A remark on two-dimensional stokes flow solutions

A transform method for determining the flow generated by the singularities of Stokes flow in a two-dimensional channel is presented. The analysis is based on a general approach to biharmonic boundary value problems in a simply connected polygon formulated by Crowdy & Fokas in this journal. The method differs from a traditional Fourier transform approach in entailing a simultaneous spectral analysis in the independent variables both along and across the channel. As an example application, we find the evolution equations for a circular treadmilling microswimmer in the channel correct to third order in the swimmer radius. Significantly, the new transform method is extendible to the analysis of Stokes flows in more complicated polygonal microchannel geometries.

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Integral method for a two-dimensional Stokes flow with shrinking holes applied to viscous sintering

Journal of Fluid Mechanics ◽

10.1017/s002211209300326x ◽

1993 ◽

Vol 257 (-1) ◽

pp. 667 ◽

Cited By ~ 37

Author(s):

G. A. L. van de Vorst

Keyword(s):

Stokes Flow ◽

Integral Method ◽

Two Dimensional ◽

Viscous Sintering

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Homogenisation of a class of fourth order equations with application to incompressible elasticity

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500014967 ◽

1992 ◽

Vol 120 (1-2) ◽

pp. 25-46 ◽

Cited By ~ 7

Author(s):

G. A. Francfort

Keyword(s):

Fourth Order ◽

Stokes Flow ◽

Dimensional Case ◽

Closure Problem ◽

Two Dimensional ◽

Elastic Materials ◽

Fourth Order Equations ◽

Incompressible Elasticity

SynopsisUpon formalising an analogy between two-dimensional Stokes flow and two-dimensional isotropic conductivity, we exhibit a class of fourth order equations which behave “isomorphically” like isotropic conductivity from the standpoint of homogenisation and from that of corresponding bounding methods on possible effective behaviours. In particular, Lipton's result on the G-closure problem for mixtures of two incompressible elastic materials is recovered in the two-dimensional case.

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Optimum profiles in two-dimensional Stokes flow

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1995.0103 ◽

1995 ◽

Vol 450 (1940) ◽

pp. 603-622 ◽

Cited By ~ 5

Keyword(s):

Circular Cylinder ◽

Cross Section ◽

Stokes Flow ◽

Unit Length ◽

Three Dimensional ◽

Variational Formula ◽

Effective Radius ◽

Two Dimensional ◽

Equivalent Radius ◽

Optimum Profile

We consider the problem of designing the section of a cylinder to minimize the drag per unit length it experiences when placed perpendicular to a uniform stream at low Reynolds number; we suppose the area of the cross-section to be given, and the flow to be two-dimensional. The relevant properties of a cylinder of general cross-section in a particular orientation can conveniently be expressed in terms of its equivalent radius; when the drag and flow at infinity are parallel, this equivalent radius is the radius of the circular cylinder giving rise to the same drag per unit length. We obtain a variational formula for this equivalent radius when the surface of the cylinder is perturbed; this shows that the optimum profile we seek must be such that the flow past it has a vorticity of constant magnitude at its surface, and this fact enables the optimum to be determined analytically. The efficacy of a particular section may be measured by its effective radius, this being the equivalent radius when the length scale is chosen to give the section an area π ; thus a circular cylinder has an effective radius of 1. The minimum possible effective radius, achieved by the optimum profile, is 0.88876. To illustrate some of the arguments we exploit in a more familiar setting, we also obtain a variational formula for the drag on a three-dimensional body in Stokes flow when its surface is perturbed.

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