Generalised Hele-Shaw flow: A Schwarz function approach

Author:

McDONALD N. R.

Abstract

An equation governing the evolution of a Hele-Shaw free boundary flow in the presence of an arbitrary external potential – generalised Hele-Shaw flow – is derived in terms of the Schwarz functiong(z,t) of the free boundary. This generalises the well-known equation ∂g/∂t= 2∂w/∂z, wherewis the complex potential, which has been successfully employed in constructing many exact solutions in the absence of external potentials. The new equation is used to re-derive some known explicit solutions for equilibrium and time-dependent free boundary flows in the presence of external potentials, including those with singular potential fields, uniform gravity and centrifugal forces. Some new solutions are also constructed that variously describe equilibrium flows with higher order hydrodynamic singularities in the presence of electric point sources and an unsteady solution describing bubbles under the combined influence of strain and centrifugal potential.

Publisher

Cambridge University Press (CUP)

Subject

Applied Mathematics

Cited by 11 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. On a two-fluid Hele-Shaw problem with an elliptical interface;Journal of Differential Equations;2021-01

2. From reflections to a uniform elliptic growth;Journal of Mathematical Analysis and Applications;2019-09

3. On a two-phase Hele-Shaw problem with a time-dependent gap and distributions of sinks and sources;Journal of Physics A: Mathematical and Theoretical;2017-12-20

4. Rotating Hele-Shaw cell with a time-dependent angular velocity;Physical Review Fluids;2017-12-19

5. Hele-Shaw Flow with a Time-Dependent Gap: The Schwarz Function Approach to the Interior Problem;Complex Analysis and Dynamical Systems VI;2016

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