VIII-5

A. Crespo, J. García and J. Jiménez

Universidad Politécnica de Madrid

Abstract

The velocity of propagation of a small perturbation in a bubbly mixture within a tube that has an elastic wall is calculated. It is found that the effect of the wall elasticity can decrease significantly the wave velocity if the volume fraction of gas is small enough. Finite non-linear transitions between uniform states are considered; the jump conditions across the shock wave are established and the stability and nature of the possible solutions is discussed according to the subsonic or supersonic conditions of the flow in a reference frame fixed to the wave. The transition across the shock wave is studied using a continuum model, coupled with the Rayleigh-Plesset equation to describe the dynamics of the bubbles.

The same model is also considered to study the case in which the undeformed tube is not of constant section but has a constriction, similar to a converging-diverging nozzle. The variation of the cross-section is assumed to be slow enough to consider the problem quasi one-dimensional, also steady state conditions are considered. In hemodynamic flows a similar problem is found when studying stenosis, although in this case no bubbles are present. A similar problem has also been studied recently by Wang and Brennen1 to describe the cavitating flow through a converging diverging nozzle, although they considered rigid walls. Quasi-analytical solutions describing the transition can be obtained in the limit of small value of the gas volume fraction and high elastic modulus; in order to predict cavitation onset, this requires a small value of the cavitation number. Although this may not be realistic in many situations it can be useful to predict tendencies. It is found that the elasticity of the wall favors the appearance of cavitation.

1. Wang, Y, C. and Brennen, C. E. "One-Dimensional bubbly cavitating flows through a converging-diverging nozzle". Journal of Fluids Engineering, ASME, vol. 120, March, 1998, pp. 166-170.