Viscous flows in a half space caused by tangential vibrations on its boundary

Studies in Applied Mathematics

Volumn 121, Issue 4, 2008, Pages 337-367

  Vladimirov, V A ^a  

^a UNIVERSITY OF YORK   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; BOUNDARY LAYERS; GEOMETRY; NAVIER STOKES EQUATIONS; VIBRATIONS (MECHANICAL);

AVERAGING METHOD; BOUNDARY LAYER THEORY; BOUNDARY VELOCITY; FLUID DOMAIN; GLOBAL SOLUTIONS; HALF SPACES; SIMPLE++; TWO-TIMING METHODS; VELOCITY FIELD; VIBRATING BOUNDARIES;

VISCOUS FLOW;

EID: 55649117297     PISSN: 00222526     EISSN: 14679590     Source Type: Journal    
DOI: 10.1111/j.1467-9590.2008.00418.x     Document Type: Article

References (45)

1. H. Schlichting, Boundary Layer Theory, McGraw-Hill book company, New York, 1960.
2. G. K. Batchelor, An Introduction to Fluid Dynamics, CUP, Cambridge, 1967.
3. J. T. Stuart, Double boundary layers in oscillatory viscous flow, Fluid Mech. 24 : 673 687 (1966).
4. J. T. Stuart, Unsteady boundary layers, in Laminar Boundary Layers, pp. 349 408, Clarendon Press, Oxford, 1963.
5. N. Riley, Steady streaming, Ann. Rev. Fluid Mech. 33 : 43 65 (2001).
6. T. von Karman, The engineer grapples with non-linear problems, Bull. Amer. Math. Soc. 46 : 615 683 (1940).
7. M. I. Vishik and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter (in Russian), Uspehi Mat. Nauk 12 : 3 122 (1957 Am. Math. Soc. Transl. ser. 2, 20 : 239 364 (1962).
8. V. A. Trenogin, Development and applications of Vishik-Lyusternik's asymptotic method (in Russian), Uspekhi Math. Nauk 25 : 123 156 (1970).
9. W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Interscience Publishers, New York, 1965.
10. A. H. Nayfeh, Perturbation Methods, John Wiley & Sons, New York, 1973.
11. D. R. Smith, Singular-Perturbation Theory. An Introduction with Applications, CUP, Cambridge, 1985.
12. L. A. Chudov, On some shortfalls of the classical boundary layer theory (in Russian), Comp. Methods Gas Dyn. 2 : 98 109 (1963).
13. I. Yu. Brailovskaja and L. A. Chudov, Two methods of construction of higher approximations with respect to a small viscosity parameter (in Russian), Comp. Methods Program. 8 : 173 185 (1967).
14. V. P. Stulov, On the laminar boundary layer equations in a two-phase medium (in Russian), Izv. Acad. Nauk SSSR Mech. Zhidkosti i Gasa 1 : 21 25 (1979).
15. V. I. Yudovich, On the asymptotic integration of the Navier-Stokes equations for a fluid under impact forces (in Russian), Izvestiya VUZov Severo-Kavkazskii region: Ser. Nat. Sci. Special issue : 126 132 (2004).
16. M. Argentina, J. Skotheim, and L. Mahadevan, Settling and swimming of flexible fluid-lubricated foils, Phys. Rev. Lett. 99 : 224503 (2007).
17. A. E. Hosoi and L. Mahadevan, Peeling, healing, and bursting in a lubricated elastic sheet, Phys. Rev. Lett. 93 : 137802 (2004).
18. S. Childress, Mechanics of Swimming and Flying, CUP, Cambridge, 1981.
19. S. Rosenblat, Torsional oscillations of a plane in a viscous fluid, J. Fluid Mech. 6 : 206 220 (1958).
20. D. J. Benney, The flow induced by a disk oscillating in its own plane, J. Fluid Mech. 18 : 385 391 (1964).
21. D. J. Benney, The flow induced by a disk oscillating about a state of steady rotation, Mech. Appl. Math. XVIII : 333 345 (1965).
22. P. G. Drazin and N. Riley, The Navier-Stokes Equations: A Classification of Flows and Exact Solutions, London Mathematical Society Lecture Note Series No. 334, CUP, Cambridge, 2006.
23. C. Y. Wang, Exact solutions of the steady-state Navier-Stokes equations, Ann. Rev. Fluid Mech. 33 : 43 65 (1991).
24. C. Y. Wang, Effect of spreading of material on the surface of a fluid - an exact solution, Int. J. Non-Linear Mech. 6 : 255 262 (1971).
25. C. Y. Wang. The three-dimensional flow due to a stretching flat surface, Phys. Fluids 27 : 1915 1917 (1984).
26. H. P. Greenspan, The Theory of Rotating Fluids, CUP, Cambridge, 1968.
27. V. A. Vladimirov, On vibrodynamics of pendulum and submerged solid, J. Math. Fluid Dyn. 7 : S226 S259 (2005).
28. V. I. Yudovich, Vibrodynamics and vibrogeometry of mechanical systems with constrains, Parts 1-3 (in Russian), Uspehi Mekhaniki 4 : 3 158 (2006).
29. A. N. Tikhonov and A. A. Samarsky, Equations of Mathematical Physics, Pergamon Press, London, 1963.
30. G. G. Stokes, On the effect of the internal friction of fluids on the motion of pendulums, Trans. Camb. Phil. Soc. 9 : 8 106 (1851).
31. F. S. Sherman, Viscous Flow, McGraw-Hill book company, New York, 1990.
32. W. H. Lyne, Unsteady viscous flow over a wavy wall, J. Fluid Mech. 50 : 33 48 (1971).
33. F. Carlsson, M. Sen, and L. Lofdahl, Fluid mixing induced by vibrating walls, European J. Mech. B/Fluids 24 : 366 378 (2005).
34. F. Carlsson, M. Sen, and L. Lofdahl, Fluid mixing due to vibrating walls, Phys. Fluids 16 : 1822 1825 (2004).
35. C. Atkinson and M. Manrique de Lara, Fluid reaction on a vibrating disc in a viscous medium, Int. J. Eng. Sci. 44 : 973 995 (2006).
36. A. Gopinath, Steady streaming due to small-amplitude torsional oscillations of a sphere in a viscous fluid, Q. J. Mech. appl. Math. 46 : 504 520 (1993).
37. G. K. Batchelor, On steady laminar flow with closed streamlines at large Reynolds number, Fluid Mech. 1 : 177 190 (1956).
38. H. K. Moffatt, Microhydrodynamics, Ecole Polytechnique, Palaiseau, 1996.
39. V. A. Vladimirov and V. F. Tarasov, Structure of flow of a viscous liquid with closed streamlines, Sov. Phys. Dokl, 27, 1 : 17 21 (1982).
40. T. J. Pedley, The Fluid Mechanics of Large Blood Vessels, CUP, Cambridge, 1980.
41. P. W. Carpenter and T. J. Pedley (Eds. Flows past Highly Compliant Boundaries and in Collapsible Tubes, Kluwer Academic Publishers, Dodrecht, 2001.
42. G. G. Stokes, On the theory of oscillatory waves, Trans. Camb. Phil. Soc., 8 : 314 326 (1847 Reprinted in Math. Phys. Pap. 1:314-26.
43. M. S. Longuet-Higgins, Mass transport in water waves, Phil. Trans. A 245 : 535 581 (1953).
44. M. S. Longuet-Higgins, Mass transport in the boundary layer at a free oscillating surface, J. Fluid Mech. 8 : 293 306 (1960).
45. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Prentice-Hall Inc., London, 1965.