Analytical formulae for source and sink flows in multiply connected domains

Abstract

Analytical expressions for the complex potentials associated with irrotational incompressible flows due to a finite collection of point sources and sinks in a circular fluid region, of arbitrary finite connectivity, are derived. By the conformal invariance of the boundary value problem, the flows due to point sources and sinks in arbitrary multiply connected fluid domains are then determined up to conformal mapping. As an application of the theory, it is combined with recent results on conformal slit mappings to derive explicit formulae for the effective size of an array of holes in a screen, or grating, following considerations presented by Tuck (Adv Appl Mech 15:89-158, 1975).

Tracer Test and Streamline Simulation for Geothermal Resources in Cuona of Tibet

“…For the completeness of this study, some related derivations and formulas are still summarized here. More details could be found in Reference [6] and other related publications [15,23,24].…”

Applying improved analytical methods for modelling flood displacement fronts in bounded reservoirs (Quitman field, east Texas)

“…1). Extension of the Riemann mapping theorem from flow around single objects situated in simply connected fluid domains to multiply connected domains of unbounded, partially bounded and fully bounded fluid regions has been elaborated in several recent landmark studies (Crowdy, 2010(Crowdy, , 2013. An unbounded fluid with one obstacle is simply connected and a bounded fluid domain with one internal obstacle is doubly connected; unbounded fluid with two obstacles are doubly connected and bounded counterparts are triply connected, and so on.…”

“…Label the outer boundary, on which |ߞ| = 1, ‫ܥ‬ and those of the ‫ܯ‬ excised disks ‫ܥ‬ , {݇ = 1, … , ‫. }ܯ‬ In general, the complex potential in ‫ܦ‬ owing to a collection of ܰ sink/source singularities can be written as (Crowdy 2013)…”

“…Bessel functions. Properties of the Schottky-Klein prime function and its applications are discussed in detail in, e.g., Crowdy (2010) & Crowdy (2013. Below, to give readers insight into the basic properties of the Prime function, we briefly give one geometrical construction.…”

Conformal Slit Maps in Applied Mathematics

“…This function has the effect of putting a point vortex of opposite circulation at infinity in order that the circulations around all the obstacles are zero. More details of point vortex and source/sink flows in multiply connected domains are given by Crowdy and Marshall[4,8,14].…”