scholarly journals A calculus for flows in periodic domains

2020 ◽  
Author(s):  
Peter J. Baddoo ◽  
Lorna J. Ayton
Keyword(s):  
Conformal Mapping ◽  
Circular Domain ◽  
Moving Boundaries ◽  
Asymptotic Approximations ◽  
Periodic Domain ◽  
Target Domain ◽  
Potential Flows ◽  
Flow Phenomena ◽  
Periodic Domains ◽  
Prime Function

AbstractPurpose: We present a constructive procedure for the calculation of 2-D potential flows in periodic domains with multiple boundaries per period window.Methods: The solution requires two steps: (i) a conformal mapping from a canonical circular domain to the physical target domain, and (ii) the construction of the complex potential inside the circular domain. All singly periodic domains may be classified into three distinct types: unbounded in two directions, unbounded in one direction, and bounded. In each case, we use conformal mappings to relate the target periodic domain to a canonical circular domain with an appropriate branch structure.Results: We then present solutions for a range of potential flow phenomena including flow singularities, moving boundaries, uniform flows, straining flows and circulatory flows.Conclusion: By using the transcendental Schottky-Klein prime function, the ensuing solutions are valid for an arbitrary number of obstacles per period window. Moreover, our solutions are exact and do not require any asymptotic approximations.

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.

An Inverse Method for Controlling the Temperature Distribution in Infinite Wedge Domain

2012 ◽  
Author(s):  
Hossein Rastgoftar ◽  
Faissal A. Moslehy
Keyword(s):  
Temperature Distribution ◽  
Conformal Mapping ◽  
Laplace Equation ◽  
Lyapunov Functional ◽  
Time Derivative ◽  
Circular Domain ◽  
Control Technique ◽  
Infinite Domain ◽  
Unit Radius ◽  
Infinite Wedge

The paper presents an analytical solution for controlling the temperature distribution in infinite wedge domain. The objective is to assign the heat flux at the boundaries of the domain such that a desired temperature distribution inside the semi-infinite domain is achieved. Since the conduction equation (Laplace equation) retains its form when the infinite domain is transformed into a finite domain by conformal mapping, the infinite domain can be transformed into a disk of unit radius. Then the Laplace equation is investigated in the domain confined by a circle of unit radius. The control technique used in this paper is based on the Lyapunov approach. A Lyapunov functional is defined over the circular domain and the control heat fluxes at the boundary of the disk are assigned such that the time derivative of the Lyapunov functional becomes negative definite. Since the conformal mapping is invertible, attaining a desired temperature distribution in the circular domain leads to achieving the desired temperature distribution in the infinite domain.

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.

Multi-Cell Homogenization of Elastic Periodic Domains

2006 ◽  
Author(s):  
Stefano Gonella ◽  
Massimo Ruzzene
Keyword(s):  
Band Gap ◽  
Dynamic Behavior ◽  
Series Expansion ◽  
Taylor Series ◽  
Dynamic Properties ◽  
Fourier Series Expansion ◽  
Periodic Domain ◽  
Low Frequencies ◽  
Gap Characteristics ◽  
Periodic Domains

Recently, much attention has been devoted to the application of homogenization methods for the prediction of the dynamic behavior of periodic domains. One of the most popular techniques consists in the application of the Fourier Transform in space which allows the application of Taylor series approximations for low frequencies/long wavelengths. This method provides continuum equations which approximate the dynamic behavior of the considered periodic domain over a range of frequencies which is defined by the order of the considered Taylor series expansion. This technique is very effective, but suffers from two major drawbacks. First, the order of the Taylor expansion, and therefore the frequency range of approximation, is limited by the corresponding order of the continuum equations and by the number of boundary conditions which may be imposed in accordance with the physical constraints on the system. Second, the approximation at low frequencies does not allow capturing the band gap characteristics of the periodic domain. An attempt at overcoming the latter can be made by applying the Fourier series expansion to a macro cell spanning two (or more) irreducible unit cells of the periodic medium. This multi-cell approach allows the description of both average and intra-cell behavior of the domain, and approximates dispersion relations and corresponding dynamic properties at low frequencies and at frequencies close to the lower band gap. The resulting continuum equations are therefore capable of reproducing in part the band-gap characteristics of the structure. The proposed methodology is tested on simple one-dimensional and two-dimensional structures, which illustrate the method and show its effectiveness.

scholarly journals Periodic domains of quasiregular maps

2017 ◽  
Vol 38 (6) ◽  
pp. 2321-2344 ◽  
Author(s):  
DANIEL A. NICKS ◽  
DAVID J. SIXSMITH
Keyword(s):  
Entire Function ◽  
Half Space ◽  
General Result ◽  
Periodic Component ◽  
Transcendental Entire Function ◽  
Periodic Domain ◽  
Fatou Set ◽  
Additional Hypothesis ◽  
Lipschitz Maps ◽  
Periodic Domains

We consider the iteration of quasiregular maps of transcendental type from $\mathbb{R}^{d}$ to $\mathbb{R}^{d}$. We give a bound on the rate at which the iterates of such a map can escape to infinity in a periodic component of the quasi-Fatou set. We give examples which show that this result is the best possible. Under an additional hypothesis, which is satisfied by all uniformly quasiregular maps, this bound can be improved to be the same as those in a Baker domain of a transcendental entire function. We construct a quasiregular map of transcendental type from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ with a periodic domain in which all iterates tend locally uniformly to infinity. This is the first example of such behaviour in a dimension greater than two. Our construction uses a general result regarding the extension of bi-Lipschitz maps. In addition, we show that there is a quasiregular map of transcendental type from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ which is equal to the identity map in a half-space.

scholarly journals Equation of motion for point vortices in multiply connected circular domains

2009 ◽  
Vol 465 (2108) ◽  
pp. 2589-2611 ◽  
Author(s):  
Takashi Sakajo
Keyword(s):  
Connected Domain ◽  
Single Point ◽  
Point Vortex ◽  
Equation Of Motion ◽  
Circular Domain ◽  
Point Vortices ◽  
Multiply Connected Domain ◽  
Multiply Connected ◽  
Circular Domains ◽  
Prime Function

The paper gives the equation of motion for N point vortices in a bounded planar multiply connected domain inside the unit circle that contains many circular obstacles, called the circular domain. The velocity field induced by the point vortices is described in terms of the Schottky–Klein prime function associated with the circular domain. The explicit representation of the equation enables us not only to solve the Euler equations through the point-vortex approximation numerically, but also to investigate the interactions between localized vortex structures in the circular domain. As an application of the equation, we consider the motion of two point vortices with unit strength and of opposite signs. When the multiply connected domain is symmetric with respect to the real axis, the motion of the two point vortices is reduced to that of a single point vortex in a multiply connected semicircle, which we investigate in detail.

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.

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 Accessory parameters in conformal mapping: exploiting the isomonodromic tau function for Painlevé VI

2018 ◽  
Vol 474 (2216) ◽  
pp. 20180080 ◽  
Author(s):  
Tiago Anselmo ◽  
Rhodri Nelson ◽  
Bruno Carneiro da Cunha ◽  
Darren G. Crowdy
Keyword(s):  
Conformal Mapping ◽  
Asymptotic Expansions ◽  
Numerical Approach ◽  
Young Diagrams ◽  
Target Domain ◽  
Conformal Maps ◽  
Circular Arc ◽  
Tau Function ◽  
Accessory Parameter ◽  
Parameter Problem

We present a novel method to solve the accessory parameter problem arising in constructing conformal maps from a canonical simply connected planar region to the interior of a circular arc quadrilateral. The Schwarz–Christoffel accessory parameter problem, relevant when all sides have zero curvature, is also captured within our approach. The method exploits the isomonodromic tau function associated with the Painlevé VI equation. Recently, these tau functions have been shown to be related to certain correlation functions in conformal field theory and asymptotic expansions have been given in terms of tuples of the Young diagrams. After showing how to extract the monodromy data associated with the target domain, we show how a numerical approach based on the known asymptotic expansions can be used to solve the conformal mapping accessory parameter problem. The viability of this new method is demonstrated by explicit examples and we discuss its extension to circular arc polygons with more than four sides.

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