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MiS Preprint
25/2001
Quasi-stationary approximation in the problem on a rotating ring
Vladislav V. Pukhnachev
Abstract
Considered is plane rotationally-symmetrical motion of viscous incompressible liquid by inertia in a ring, which boundaries are free. Corresponding initial boundary-value problem for the Navier-Stokes equations was studied in [1] and also earlier in [2], where the case of zero surface tension was considered.
The problem on a rotating ring represents a rich in content and at the same time simple enough object for grounding of approximate methods in theory of viscous flows with free boundaries. An asymptotics of the solution of this problem at large Reynolds numbers was built in [3] on the basis of the scheme suggested in [4]; the closeness of asymptotic and exact solutions was proved at finite time interval.
The analysis of quasi-stationary approximation in the problem on a rotational ring is the aim of the present work. The equations of quasi-stationary approximation for the general problem on motion of isolated volume of viscous incompressible capillary liquid were derived in [5] from the exact equations with the help of expansion by the small parameter of quasi-stationarity equal to the ratio of the Stokes time to the capillary one. The problem contains one more dimensionless parameter proportional to the modulus of conserving angular moment of liquid volume; this parameter can be also considered as a small one. In dependence on the relation between these parameters one can obtain three variants of limit problem: traditional and two new ones. Built in [5] is formal asymptotics of solutions of the problem arising at tending of the quasi-statinarity parameter to zero. The question of grounding of quasi-stationary approximation was open until recently. Presented in [6] is the ground for the traditional variant of limit problem. Unfortunately the standard model of quasi-stationary approximation is insipid in the problem on a rotational ring.
The problem mentioned above represents the first nontrivial example of such motion of viscous liquid that its topology can be changed with time - a ring turns into a disk in finite period of time in irreversible way [1] when the surface tension is large enough. The usefulness of quasi-stationary approximation for the description of the process of modification of topology of the flow domain in the considered problem in simple terms - by the analysis of the solutions of ordinary autonomous differential equation of the first order - is the remarkable peculiarity.
References (excerpt):[1] Lavrentyeva O.M. Limiting regimes of the rotating viscous ring motion Dinamika Sploshnoi Sredy, Novosibirsk, 1980. Vol. 44, pp. 15-34 (in Russian). [2] Bytev V.O. Unsteady motion of ring of viscous incompressible liquid with free boundaries Prikladnaya Mekhanika i Tekhnicheskaya Fizika, 1970. No 3, pp. 82-88 (in Russian). [3] Pukhnachov V.V. Nonclassical problems of the boundary layer theory Novosibirsk State University, Novosibirsk, 1979 (in Russian). [4] Batishev V.A., Srubshik L.S. On asymptotics of free boundary of liquid at vanishing viscosity Doklady Akademii Nauk SSSR, 1975. Vol. 222. No 4, pp. 782 - 785 (in Russian). [5] Pukhnachov V.V. Quasistationary approximation in problem on motion of isolated volume of viscous incompressible capillary liquid Prikladnaya Matematika i Mekhanika, 1998. Vol. 62. No 6, pp. 1002-1013 (in Russian). [6] Solonnikov V.A. On the justification of quasistationary approximation in the problem of motion of a viscous capillary drop Interfaces and Free Boundaries, 1999. Vol. 1. No. 2, pp. 125-174.
