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On the slow motion of a self-propelled rigid body in a viscous incompressible fluid. (English) Zbl 1121.76321

Summary: In this paper we study the Stokes approximation of the self-propelled motion of a rigid body in a viscous liquid that fills all the three-dimensional space exterior to the body. We prove the existence and uniqueness of strong solution to the coupled systems of equations describing the motion of the system body-liquid, for any time and any regular distribution of velocity on the boundary of the body. For the corresponding stationary problem we derive \(L^p\)-estimates for the solution in terms of the data. Finally, we prove that every steady solution is attainable as the limit, when \(t\to\infty\), of an unsteady self-propelled solution which starts from rest.

MSC:

76D07 Stokes and related (Oseen, etc.) flows
35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76Z10 Biopropulsion in water and in air
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[1] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101
[2] Blake, J., A finite model for ciliated micro-organisms, J. Biomech., 6, 133-140 (1973)
[3] Crandall, M. G.; Pazy, A.; Tartar, L., Israel J. Math., 32, 363-374 (1979)
[4] Danielson, D. A., Vectors and Tensors in Engineering and Physics (1997), Perseus Books: Perseus Books Cambridge, MA · Zbl 0768.73001
[5] Farwig, R.; Sohr, H., The stationary and non-stationary Stokes system in exterior domains with non-zero divergence and non-zero boundary values, Math. Methods Appl. Sci., 17, 269-291 (1994) · Zbl 0798.35125
[6] Finn, R., On the exterior stationary problem for the Navier-Stokes equations and associated perturbation problems, Arch. Rational Mech. Anal., 19, 363-406 (1965) · Zbl 0149.44606
[7] Galdi, G. P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Linearized Steady Problems. An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Linearized Steady Problems, Springer Tracts in Natural Philosophy, 38 (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0949.35004
[8] Galdi, G. P., On the steady, translational self-propelled motion of a symmetric body in a Navier-Stokes fluid, Quad. Mat. II Univ. Napoli, 1, 97-169 (1997) · Zbl 0974.76020
[9] Galdi, G. P., Slow motion of a body in a viscous incompressible fluid with application to particle sedimentation, Quad. Mat. II Univ. Napoli, 2, 1-35 (1998) · Zbl 0930.76021
[10] Galdi, G. P., On the steady self-propelled motion of a body in a viscous incompressible fluid, Arch. Rational Mech. Anal., 148, 53-88 (1999) · Zbl 0957.76012
[11] Goldstein, J. A., Semigroups of Linear Operators and Applications (1985), Oxford University Press: Oxford University Press New York · Zbl 0592.47034
[12] Gray, J., Animal Locomotion (1968), Weidenfeld and Nicholson: Weidenfeld and Nicholson London
[13] Grobbelaar-Van Dalsen, M.; Sauer, N., Dynamic boundary conditions for the Navier-Stokes equations, Proc. Roy. Soc. Edinburgh Sect. A, 113, 1-11 (1989) · Zbl 0682.76018
[14] Hancock, G. J., The self-propulsion of microscopic organisms through liquids, Proc. Roy. Soc. London Ser. A, 217, 96-121 (1953) · Zbl 0050.19302
[15] Happel, V.; Brenner, H., Low Reynolds Number Hydrodynamics (1965), Prentice-Hall
[16] Koiller, J.; Ehlers, K.; Montgomery, R., Problems and progress in microswimming, J. Nonlinear Sci., 6, 507-541 (1996) · Zbl 0867.76099
[17] Lamb, H., Hydrodynamics (1932), Cambridge University Press · JFM 26.0868.02
[18] Lighthill, J., Mathematical Biofluid Dynamics (1975), SIAM
[19] Milne-Thomson, L. M., Theoretical Aerodynamics (1952), Van Nostrandt: Van Nostrandt New York · Zbl 0047.43604
[20] Pukhnacev, V. V., Asymptotics of a velocity field at considerable distances from a self-propelled body, J. Appl. Mech. Tech. Phys., 30, 52-60 (1989)
[21] Pukhnacev, V. V., The Problem of Momentumless Flow for the Navier-Stokes Equations. The Problem of Momentumless Flow for the Navier-Stokes Equations, Lecture Notes in Mathematics, 1431 (1990), Springer-Verlag: Springer-Verlag New York, pp. 87-94 · Zbl 0708.35066
[22] Sennitskiı̆, V. L., Liquid flow around a self-propelled body, J. Appl. Mech. Tech. Phys., 3, 15-27 (1978)
[23] Sennitskiı̆, V. L., An example of axisymmetric fluid flow around a self-propelled body, J. Appl. Mech. Tech. Phys., 4, 31-36 (1984)
[24] Sennitskiı̆, V. L., Self-propulsion of a body in a fluid, J. Appl. Mech. Tech. Phys., 31, 266-272 (1990)
[25] Shapere, A.; Wilczek, F., Geometry of self-propulsion at low Reynolds number, J. Fluid Mech., 198, 557-585 (1989) · Zbl 0674.76114
[26] A.L. Silvestre, On the unsteady self-propelled motion of a rigid body in a viscous liquid and on the attainability of steady symmetric self-propelled motions, J. Math. Fluid Mech., to appear; A.L. Silvestre, On the unsteady self-propelled motion of a rigid body in a viscous liquid and on the attainability of steady symmetric self-propelled motions, J. Math. Fluid Mech., to appear · Zbl 1022.35041
[27] Sleigh, M. A., The Biology of Cilia and Flagella (1962), Pergamon: Pergamon Oxford
[28] Tanabe, H., Equations of Evolution (1979), Pitman: Pitman London
[29] Taylor, G. I., Analysis of the swimming of microscopic organisms, Proc. Roy. Soc. London Ser. A, 209, 447-461 (1951) · Zbl 0043.40302
[30] G.I. Taylor, Low Reynolds number flow, Videotape, Encyclopaedia Britannica Educational Corporation; G.I. Taylor, Low Reynolds number flow, Videotape, Encyclopaedia Britannica Educational Corporation
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