant increase in the effective viscosity. We present experimental evidence that this particle accretion introduces stability considerations that would otherwise not exist if the suspension remained uniform. For conditions favoring global stability, a local fingering instability can occur in the accretion band. For conditions favoring global instability, accretion can suppress the global growth of viscous fingers at the interface.
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