Conformal Mappings between Canonical Multiply Connected Domains

Crowdy, Darren; Marshall, J. S.

doi:10.1007/bf03321118

Cited by 109 publications

(110 citation statements)

References 6 publications

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“…24 The parameters , 2 , and in Eq. ͑32͒ are all functions of time and can be determined implicitly ͑e.g., using Newton's method͒ by enforcing the geometric conditions on ␤, , c, and l defining Fig.…”

Section: A Conformal Mapmentioning

confidence: 99%

Fluid-structure interaction of two bodies in an inviscid fluid

2010

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The interaction of two arbitrary bodies immersed in a two-dimensional inviscid fluid is investigated. Given the linear and angular velocities of the bodies, the solution of the potential flow problem with zero circulation around both bodies is reduced to the determination of a suitable Laurent series in a conformally mapped domain that satisfies the boundary conditions. The potential flow solution is then used to determine the force and moment acting on each body by using generalized Blasius formulas. The current formulation is applied to two examples. First, the case of two rigid circular cylinders interacting in an unbounded domain is investigated. The forces on two cylinders with prescribed motion ͑forced-forced͒ is determined and compared to previous results for validation purposes. We then study the response of a single "free" cylinder due to the prescribed motion of the other cylinder ͑forced-free͒. This forced-free situation is used to justify the hydrodynamic benefits of drafting in aquatic locomotion. In the case of two neutrally buoyant circular cylinders, the aft cylinder is capable of attaining a substantial propulsive force that is the same order of magnitude of its inertial forces. Additionally, the coupled interaction of two cylinders given an arbitrary initial condition ͑free-free͒ is studied to show the differences of perfect collisions with and without the presence of an inviscid fluid. For a certain range of collision parameters, the fluid acts to deflect the cylinder paths just enough before the collision to drastically affect the long time trajectories of the bodies. In the second example, the flapping of two plates is explored. It is seen that the interactions between each plate can cause a net force and torque at certain instants in time, but for idealized sinusoidal motions in irrotational potential flow, there is no net force and torque acting at the system center.

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Section: A Conformal Mapmentioning

confidence: 99%

Fluid-structure interaction of two bodies in an inviscid fluid

2010

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“…To name just a few of the problems tackled within this framework, it is now known how to represent all the important objects of potential theory (such as Green's functions and harmonic measures) in terms of the S-K prime function [5]; the Hamiltonians, or Kirchhoff-Routh path functions, associated with point vortex motion in multiply connected domains can also be conveniently expressed in terms of it [6]; the boundaries of a class of domains known as quadrature domains, which themselves have abundant applications in both pure and applied mathematics, can be uniformized in terms of the prime function [7]; conformal slit maps, also useful in a variety of applications (e.g. in Hele-Shaw flows [1,8]) have very simple representations in terms of the prime function [9]. Indeed, the multiply connected generalization of the classical Schwarz-Christoffel formula giving the conformal mapping from canonical circular domains to multiply connected polygonal regions has recently been found and has a concise and convenient form when written using the prime function [10,11].…”

Section: Introductionmentioning

confidence: 99%

“…(For the complete set of formulae for the five conformal slit maps of non-mixed type, see [9].) It is therefore natural to ask whether similar formulae exist for conformal slit maps of mixed type.…”

Section: Introductionmentioning

confidence: 99%

Secondary Schottky–Klein prime functions associated with multiply connected planar domains

2015

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In recent years, a general mathematical framework for solving applied problems in multiply connected domains has been developed based on use of the Schottky-Klein (S-K) prime function of an underlying compact Riemann surface known as the Schottky double of the domain. In this paper, we describe additional function-theoretic objects that are naturally associated with planar multiply connected domains and which we refer to as secondary S-K prime functions. The basic idea develops, and extends, an observation of Burnside dating back to 1892. Applications of the new functions to represent conformal slit maps of mixed type that have been a topic of recent interest in the literature are given. Other possible applications are also surveyed.

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“…DeLillo et al [11] also generalized the Schwarz-Christoffel mapping to multiply connected domains, and derived a formula for conformal mapping from an unbounded circular domain onto an unbounded polygonal domain. Crowdy and Marshall [9] constructed analytical formulae for conformal mapping from a bounded circular domain onto the five canonical slit domains above mentioned, and DeLillo et al [12] from an unbounded circular domain onto the circular and radial slit domain. Crowdy [8] constructed also an analytical solution for uniform potential flows past multiple cylinders.…”

Section: Introductionmentioning

confidence: 99%

Numerical conformal mappings onto the linear slit domain

Amano

Okano

Ogata

et al. 2012

Japan J. Indust. Appl. Math.

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We propose a numerical method for the conformal mapping of unbounded multiply connected domains exterior to closed Jordan curves C 1 , . . . , C n onto a canonical linear slit domain, which is the entire plane with linear slits S 1 , . . . , S n of angles θ 1 , . . . , θ n arbitrarily assigned to the real axis, respectively. If θ 1 = · · · = θ n = θ then it is the well-known parallel slit domain, which is important in the problem of potential flows past obstacles. In the method, we reduce the mapping problem to a boundary value problem for an analytic function, and approximate it by a linear combination of complex logarithmic functions based on the charge simulation method. Numerical examples show the effectiveness of our method.

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Conformal Mappings between Canonical Multiply Connected Domains

Cited by 109 publications

References 6 publications

Fluid-structure interaction of two bodies in an inviscid fluid

Fluid-structure interaction of two bodies in an inviscid fluid

Secondary Schottky–Klein prime functions associated with multiply connected planar domains

Numerical conformal mappings onto the linear slit domain

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