Conformal mappings to a doubly connected polycircular arc domain

2007 ◽  
Vol 463 (2084) ◽  
pp. 1885-1907 ◽  
Author(s):  
D.G Crowdy ◽  
A.S Fokas
Keyword(s):  
Flow Problem ◽  
Explicit Construction ◽  
Polygonal Domain ◽  
Classical Formula ◽  
Circular Arc ◽  
Concentric Annulus ◽  
Doubly Connected Domain ◽  
Analogous Formula ◽  
Important Open Problem ◽  
Arc Mapping

The explicit construction of the conformal mapping of a concentric annulus to a doubly connected polygonal domain was first reported by Akhiezer in 1928. The construction of an analogous formula for the case of a polycircular arc domain, i.e. for a doubly connected domain whose boundaries are a union of circular arc segments, has remained an important open problem. In this paper, we present this explicit formula. We first introduce a new method for deriving the classical formula of Akhiezer and then show how to generalize the method to the case of a doubly connected polycircular arc domain. As an analytical check of the formula, a special exact solution for a doubly connected polycircular arc mapping is derived and compared with that obtained from the more general construction. As an illustrative example, a doubly connected polycircular arc domain arising in a classic potential flow problem considered in the last century by Lord Rayleigh is considered in detail.

On geometry of the (generalized) G2-manifolds

2015 ◽  
Vol 30 (20) ◽  
pp. 1550112 ◽  
Author(s):  
Sen Hu ◽  
Zhi Hu
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Complex Structure ◽  
Vertical Direction ◽  
Classical Case ◽  
Explicit Construction ◽  
Text Structure ◽  
Classical Formula ◽  
Text Structures ◽  
G2 Manifolds

In this paper, we first understand the classical [Formula: see text]-structure and [Formula: see text]-geometry from the viewpoint of spinor, which is a more familiar framework for physicists. Explicit construction of invariant spinor is given via the Dirac gamma matrices. We introduce a notion of multispinor bundle associated with invariant spinor and differential operator on the section of this bundle. Then we study the vector fields satisfy some additional properties on [Formula: see text]-manifold, more precisely, we prove some no-go theorems corresponding to the vector field preserving the associated 4-form on [Formula: see text]-manifold, and we also consider the nowhere-vanishing vector field which induces an integrable complex structure on the vertical direction of tangent bundle. Next we discuss the relation between the variation of metric and that of effective action on the moduli space of integrable [Formula: see text]-structures. In the last section, we deal with the structure operators on generalized [Formula: see text]-manifold after describing the integrability of generalized [Formula: see text]-structure, some identities of structure operators are derived, which are analogues of Kähler-type and Weitzenböck-type identities under the classical case. And finally, we introduce a flow of which a generalized [Formula: see text]-manifold can be realized as the fixed point.

Temperature field evaluation of a two-dimensional partial heat flow problem through Green's Functions

2019 ◽  
Author(s):  
Guilherme Ramalho Costa ◽  
José Aguiar santos junior ◽  
José Ricardo Ferreira Oliveira ◽  
Jefferson Gomes do Nascimento ◽  
Gilmar Guimaraes
Keyword(s):  
Temperature Field ◽  
Heat Flow ◽  
Green's Functions ◽  
Flow Problem ◽  
Green’S Functions ◽  
Field Evaluation ◽  
Two Dimensional

Effect of Viscous Dissipation on MHD Williamson Nanofluid Flow in a Porous Medium

2019 ◽  
Vol 8 (3) ◽  
pp. 5795-5802 ◽  
Keyword(s):  
Porous Medium ◽  
Steady State ◽  
Numerical Solution ◽  
Viscous Dissipation ◽  
Flow Problem ◽  
Numerical Study ◽  
Steady State Flow ◽  
Nanofluid Flow ◽  
Shooting Technique ◽  
Order Four

The main objective of this paper is to focus on a numerical study of viscous dissipation effect on the steady state flow of MHD Williamson nanofluid. A mathematical modeled which resembles the physical flow problem has been developed. By using an appropriate transformation, we converted the system of dimensional PDEs (nonlinear) into coupled dimensionless ODEs. The numerical solution of these modeled ordinary differential equations (ODEs) is achieved by utilizing shooting technique together with Adams-Bashforth Moulton method of order four. Finally, the results of discussed for different parameters through graphs and tables.

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