Three–phase eccentric annulus subjected to a potential field induced by arbitrary singularities

Abstract

Abstract. An infinite planar, three-component heterogeneous medium with a pair of circles as interfaces between homogeneous zones forming an eccentric annulus is considered for refraction of a potential field on the two interfaces. The velocity field is generated by an arbitrary system of singularities of arbitrary order, in congruity with the MilneThomson case of a two-component medium and a single circular interface. An exact analytical solution of the corresponding R-linear conjugation problem of two Laplacian fields in the eccentrical annulus structure is derived in the class of piecewise meromorphic functions with fixed principal part. Three general cases of loci of the singularities with respect to the interfaces are investigated. Flow nets (isobars and streamlines) are presented.

An ℝ-linear conjugation problem for two concentric annuli

“…Let us consider the last summands of functions (16). Omitting for the sake of simplicity almost all indexes, we derive…”

“…Only for some specific structures it is possible to do. For example, the problem of the perturbation of a given complex potential by inserting distinct inclusions into an isotropic medium was solved for circular [12], elliptical [17], parabolic [13], hyperbolic [10], circular and elliptical annuli inclusions [16] and [4]. Much more progress can be made if all inclusions are perfectly resisting [2].…”

“…By the generalized Liouville theorem Φ k (z) = C k /z, where C 1 , C 2 , C 3 are constants to be determined. From (14), (16), (17) follows…”

A Well in a ‘Target’ Stratum of a Two-Layered Formation: The Muskat–Riesenkampf Solution Revisited

“…1, consider the periodic line sinks (horizontal drains) and find solution to the flow problem (2)-(3). In order to diversify the techniques, instead of the method of images employed by Anderson (2000) we utilize the theory of boundary value problem of R-linear conjugation, which has recently been implemented in similar problems of potential flows through piece-wise homogeneous media (Obnosov 1996(Obnosov , 1999(Obnosov , 2006(Obnosov , 2009a(Obnosov , 2010.…”

Topology of Steady Heat Conduction in a Solid Slab Subject to a Nonuniform Boundary Condition: The Carslaw–jaeger Solution revisited

“…Analytical solutions to potential field problems, where the intricate topology of 2D flow nets is controlled by internal heterogeneities of the domain, but the boundary conditions are uniform or follow from the symmetry principle in cases of internal heat sources/sinks, have recently been obtained [6,[9][10][11][12]. In these cases, an elementary cell, where the potential problems were solved, represents a flow tube, which consists of two isotherms and two adiabatic lines.…”