Oscillations of Visco-Elastic and Relaxing Media

Abstract

As we mentioned in the Introduction to Part IV, this chapter deals with the problem on small oscillations of a visco-elastic or relaxing fluid. In Section 11.1 we will consider the most basic problem: The oscilations of a visco-elastic fluid in an arbitrary completely filled container. Based on the connection between the tensors of viscous stresses and deformation velocities, we formulate an initial boundary-value problem. This problem leads to a differential equation of the first order in a Hilbert space. Using the dissipativity of the operator coefficient of this equation we prove the theorem on correct solvability of the initial problem. Further, we study the normal oscillations of the system, prove that the spectrum is discrete, and show the existence of its limit points, situated on the positive semiaxis and caused by the presence of visco-elastic forces.