Cauchy integral formula for generalized analytic functions in hydrodynamics

Zabarankin, Michael

doi:10.1098/rspa.2012.0335

Cited by 9 publications

(13 citation statements)

References 28 publications

(66 reference statements)

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“…For ν ∈ [0, 1/2), (3.8) has no non-zero homogeneous solutionthis can be proved similarly to either theorem 10 in [33] or theorem 3.2 in [37].…”

Section: Proposition 33 (Resistance Force In Axisymmetric Translatiomentioning

confidence: 95%

“…The representations (3.3) and (3.4) extend formulae (36) and (37) in [33] for the velocity and pressure of axisymmetric Stokes flows, whose mathematical model is identical to the Navier equations (1.1) with ν = 1/2.…”

Section: Minimum-resistance Shape In Axisymmetric Translationmentioning

confidence: 99%

“…For ν = 1/2, the system (4.10)-(4.12) reduces to (37)- (40) in [34] that corresponds to the transversal translation of a rigid body of revolution in a viscous incompressible fluid under the zero Reynolds number assumption. 10)-(4.12) is solved similar to (3.8)…”

Section: Proposition 41 (Resistance Force In Transversal Translationmentioning

confidence: 99%

“…14 Ω + and Ω − in (B 4) are given by (2.22) in [37], in which the multiplier 1/κ 2 was accidentally missing. imposed arbitrary displacement boundary conditions.…”

Section: Appendix a Proof Of The Relationship (25)mentioning

confidence: 99%

“…In contrast to the Kolosov formulae [27,28] (also called Kolosov-Muskhelishvili formulae) and their three-dimensional analogues [35,36], those representations contain no derivatives of the involved generalized analytic functions and, as a result, have an advantage in obtaining closedform solutions and in deriving BIEs through the generalized Cauchy integral formula [37]. As shown in [14,23,24], similar derivative-free representations provide a simple coordination of all the parts of the optimization scheme: for axisymmetric Stokes and Oseen flows, the drag of the body, the optimality conditions for the minimum-drag shape and the corresponding BIEs are all formulated in terms of the same generalized analytic function.…”

Section: Introductionmentioning

confidence: 99%

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Minimum-resistance shapes in linear continuum mechanics

Zabarankin

2013

Proc. R. Soc. A.

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A necessary optimality condition for the problem of the minimum-resistance shape for a rigid threedimensional inclusion displaced in an unbounded isotropic elastic medium subject to a constraint on the volume of the inclusion is obtained through Betti's reciprocal work theorem. It generalizes Pironneau's optimality condition for the minimum-drag shape for a rigid body immersed into a uniform Stokes flow and is specialized for axisymmetric inclusions in axisymmetric and transversal translations. In both cases of translation, the three-dimensional displacement field is represented in terms of generalized analytic functions, and the three-dimensional elastostatics problem is reduced to boundary-integral equations (BIEs) via the generalized Cauchy integral formula. Minimum-resistance shapes are found in the semi-analytical form of functional series from an iterative procedure coupling the optimality condition and the BIEs. They are compared with the minimumresistance spheroids and with the minimum-resistance spindle-shaped and lens-shaped bodies. Remarkably, in the axisymmetric translation, the minimumresistance shapes transition from spindle-like shapes to almost prolate spheroidal shapes as the Poisson ratio changes from 1/2 to 0, whereas in the transversal translation, they are close to oblate spheroidal shapes for any Poisson ratio.

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“…For ν ∈ [0, 1/2), (3.8) has no non-zero homogeneous solutionthis can be proved similarly to either theorem 10 in [33] or theorem 3.2 in [37].…”

Section: Proposition 33 (Resistance Force In Axisymmetric Translatiomentioning

confidence: 95%

Section: Minimum-resistance Shape In Axisymmetric Translationmentioning

confidence: 99%

Section: Proposition 41 (Resistance Force In Transversal Translationmentioning

confidence: 99%

“…14 Ω + and Ω − in (B 4) are given by (2.22) in [37], in which the multiplier 1/κ 2 was accidentally missing. imposed arbitrary displacement boundary conditions.…”

Section: Appendix a Proof Of The Relationship (25)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Minimum-resistance shapes in linear continuum mechanics

Zabarankin

2013

Proc. R. Soc. A.

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Quaternionic Functions and Their Applications in a Viscous Fluid Flow

Grigor’ev

2017

Complex Anal. Oper. Theory

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Liquid toroidal drop in compressional Stokes flow

2015

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The deformation of an immiscible toroidal drop embedded in axisymmetric compressional Stokes flow is analysed via the boundary integral formulation in the case of equal viscosity. Numerical simulations are performed for the drop having initially the shape of a torus with circular cross-section. The quasi-stationary dynamic simulations reveal that, when the viscous forces, proportional to the intensity of the flow, are relatively weak compared with the surface tension (the ratio of these forces is characterized by the capillary number, Ca), three different scenarios of drop evolution are possible: indefinite expansion of the liquid torus, contraction to the centre and a stationary toroidal shape. When the intensity of the flow is low, the stationary shapes are shown to be close to circular tori. Once the outer flow strengthens, the cross-section of the stationary torus assumes first an elliptic and then an egg-like shape. For the capillary number greater than a critical value, Ca cr , toroidal stationary shapes were not found. Remarkably, Ca cr is close to the critical capillary number found previously for a simply connected drop flattened in compressional flow. Thus, a new example of non-uniqueness of stationary drop shape in viscous flow is obtained. Approximate stationary solutions in the form of tori with circular and elliptic cross-sections are obtained by minimizing the normal velocity over the drop interface. They are shown to be in good agreement with the stationary shapes from quasi-dynamic simulations for the corresponding intervals of the capillary number.

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Cauchy integral formula for generalized analytic functions in hydrodynamics

Cited by 9 publications

References 28 publications

Minimum-resistance shapes in linear continuum mechanics

Minimum-resistance shapes in linear continuum mechanics

Quaternionic Functions and Their Applications in a Viscous Fluid Flow

Liquid toroidal drop in compressional Stokes flow

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