SHOCK WAVES IN FLOW OF AN INCOMPRESSIBLE LIQUID IN COLLAPSING PIPES: APPLICATION TO LARGE BLOOD VESSELS

8. L. I. Sedov, Mechanics of a Continuous Medium, Vol. i [in Russian], Nauka, Moscow (1983).
SHOCK WAVES IN FLOW OF AN INCOMPRESSIBLE LIQUID IN COLLAPSING
PIPES: APPLICATION TO LARGE BLOOD VESSELS
Yu. Z. Saakyan UDC 532.542
Flow of a liquid in collapsing pipes is of great interest for problems in the
mechanics of blood circulation, since collapse can take place in many blood
vessels. This effect forms the basis for a large number of diagnostic and
therapeutic methods, and also for methods of investigating the system of
blood circulation. Consequently the mechanics of collapsing pipes has been
studied intensively of late [I], but the available studies are far from
exhausting the theoretical or the applied aspects of the problem. This applies
also to the study of discontinuous solutions such as shock waves which
describe steep fronts of opening or narrowing of a blood vessel~ The most
studied phenomenon is unsteady flow caused by change in the external pressure
[2]. There is an explanation in [3-6] of the effect on the process of
formation of discontinuities in collapsing pipes due to such factors as
friction on the wall, distributed lateral outflow, the presence of a stagnant
zone in the flow, and viscoelasticity of the wall. The origin of some acoustic
phenomena in the arteries is connected by some with the propagation of discontinuities;
these phenomena include Korotkov sounds, used in the determination
of the arterial pressure of blood [i, 7]~ The present study considers quasione-
dimensional flow of a viscous incompressible liquid in a collapsing pipe
of finite length and made of a nonlinear viscoelastic material; there is a
study of the conditions in which discontinuities arise in such systems, and
an investigation of the structure of shock waves with allowance for the
effect of the surrounding tissues.
i. The equations of motion of a
cross section area A and perimeter F
liquid with constant density p along a pipe with
will be written in the form
aA+_a (1.1)
at Ox
Ou Oa I Op
P "~'+u'-~x/+--=-qDox (1o2)
Here x is the axial coordinate (0 ~ x ~ L, L is the length of the pipe), ~=(F~JA);
u, p are the mean axial velocity and pressure over the cross section, ~ is the specific
outflow of the liquid across the lateral surface of the pipe, and ~w is the mean viscous
tangential stress on it over the perimeter.
We assume, in accordance with [8], that the dependence of ~ on the parameters of the
problem is of the form ~ = a(p -- Pc) + 7uA, where Pc is the pressure at the outlet of the
pipe, and ~ > 0 and 7 > 0 are constant coefficients. For ~w we postulate a linear dependence
on u, corresponding to an instantaneous Poiseuille velocity profile for the flow
of a liquid with a dynamic coefficient of viscosity ~ along a pipe of elliptical cross
section: q=4xB(6+|/6)(u/A). 6(F, A) being the ratio of the minor and major axes of the
ellipse.
Equations (iol) and (1.2) must be supplemented by a relationship connecting the pressure
p with the cross section area A and reflecting the theological properties and nature
of the motion of the pipe and the medium surrounding it. We shall assume that the pipe
is located in an incompressible anisotropic continuous medium of density Pm, occupying
in the nondeformed state the area between coaxial cylinders of radii R 0 and r 0 (r 0 is at
the same time the nondeformed radius of the mean surface of the shell modeling the pipe).
We assume, moreover, that on a certain segment [i < x < s an external pressure pe(X) is
Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza,
No. 6, pp. 44,50, November-December, i987. Original article submitted March 6, 1987.
862 0015-4628/87/2206-0862512.50 9 1988 Plenum Publishing CorpOration