# Governing equations for wave propagation in prestressed joined dissimilar elastic tubes containing fluid flow: with an example for a tapered section

Hart V.G. and Shi J. (1995) Governing equations for wave propagation in prestressed joined dissimilar elastic tubes containing fluid flow: with an example for a tapered section. International Journal of Engineering Science, 33 8: 1121-1138. doi:10.1016/0020-7225(94)00115-Z

Author Hart V.G.Shi J. Governing equations for wave propagation in prestressed joined dissimilar elastic tubes containing fluid flow: with an example for a tapered section International Journal of Engineering Science   Check publisher's open access policy 0020-7225 1995-01-01 Article (original research) 10.1016/0020-7225(94)00115-Z Not yet assessed 33 8 1121 1138 18 eng 2200 Engineering Understanding reflection and transmission of stress and deformation waves at a joint of dissimilar tubes joined longitudinally and containing pulsatile fluid flow is important in arterial surgery. In the present paper we first linearize the governing nonlinear equations for the motion of the fluid and the tube, together with the interaction conditions between the fluid and the tube wall as well as the joint conditions between the different tubes. The tube walls are assumed to be orthotropic elastic and thin compared with the radius of the tube; and the fluid is incompressible and Newtonian. The joint conditions hold for both fluid and tube wall. The linearization is done by a perturbation scheme for small motion, superposed on a finitely prestressed equilibrium state of the joined tubes, whose cross section varies along the axis. Due to this variation in cross section these equations, although linear, have very complicated coefficients and are difficult to solve. In general a numerical method is required to obtain a solution. However, if the mean pressure is zero or very small and/or these equations can be reduced to or approximated by simple form, then analytic wave form solutions can be found. The reflection and transmission have been investigated previously by the authors when the radius is constant along the whole joined tube, or along each tube but with an abrupt joint. In the second part of this paper we will seek analytic solutions for another simplified case: wave propagation along a slowly tapered prestressed tube and the reflection and transmission by a segment of slowly tapered tube. In the case of the tapered tube, the tube is restricted so that its axial displacement vanishes. It is found that the amplitude of the pressure wave increases or decreases along a tapered section according as the effect of tapering is greater or less respectively than the effect of viscosity. Also a change of mechanical properties of the tube can lead to an increase in pressure wave amplitude. C1 Provisional Code Unknown

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