Vibrational Flows of Viscous Incompressible Fluids for High Reinolds Numbers

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system’s guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. DOI: 10.23671/VNC.2019.2.32112 Vibrational Flows of Viscous Incompressible Fluids for High Reinolds Numbers Ilin, K. I. , Morgulis, A. B. Vladikavkaz Mathematical Journal 2019. Vol. 21. Issue 2. Abstract: The article presents the high-frequency asymptotics of the Navier-Stokes system, which describes the motion of a viscous incompressible fluid in the region bounded by a vibrating surface. The boundary conditions require the coincidence of the velocity vectors of the material particle of the fluid and the point of the boundary in which the particle is located. Consequently, the fluid is not allowed either to slip along the boundary (the no-slip condition) or to penetrate through it. It is assumed that the motion of the boundary surface is given and periodic in time, and the domain confined within it stays at rest on average but, generally speaking, can be changing its shape. The frequency of oscillations of the boundary tends to infinity, and the amplitude tends to zero, but the ratio of the amplitude to the Stokes's layer thickness remains of the order of unity. The main result is the explicit form of the equations and boundary conditions that determine the mean flow in the most general case, without special assumptions about the problem data. On this basis, a number of specific flows have been investigated, in particular, a flow in a circular pipe, caused by the normal vibration of its walls. Keywords: Navier-Stokes system, high-frequency asymptotic, vibration, mean flow, steady streaming Language: Russian Download the full text For citation: Ilin, K. I., Morgulis, A. B. Vibrational Flows of Viscous Incompressible Fluids for High Reinolds Numbers, Vladikavkaz Math. J., 2019, vol. 21, no. 2, pp. 5-17 (in Russian). DOI 10.23671/VNC.2019.2.32112 + References 1. Craik, A. D. D. and Leibovich, S. A Rational Model for Langmuir Circulations, J. Fluid Mech., 1976, vol. 73, no. 3, pp. 401-426. DOI: 10.1017/S0022112076001420. 2. Duck, P. W. and Smith, F. T. Steady Streaming Induced Between Oscillating Cylinders, J. Fluid Mech., 1979, vol. 91, no. 1, pp. 93-110. DOI: 10.1017/S0022112079000057. 3. Haddon, E. W. and Riley, N. The Steady Streaming Induced Between Oscillating Circular Cylinders, The Quarterly Journal of Mechanics and Applied Mathematics, 1979, vol. 32, no. 3, pp. 265-282. DOI: 10.1093/qjmam/32.3.265. 4. Gopinath, A. Steady Streaming due to Small-Amplitude Torsional Oscillations of a Sphere in a Viscous Fluid, The Quarterly Journal of Mechanics and Applied Mathematics, 1993, vol. 46, no. 3, pp. 501-520. DOI: 10.1093/qjmam/46.3.501. 5. Levenshtam, V. B. Asymptotic Expansion of the Solution of a Problem of Vibrational Convection, Computational Mathematics and Mathematical Physics, 2000, vol. 40, no. 9, pp. 1357-1365. 6. Vladimirov, V. A. Viscous Flows in a Half Space Caused by Tangential Vibrations on its Boundary, Studies in Applied Mathematics, 2008, vol. 121, no. 4, pp. 337-367. DOI: 10.1111/j.1467-9590.2008.00418.x. 7. Ilin, K. and Morgulis, A. On The Steady Streaming Induced By Vibrating Walls, SIAM Journal on Applied Mathematics, 2012, vol. 72, no. 5, pp. 1406-1427. DOI: 10.1137/110859634. 8. Riley, N. Steady Streaming, Annual Review of Fluid Mechanics, 2001, vol. 33, pp. 43-65. DOI: 10.1146/annurev.fluid.33.1.43. 9. Longuet-Higgins, M. S. Mass Transport in Water Waves, Philos. Trans. Roy. Soc. London. Series A. Mathematical and Physical Sciences, 1953, vol. 245, no. 903, pp. 535-581. 10. Longuet-Higgins, M. S. Peristaltic Pumping in Water Waves, J. Fluid Mech., 1983, vol. 137, pp. 393-407. DOI: 10.1017/S0022112083002475. 11. Hollerbach, R. et al. The Flow Around a Torsionally Oscillating Sphere, Physics of Fluids, 2002, vol. 14, no. 12, pp. 4192-4205. DOI: 10.1063/1.1518029. ← Contents of issue


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