Darren Crowdy

Imperial College London, Mathematics, Faculty Member

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My research interests are in applied complex analysis, in particular, using complex analysis as a constructive tool in solving problems arising in different applications from areas of classical physics such as fluid dynamics to more modern areas of mathematical physics. I have special interest in problems involving multiply connected geometries and I'm particularly interested in the function theory associated therewith.
Supervisors: Philip Saffman, FRS
Phone: 020 7594 8587

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Papers by Darren Crowdy

Stuart vortices on a sphere

The exact steady two-dimensional solutions of the Euler equations due to are generalized to the s... more The exact steady two-dimensional solutions of the Euler equations due to are generalized to the surface of a sphere. The solutions are parametrized by three parameters N, θ 0 and Ω max ; the integer N > 1 denotes the number of smooth vorticity extrema, with extremal vorticity Ω max , that are equally spaced in longitudinal angle around a latitude circle at latitudinal spherical polar angle θ 0 . The solutions have two equal point-vortex singularities at the north and south spherical poles. Like Stuart's, the solutions are exact and explicit. The solutions are expected to be useful in geophysical and astrophysical applications where curvature effects are important.

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Calculating the lift on a finite stack of cylindrical aerofoils

format_quoteMultiple aerofoils experience non-trivial interference forces, with their dynamics influenced by shape and relative circulations.format_quote

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Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions

A formula for the generalized Schwarz-Christoffel conformal mapping from a bounded multiply conne... more A formula for the generalized Schwarz-Christoffel conformal mapping from a bounded multiply connected circular domain to an unbounded multiply connected polygonal domain is derived. The formula for the derivative of the mapping function is shown to contain a product of powers of Schottky-Klein prime functions associated with the circular preimage domain. Two analytical checks of the new formula are given. First, it is compared with a known formula in the doubly connected case. Second, a new slit mapping formula from a circular domain to the triply connected region exterior to three slits on the real axis is derived using separate arguments. The derivative of this independently-derived slit mapping formula is shown to correspond to a degenerate case of the new Schwarz-Christoffel mapping. The example of the mapping to the triply connected region exterior to three rectangles centred on the real axis is considered in detail.

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Green's functions for Laplace's equation in multiply connected domains

Analytical formulae for the first-type Green's function for Laplace's equation in multiply connec... more Analytical formulae for the first-type Green's function for Laplace's equation in multiply connected circular domains are presented. The method is constructive and relies on the use of a special function known as the Schottky-Klein prime function associated with multiply connected circular domains. It is shown that all the important functions of potential theory, including the first-type Green's function, the modified Green's functions and the harmonic measures of a domain, can be written in terms of this prime function. A broad range of representative examples are given to demonstrate the efficacy of the method as well as a quantitative comparison, in special cases, with the results obtained using other independent methods.

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Geometric function theory: a modern view of a classical subject

A numerical algorithm is presented for the computation of the Schottky-Klein prime function on th... more A numerical algorithm is presented for the computation of the Schottky-Klein prime function on the Schottky double of multiply connected circular domains in the plane. While there exist classical formulae for the Schottky-Klein prime function in the form of infinite products over a Schottky group, such products are not convergent for all choices of multiply connected circular domains. The prime function itself, however, is a well-defined function for any multiply connected circular domain. The present algorithm facilitates the evaluation of this prime function when the planar domains are such that the classical infinite product representation is either not convergent or so slowly convergent as to be impracticable.

format_quoteA robust algorithm computes the Schottky-Klein prime function using Fourier-Laurent expansions without requiring convergence of sum over Schottky group.format_quote

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Constructing Multiply Connected Quadrature Domains

Multiply connected bounded quadrature domains, with finite connectivity, are reconstructed from t... more Multiply connected bounded quadrature domains, with finite connectivity, are reconstructed from their quadrature data using conformal mappings that are ratios of products of Schottky-Klein prime functions. This method provides the natural generalization of the conformal maps to simply and doubly connected quadrature domains constructed by the first author in a number of physical applications. The efficacy of the method is demonstrated by the explicit construction of a range of examples as well as by comparison with alternative constructive methods recently introduced by Crowdy [R.

format_quoteNew conformal mapping method using Schottky-Klein functions reconstructs multiply connected bounded quadrature domains, enhancing applicability in physical scenarios.format_quote

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