Estimates of the Evolucion of Small Perturbations by Radial Spreading (Drain) of a Viscous Ring
D. V. Georgievskii, G. S. Tlyustangelov
Lomonosov Moscow State University, Moscow, 119991, Russia
Keywords: растекание, сток, вязкая жидкость, возмущение, метод интегральных соотношений, неравенства Фридрихса, оценки устойчивости, невязкий предел, spreading, drain, viscous fluid, perturbation, method of integral relations, Friedrichs inequalities, stability estimates, inviscid limit
Abstract
The evolution of small perturbations of the kinematic and dynamic characteristics of the radial flow of a flat ring filled with a homogeneous Newtonian fluid or an ideal incompressible fluid is studied. When the flow rate is specified as a function of time, the basic motion is completely defined by the incompressibility condition regardless of the properties of the medium. For the streamfunction, we obtained a biparabolic equation with four homogeneous boundary conditions, which simulate adherence to the expanding (narrowing) walls of the ring. Upper-bound estimates of the perturbation are obtained using the method of integral relations for quadratic functionals. The case of exponential decay of initial perturbations is considered on a finite or infinite time interval. Justified The admissibility of the inviscid limit in the given problem is substantiated, and and both upper- and lower-bound estimates for this limit are obtained.
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