Quasi-stationary approximation in the problem on a rotating ring
Vladislav V. Pukhnachev
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Submission date: 25. May. 2001
published in: Siberian mathematical journal, 43 (2002) 3, p. 525- 548
DOI number (of the published article): 10.1023/A:1015423904785
with the following different title: Quasistationary approximation in the rotating ring problem
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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  and also earlier in , 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  on the basis of the scheme suggested in ; 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  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  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  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  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.
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