Irrotational Motion of an Incompressible Fluid: Laplace’s Equation

The average orthodox mathematician who writes or reads books about hydrodynamics is so keen about getting a peg on which he can hang what he calls examples that he quite forgets to apply his formulae to compressible fluids. It appeared to me on reading Colonel De Villamil's letter that a useful purpose would be served by writing the present note; it contains nothing but what anyone could write out for himself, but it was only recently that I made the attempt myself and I should be surprised if many readers have seen the same thing before. We know that in the irrotational motion of an incompressible fluid the velocity potential satisfies Laplace's equation, the present object is to find the corresponding equation for a compressible fluid, to grind it out at full length, and (what is more important than grinding it out) to interpret the result.

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Stability analysis of the method of fundamental solutions with smooth closed pseudo-boundaries for Laplace’s equation: better pseudo-boundaries

Numerical Algorithms ◽

10.1007/s11075-021-01150-5 ◽

2021 ◽

Author(s):

Li-Ping Zhang ◽

Zi-Cai Li ◽

Ming-Gong Lee ◽

Hung-Tsai Huang

Keyword(s):

Stability Analysis ◽

Fundamental Solutions ◽

Method Of Fundamental Solutions ◽

Laplace’S Equation ◽

Laplace's Equation

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Central difference regularization method for the Cauchy problem of the Laplace’s equation

Applied Mathematics and Computation ◽

10.1016/j.amc.2006.01.056 ◽

2006 ◽

Vol 181 (1) ◽

pp. 675-684 ◽

Cited By ~ 22

Author(s):

Xiang-Tuan Xiong ◽

Chu-Li Fu

Keyword(s):

Cauchy Problem ◽

Regularization Method ◽

Laplace’S Equation ◽

Central Difference ◽

Laplace's Equation ◽

The Cauchy Problem

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Approximate analytic solution of the Dirichlet problems for Laplace’s equation in planar domains by a perturbation method

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2011.10.072 ◽

2012 ◽

Vol 63 (1) ◽

pp. 60-67 ◽

Cited By ~ 2

Author(s):

E. Di Costanzo ◽

A. Marasco

Keyword(s):

Perturbation Method ◽

Analytic Solution ◽

Approximate Analytic Solution ◽

Laplace’S Equation ◽

Dirichlet Problems ◽

Laplace's Equation ◽

Planar Domains

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An efficient finite element method for treating singularities in Laplace's equation

Journal of Computational Physics ◽

10.1016/0021-9991(91)90242-d ◽

1991 ◽

Vol 96 (2) ◽

pp. 391-410 ◽

Cited By ~ 35

Author(s):

Lorraine G Olson ◽

Georgios C Georgiou ◽

William W Schultz

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Laplace’S Equation ◽

Laplace's Equation ◽

Element Method

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A note on the estimation of the optimum successive overrelaxation parameter for Laplace's equation

The Computer Journal ◽

10.1093/comjnl/10.4.400 ◽

1968 ◽

Vol 10 (4) ◽

pp. 400-401 ◽

Cited By ~ 6

Author(s):

T. J. Randall

Keyword(s):

Laplace’S Equation ◽

Successive Overrelaxation ◽

Laplace's Equation

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Note on the electric capacity coefficients of spheres

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1912.0101 ◽

1912 ◽

Vol 87 (598) ◽

pp. 485-487

Keyword(s):

General Solution ◽

Laplace’S Equation ◽

Laplace's Equation ◽

Electric Capacity ◽

Physical Problems

In the 'Proceedings' of the Society, vol. 87, p. 109, Mr. Jeffery obtains a general solution of Laplace’s equation in a form suitable for physical problems in connection with two spheres. As an illustration he applies his solution to the problem of finding the capacity coefficients of two equal spheres, obtaining a result which he shows to be equivalent to one of Maxwell’s series formulæ. He then computes a table of the numerical values of these coefficients.

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