One-dimensional waves in liquids containing bubbles flowing along a tube with an elastic wall.

The velocity of propagation of a small perturbation in a bubbly mixture within a tube that has an elastic deformable wall is calculated; the wall elasticity can decrease significantly the wave velocity. 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. 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. A similar problem has also been studied recently by Wang and Brennen [1] to describe the cavitating flow through a converging diverging nozzle. Quasi-analytical solutions describing the transition can be obtained in the limit of small value of the gas volume fraction and high elastic modulus. It is found that the elasticity of the wall favors the appearance of cavitation.