Wave motions of a liquid in tubes of viscoelastic material. Waves of small amplitude

Abstract

The pliability of the walls of blood vessels is one of the essential factors determining the movement of the blood. Although there is an extensive literature devoted to this problem (see, for example, the collection [1] and the bibliography it contains), little attention has been devoted to the physiologically important specific effects related to the presence of muscular layers in the vascular wall. These effects may be roughly characterized as a significant variation in the rheological properties of the wall resulting from the activity of the vascular musculature. The latter, in turn, is controlled not only by external, nonmechanical factors but also by the stressed state of the wall itself (see, for example, [2]). Experiments indicate that in the vascular walls there are time-dependent effects characteristic of viscoelastic bodies [1] and a complex static “pressure-radius” characteristic, which may have a decreasing segment [3, 4]. Both of these facts can be reasonably interpreted within the framework of the general continual model for muscle tissue presented in [5]. In what follows, we shall study in a quasi-one-dimensional approximation the combined effect of the above-mentioned properties of the wall material on the behavior of small disturbances imposed on the stationary flow in the tube. We consider the possible types of instabilities. It is assumed that the inertial effects are insignificant and that the rheological law applicable to the wall is a nonlinear generalization of the Poynting model.