Estimates of the evolucion of small perturbations by radial spreading (drain) of a viscous ring

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.