A generalized Milne-Thomson theorem

Obnosov, Yu.V.

doi:10.1016/j.aml.2005.08.006

Cited by 18 publications

(9 citation statements)

References 5 publications

(3 reference statements)

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“…Instead of a tall barrier I in figure 2a, examined in this study, we can tackle a thin but broad barrier II. The recent progress in finding analytical solutions to flow problems with arbitrary driving singularities in piece-wise homogeneous porous media (Obnosov 2006(Obnosov , 2009(Obnosov , 2011 makes the extension to seepage regimes with a macro-volume of LNAPL attached to low-permeable (rather than impermeable) baffles and to ambient groundwater moving from injection to abstraction wells promising, rather than to vertical infiltration as in this study.…”

Section: Discussionmentioning

confidence: 99%

Accumulation of a light non-aqueous phase liquid on a flat barrier baffling a descending groundwater flow

Kacimov

2012

Proc. R. Soc. A.

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The pioneering solution of Zhukovskii for a steady two-dimensional flow of an ideal heavy fluid with a nonlinear free boundary condition is extended to a Darcian flow of groundwater encumbered by an impermeable barrier. The stoss or/and lee sides of the barrier are covered by a macrovolume of a liquid contaminant. Explicit parametric equations of the sharp interface are obtained by inversion of the hodograph domain. Zhukovskii's gas-finger shape is shown to be a particular case of our new class of free surfaces. For a cap of a light liquid, partially covering the roof, from the given crosssectional area of the cap, the affixes of the conformal mapping are found as a solution of a system of two nonlinear equations. The horizontal width and vertical height of the cap are determined. If the dimensionless incident velocity is higher than the density contrast, then the interface (cap boundary) cusps at its apex. For a relatively small velocity, the interface spreads to the vertices of the barrier, the apex zone remaining blunt shaped. We depict all the relevant domains and plot the flow nets using computer algebra routines.Keywords: nonlinear free boundary problem; two-dimensional flow of heavy ideal fluid -seepage; holomorphic functions; hodograph; conformal mappings Historical motivation from Zhukovskii's solutionThere are very few analytical solutions to the problem of steady two-dimensional motion of an ideal irrotational heavy fluid with a free surface. This is caused by the necessity to satisfy the Bernoulli equationalong an unknown isobaric streamline (free surface). In equation (1.1a), V is the magnitude of the velocity vector V = V e iq , q is the angle that V makes with the horizontal axis x, g = 9.8 m s −2 and y is the vertical coordinate oriented upwards, against gravity.

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Section: Discussionmentioning

confidence: 99%

Accumulation of a light non-aqueous phase liquid on a flat barrier baffling a descending groundwater flow

Kacimov

2012

Proc. R. Soc. A.

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“…As a continuation of investigation of two-phase [12] and three-phase ( [14], p. 92) concentric circular structures we have given a constructive explicit solution of the corresponding four-phase problem. It was shown that the same basic idea as in the above cited papers is also working here.…”

Section: Resultsmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

An ℝ-linear conjugation problem for two concentric annuli

Kazarin

2015

Lobachevskii J Math

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We consider an infinite planar four-phase heterogeneous medium with three concentric circles as a boundary between isotropic medium's components of distinct resistivities/conductivities. It is supposed that the velocity field in this structure is generated by a finite set of arbitrary multipoles. We distinguish two cases when multipoles are inside of medium's components or at the interface. An exact analytical solution of the corresponding R-linear conjugation boundary value problem is derived for both cases. Examples of flow nets (isobars and streamlines) are presented.

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“…1, rather than in the 'target' layer of Figs. 1 and 2, can be easily tackled as in Obnosov (2006Obnosov ( , 2009a, as well as the case when the singularity is exactly on the interface. More challenging is the problem for periodic singularities with other interfaces (e.g., circular).…”

Section: Discussionmentioning

confidence: 99%

“…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.…”

Section: Array Of Line Sinksmentioning

confidence: 99%

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

Kasimova

Kacimov

2010

Transp Porous Med

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Explicit analytical solutions are obtained in terms of hydraulic head (pressure) and Darcian velocity for a steady Darcian flow to a point/line sink and array of sinks with refraction of streamlines on a horizontal interface between two layers of constant hydraulic conductivities. The sinks are placed in a 'target' layer between a constant potential plane and interface. An equipotential surface, encompassing the sink represents a horizontal or vertical well, is reconstructed as a quasi-cylinder or quasi-sphere. The method of electrostatic images and theory of holomorphic functions are employed for obtaining series expansion solutions of two conjugated Laplace equations. If the conductivity of the 'target' layer is less than that of the super/sub-stratum, then there is a minimum of the flow rate into the well of a given size. Applications to agricultural drainage and surface DC-electrical resistivity surveying are discussed.

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A generalized Milne-Thomson theorem

Abstract: Using analytic continuation theory, a new simple proof of a standard generalized circle theorem is given. Additionally, new cases involving complex coefficients in the boundary condition and allowing for an arbitrary singularity of a given complex potential at the interface are considered.

Cited by 18 publications

References 5 publications

Accumulation of a light non-aqueous phase liquid on a flat barrier baffling a descending groundwater flow

Accumulation of a light non-aqueous phase liquid on a flat barrier baffling a descending groundwater flow

An ℝ-linear conjugation problem for two concentric annuli

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

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