Fluid wave motion in pipes of viscoelastic material. Stationary nonlinear waves
Steady blood flows in the small vessels can lose stability under conditions of realizing a dropping section on the curve of the pressure as a function of the radius. There are experiments verifying the existence of such a section . The viscoelastic properties of the blood vessel walls, which have been determined in tests  and predicted by rheological models of the muscle tissue (see , for instance), play a major part in oscillation of the perturbations. A linear analysis of the flow stability in vessels with S- and N-shaped static characteristics is given in . The formation of stationary waves of finite amplitude can be the result of the development of instability in nonlinear systems. Such motions are investigated below in an inertialess hydraulic approximation. Nonlinear model equations, relating the final nonstationary pressure perturbations and the deformations, are involved in the closure of the hydraulic description. A class of bounded solutions containing periodic waves and solitons is studied for S-type vessels. Existence conditions for the periodic solutions are determined in the case of the N-shaped characteristic; a description is given for the periodic waves in a harmonic approximation. It is detected that the lengths of bounded stationary waves in vessels of both types correspond to the linear instability domain.
KeywordsSoliton Periodic Wave Stationary Wave Viscoelastic Material Rheological Model
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