scholarly journals Multiple bubbles and fingers in a Hele-Shaw channel: complete set of steady solutions

2015 ◽  
Vol 780 ◽  
pp. 299-326 ◽  
Author(s):  
Giovani L. Vasconcelos
Keyword(s):  
Surface Tension ◽  
Conformal Mapping ◽  
Circular Domain ◽  
Multiply Connected ◽  
Transcendental Functions ◽  
Region Exterior ◽  
Complete Set ◽  
Multiple Bubbles ◽  
Steady Solutions ◽  
Connected Circular Domain

Analytical solutions for both a finite assembly and a periodic array of bubbles steadily moving in a Hele-Shaw channel are presented. The particular case of multiple fingers penetrating into the channel and moving jointly with an assembly of bubbles is also analysed. The solutions are given by a conformal mapping from a multiply connected circular domain in an auxiliary complex plane to the fluid region exterior to the bubbles. In all cases the desired mapping is written explicitly in terms of certain special transcendental functions, known as the secondary Schottky–Klein prime functions. Taken together, the solutions reported here represent the complete set of solutions for steady bubbles and fingers in a horizontal Hele-Shaw channel when surface tension is neglected. All previous solutions under these assumptions are particular cases of the general solutions reported here. Other possible applications of the formalism described here are also discussed.

Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions

2007 ◽  
Vol 142 (2) ◽  
pp. 319-339 ◽  
Author(s):  
DARREN CROWDY
Keyword(s):  
Real Axis ◽  
Mapping Function ◽  
Circular Domain ◽  
Connected Region ◽  
Polygonal Domain ◽  
The Real ◽  
Multiply Connected ◽  
Region Exterior ◽  
New Formula ◽  
Connected Circular Domain

AbstractA 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.

scholarly journals Multiple steadily translating bubbles in a Hele-Shaw channel

2014 ◽  
Vol 470 (2163) ◽  
pp. 20130698 ◽  
Author(s):  
Christopher C. Green ◽  
Giovani L. Vasconcelos
Keyword(s):  
Steady Motion ◽  
Flow Region ◽  
Circular Domain ◽  
Multiply Connected Domains ◽  
Moving Frames ◽  
Polygonal Domains ◽  
Multiply Connected ◽  
Region Exterior ◽  
Prime Function ◽  
Connected Circular Domain

Analytical solutions are constructed for an assembly of any finite number of bubbles in steady motion in a Hele-Shaw channel. The solutions are given in the form of a conformal mapping from a bounded multiply connected circular domain to the flow region exterior to the bubbles. The mapping is written as the sum of two analytic functions—corresponding to the complex potentials in the laboratory and co-moving frames—that map the circular domain onto respective degenerate polygonal domains. These functions are obtained using the generalized Schwarz–Christoffel formula for multiply connected domains in terms of the Schottky–Klein prime function. Our solutions are very general in that no symmetry assumption concerning the geometrical disposition of the bubbles is made. Several examples for various bubble configurations are discussed.

scholarly journals Generalization of the Schwarz–Christoffel mapping to multiply connected polygonal domains

2014 ◽  
Vol 470 (2166) ◽  
pp. 20130848 ◽  
Author(s):  
Giovani L. Vasconcelos
Keyword(s):  
Conformal Mapping ◽  
Circular Domain ◽  
Polygonal Domains ◽  
Multiply Connected ◽  
Slit Domain ◽  
Combined Use ◽  
Polygonal Region ◽  
Upper Half Plane ◽  
Indefinite Integral ◽  
Detailed Derivation

A generalization of the Schwarz–Christoffel mapping to multiply connected polygonal domains is obtained by making a combined use of two preimage domains, namely, a rectilinear slit domain and a bounded circular domain. The conformal mapping from the circular domain to the polygonal region is written as an indefinite integral whose integrand consists of a product of powers of the Schottky-Klein prime functions, which is the same irrespective of the preimage slit domain, and a prefactor function that depends on the choice of the rectilinear slit domain. A detailed derivation of the mapping formula is given for the case where the preimage slit domain is the upper half-plane with radial slits. Representation formulae for other canonical slit domains are also obtained but they are more cumbersome in that the prefactor function contains arbitrary parameters in the interior of the circular domain.

The Schwarz–Christoffel mapping to bounded multiply connected polygonal domains

2005 ◽  
Vol 461 (2061) ◽  
pp. 2653-2678 ◽  
Author(s):  
Darren Crowdy
Keyword(s):  
Mapping Function ◽  
Circular Domain ◽  
Schottky Group ◽  
Polygonal Domain ◽  
Schottky Groups ◽  
Image Domain ◽  
Polygonal Domains ◽  
Multiply Connected ◽  
Connected Circular Domain ◽  
Natural Way

A formula for the generalized Schwarz–Christoffel mapping from a bounded multiply connected circular domain to a bounded multiply connected polygonal domain is derived. The theory of classical Schottky groups is employed. The formula for the derivative of the mapping function contains a product of powers of Schottky–Klein prime functions associated with a Schottky group relevant to the circular pre-image domain. The formula generalizes, in a natural way, the known mapping formulae for simply and doubly connected polygonal domains.

scholarly journals Periodic Schwarz–Christoffel mappings with multiple boundaries per period

2019 ◽  
Vol 475 (2228) ◽  
pp. 20190225 ◽  
Author(s):  
Peter J. Baddoo ◽  
Darren G. Crowdy
Keyword(s):  
Circular Domain ◽  
Target Domain ◽  
Multiply Connected ◽  
Single Period ◽  
Straight Line ◽  
Finite Set ◽  
Periodic Configuration ◽  
Parameter Problem ◽  
Prime Function ◽  
Connected Circular Domain

We present an extension to the theory of Schwarz–Christoffel (S–C) mappings by permitting the target domain to be a single period window of a periodic configuration having multiple polygonal (straight-line) boundaries per period. Taking the arrangements to be periodic in the x -direction in an ( x ,  y )-plane, three cases are considered; these differ in whether the period window extends off to infinity as y  →  ± ∞, or extends off to infinity in only one direction ( y  →  + ∞ or y  →  − ∞), or is bounded. The preimage domain is taken to be a multiply connected circular domain. The new S–C mapping formulae are shown to be expressible in terms of the Schottky–Klein prime function associated with the circular preimage domains. As usual for an S–C map, the formulae are explicit but depend on a finite set of accessory parameters. The solution of this parameter problem is discussed in detail, and illustrative examples are presented to highlight the essentially constructive nature of the results.

scholarly journals Unsteady flow induced by a withdrawal point beneath a free surface

2005 ◽  
Vol 47 (2) ◽  
pp. 185-202 ◽  
Author(s):  
T. E. Stokes ◽  
G. C. Hocking ◽  
L. K. Forbes
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Unsteady Flow ◽  
Critical Values ◽  
The Other ◽  
Withdrawal Rate ◽  
Steady Solutions

AbstractThe unsteady axisymmetric withdrawal from a fluid with a free surface through a point sink is considered. Results both with and without surface tension are included and placed in context with previous work. The results indicate that there are two critical values of withdrawal rate at which the surface is drawn directly into the outlet, one after flow initiation and the other after the flow has been established. It is shown that the larger of these values corresponds to the point at which steady solutions no longer exist.

Sign in / Sign up

Export Citation Format

Share Document