Hilbert formulas for 𝑟-analytic functions and the Stokes flow about a biconvex lens

Abstract

Abstract. The so-called r-analytic functions are a subclass of p-analytic functions and are defined by the generalized Cauchy-Riemann system with p(r, z) = r. In the system of toroidal coordinates, the real and imaginary parts of an r-analytic function are represented by Mehler-Fock integrals with densities, which are assumed to be meromorphic functions. Hilbert formulas, establishing relationships between those functions, are derived for the domain exterior to the contour of a biconvex lens in the meridional cross-section plane. The derivation extends the framework of the theory of Riemann boundary-value problems, suggested in our previous work, to solving the three-contour problem for the case of meromorphic functions with a finite number of simple poles. For numerical calculations, Mehler-Fock integrals with Hilbert formulas reduce to the form of regular integrals. The 3D problem of the axially symmetric steady motion of a rigid biconvex lens-shaped body in a Stokes fluid is solved, and the Hilbert formula for the real part of an r-analytic function is used to express the pressure in the fluid via the vorticity analytically. As an illustration, streamlines and isobars about the body, the vorticity and pressure at the contour of the body and the drag force exerted on the body by the fluid are calculated.

Liquid toroidal drop in compressional flow with arbitrary drop-to-ambient fluid viscosity ratio

“…' The functions Ψ ± and Ψ ∞ satisfy the incompressibility equation identically, i.e. div u ± ≡ 0 in D ± , and the first equation in (2.1) implies 4 In this case, the pressure can be found in closed form through Hilbert formulae for r-analytic functions, see [20][21][22]. 5 A general representation of the Stokes stream function in the toroidal coordinates is given by (2.58) in [29], where {cos νη, cos νη} should be {cos νη, sin νη}.…”

“…one-phase Stokes flow problem for solid spindle, two-spheres, lens and torus, for which stream functions admit closed-form solutions 4 [20][21][22][23][24][25][26][27][28]. 5 The second approach is less complex, because the kinematic condition is considerably simpler than the normal stress boundary condition and does not involve pressure.…”

Asymmetric Creeping Motion of a Rigid Spindle-Shaped Body in a Viscous Fluid

“…As in the proof of Theorem 2, let S be the sphere of large radius R 0 , and let (R, ϑ, ϕ) be the system of the spherical coordinates. For S, we have dS = R 2 0 sin ϑ dϑ dϕ, r = R 0 e R , and n = e R , and thus, (22) reduces to…”

“…On the other hand, the 3D Stokes flow problem corresponding to axially symmetric translation of bodies of revolution can be reduced to determining a biharmonic stream function [7,19] and has been solved for sphere [17], prolate and oblate spheroids [12,7], circular disk [12,7], spherical cap [13,2,19], two spheres [16], torus [14,21], spindle-shaped body [15,23], and lens-shaped body [13,22]. However, it is well known that the stream function approach cannot be extended to solving asymmetric 3D Stokes flow problems.…”

Cauchy integral formula for generalized analytic functions in hydrodynamics

“…For example, if in the rz-half plane, a sphere of radius a is parametrized by z(t) = ae pit/2 , t ∈ [−1, 1], then G 1 (t) = −3v z i/(2a)e −pit/2 , t ∈ [−1, 1] and F z = −6pm v z a, which is the well-known Stokes formula for the sphere drag. Zabarankin (2008a) showed that for prolate and oblate spheroids, biconvex lens and torus of circular cross section, solutions of the boundary-integral equation (3.4) coincide with corresponding analytical solutions in Happel & Brenner (1983), Zabarankin & Krokhmal (2007) and Zabarankin &Ulitko (2006). Also, Zabarankin (2008a) solved (3.4) for solid bi-spheroids (two separate spheroids of equal size) and torus of elliptical cross section, whereas Zabarankin & Molyboha (2010) used (3.4) to find minimum-drag shapes for solid bodies of revolution subject to constraints on body's volume and body's shape.…”