| Abstract: | The deformation stability relative to small perturbations is analyzed for weakly inhomogeneous viscous media on the assumption
that both the main flow and perturbation field are three-dimensional. To test the damping or growth of initial perturbations,
sufficient estimates based on the use of variational inequalities in different function spaces (energy estimates) are obtained.
The choice of function space determines the measures of the parameter deviations, which may be different for the initial and
current parameters. The unperturbed process chosen is a fairly arbitrary unsteady flow of homogeneous incompressible viscous
fluid in a three-dimensional region of Eulerian space. At the initial instant, not only the kinematics of the motion but also
the density and viscosity of the fluid are disturbed and the medium is therefore called weakly inhomogeneous.
On the basis of the integral relation methods developed in recent years, sufficient integral estimates are obtained for lack
of perturbation growth in the mean-square sense (in theL
2 space measure). The rate of growth or damping of the kinematic perturbations depends linearly on the initial variations of
the kinematics, density and viscosity. Illustrations of the general result are given.
Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 56–67, March–April, 2000.
The work was supported by the Russian Foundation for Basic Research (projects No. 99-01-00125 and No. 99-01-00250) and by
the Federal Special “Integration” Program (project No. 426). |
