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

Zabarankin, Michael; Ulitko, Andrei

doi:10.1090/S0033-569X-06-01011-7

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

Authors: Michael Zabarankin and Andrei F. Ulitko
Journal: Quart. Appl. Math. 64 (2006), 663-693
MSC (2000): Primary 30E20, 35Q15, 35Q30, 76D07
DOI: https://doi.org/10.1090/S0033-569X-06-01011-7
Published electronically: September 14, 2006
MathSciNet review: 2284465
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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.

Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756

Beltrami Beltrami E. (1911) Sulle Funzioni Potenziali di Sistemi Simmetrici Intorno ad un Asse. Opere Mathematiche, Vol. 3, pp. 115–128.

Lipman Bers, Theory of pseudo-analytic functions, Institute for Mathematics and Mechanics, New York University, New York, 1953. MR 0057347
Lipman Bers and Abe Gelbart, On a class of differential equations in mechanics of continua, Quart. Appl. Math. 1 (1943), 168–188. MR 8556, DOI https://doi.org/10.1090/S0033-569X-1943-08556-1
W. D. Collins, A note on the axisymmetric Stokes flow of viscous fluid past a spherical cap, Mathematika 10 (1963), 72–78. MR 154496, DOI https://doi.org/10.1112/S0025579300003387
F. D. Gakhov, Boundary value problems, Pergamon Press, Oxford-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966. Translation edited by I. N. Sneddon. MR 0198152

Ghosh Ghosh, S. (1927). On the Steady Motion of a Viscous Liquid due to Translation of a Torus Parallel to its Axis. Bulletin of the Calcutta Mathematical Society, Vol. 18, pp. 185–194.

O. G. Goman, Representation in terms of $p$-analytic functions of the general solution of equations of the theory of elasticity of a transversely isotropic body, Prikl. Mat. Mekh. 48 (1984), no. 1, 98–104 (Russian); English transl., J. Appl. Math. Mech. 48 (1984), no. 1, 62–67 (1985). MR 792756, DOI https://doi.org/10.1016/0021-8928%2884%2990108-4

Goren Goren, S.L., O’Neill, M.E. (1980). Asymmetric Creeping Motion of an Open Torus. Journal of Fluid Mechanics, Vol. 101, Part 1, pp. 97–110.

Happel Happel, J., Brenner, H. (1983). Low Reynolds Number Hydrodynamics. Springer, New York, 572 pp.

R. Hill and G. Power, Extremum principles for slow viscous flow and the approximate calculation of drag, Quart. J. Mech. Appl. Math. 9 (1956), 313–319. MR 81109, DOI https://doi.org/10.1093/qjmam/9.3.313
E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Chelsea Publishing Company, New York, 1955. MR 0064922

Johnson Johnson, R.E., Wu, T.Y. (1979). Hydrodynamics of Low-Reynolds-number Flow. Part 5. Motion of a Slender Torus. Journal of Fluid Mechanics, Vol. 95, Part 2, pp. 263–277.

Horace Lamb, Hydrodynamics, 6th ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993. With a foreword by R. A. Caflisch [Russel E. Caflisch]. MR 1317348
L. E. Payne, On axially symmetric flow and the method of generalized electrostatics, Quart. Appl. Math. 10 (1952), 197–204. MR 51060, DOI https://doi.org/10.1090/S0033-569X-1952-51060-6
L. E. Payne and W. H. Pell, The Stokes flow problem for a class of axially symmetric bodies, J. Fluid Mech. 7 (1960), 529–549. MR 115471, DOI https://doi.org/10.1017/S002211206000027X
W. H. Pell and L. E. Payne, The Stokes flow about a spindle, Quart. Appl. Math. 18 (1960/61), 257–262. MR 120967, DOI https://doi.org/10.1090/S0033-569X-1960-0120967-8
W. H. Pell and L. E. Payne, On Stokes flow about a torus, Mathematika 7 (1960), 78–92. MR 143413, DOI https://doi.org/10.1112/S0025579300001601
Teoriya i primenenie $p$-analiticheskikh i $(p,\,q)$-analiticheskikh funktsiĭ, Izdat. “Naukova Dumka”, Kiev, 1973 (Russian). Obobshchenie teorii analiticheskikh funktsiĭ kompleksnogo peremennogo. (Russian) [Generalization of the theory of analytic functions of a complex variable]; Second edition, revised and augmented. MR 0352489

Sneddon Sneddon, I.N. (1972). The Use of Integral Transforms. McGraw-Hill, 539 pp.

Stimson Stimson, M., Jeffery, G.B. (1926). The Motion of Two-spheres in a Viscous Fluid. Proceedings of Royal Society, London, A, Vol. 111, pp. 110–116.

Stokes Stokes, G.G. (1880). Mathematical and Physical Papers. Vol. I, Cambridge University Press, 328 pp.

Takagi Takagi, H. (1973). Slow Viscous Flow due to the Motion of a Closed Torus. Journal of Physical Society of Japan, Vol. 35, No. 4, pp. 1225–1227.

Ulitko Ulitko, A.F. (2002). Vectorial Decompositions in the Three-Dimensional Theory of Elasticity. Akademperiodika, Kiev, 342 pp. (in Russian)

I. N. Vekua, Generalized analytic functions, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962. MR 0150320
V. S. Vladimirov, Equations of mathematical physics, Pure and Applied Mathematics, vol. 3, Marcel Dekker, Inc., New York, 1971. Translated from the Russian by Audrey Littlewood; Edited by Alan Jeffrey. MR 0268497

Wakiya80 Wakiya, S. (1980). Axisymmetric Stokes Flow about a Body Made of Intersection of Two Spherical Surfaces. Archiwum Mechaniki Stosowanej, Vol. 32, No. 5, pp. 809–817.

Wakiya76 Wakiya, S. (1976). Axisymmetric Flow of a Viscous Fluid near the Vertex of a Body. Fluid Mechanics, Vol. 78, Part 4, pp. 737–747.

Wakiya74 Wakiya, S. (1974). On the Exact Solution of the Stokes Equations for a Torus. Journal of Physical Society of Japan, Vol. 37, No. 3, pp. 780–783.

Wakiya67 Wakiya, S. (1967). Slow Motion of a Viscous Fluid around Two Spheres. Journal of Physical Society of Japan, Vol. 22, No. 4, pp. 1101–1109.

Hilbert Zabarankin M., Ulitko A.F. (2006). Hilbert Formulas for $r$-Analytic Functions in the Domain Exterior to Spindle. SIAM Journal on Applied Mathematics, Vol. 66, No. 4, pp. 1270–1300.

thesis Zabarankin, M. (1999). Exact Solutions to Displacement Boundary-value Problems for an Elastic Medium with a Spindle-shaped Inclusion. Ph.D. Thesis, National Taras Shevchenko University of Kiev, Kiev, 180 pp. (in Russian).

M. Yu. Zabarankin, A unified approach to the solution of a generalized system of equations of Cauchy-Riemann type, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 5 (1999), 30–33 (Russian, with English summary). MR 1727563

Zabarankin Zabarankin, M., Ulitko, A.F. (1999). The Stokes Flow about a Spindle in Axisymmetric Case. Bulletin of National Taras Shevchenko University of Kiev in the field of Mathematics and Mechanics, Issue 3, pp. 58–66 (in Ukrainian).