SHOCK WAVES IN LIQUID WITH SOLUBLE GAS BUBBLES

Abstract

ABSTRACT : In this thesis we present a study of growth of spherical soluble gas bubble in liquid controlled by diffusion by using basic equations that describing the behaviour of a soluble gas bubble in a liquid. And we find analytical self-similar solution by transformation of partial differential diffusion equation to ordinary differential equation. This is possible when RR= constant, where R is the bubble radius and R is the time derivative of the radius. Also we find expressions for the bubble growth, the parameter h which control-ling the radial motion, and curves are presented for parameter h and Co-Cs P P g for different values of non-dimensional parameter ɛ = PL Where C is the mass concentration of gas in the liquid, the subscript o and s refers to the value at remote from the bubble and saturation respectively. p is the density, the subscript g and L refers to the parameters of gas and liquid. Structure of stationary shock waves in a liquid with soluble gas bubbles are dis-cussed by using the equations for a stationary one-dimensional flow. And we find the solution for shock-wave structure near critical point by investigating the solution behaviour before the shock wave. To investigate the asymptotic beha-viour of the solution of system of equations for a one dimensional stationary flow of liquid with soluble gas bubbles in the neighbourhood of the initial equi-librium state, these equations need to be linearized with respect to the values of the parameters at the point 0. Their solution is sought in the form of an exponential damped as x→∞ Ф-Фо (1+A(4) expkox), Re ko After linearization we obtain from system of equations for a one-dimensional stationary flow of liquid with soluble gas bubbles a system of linear homogene-ous equations in the amplitudes A(Ф). 0 0 The condition of the existence of non-trivial solution to this system leads to an algebraic equation of sixth order with respect to K=kV, where ko is ex-ponention coefficient and V is the dimensionless wave velocity. It has been proved that the desired solution exists and is unique. This proves that in the pres-ence of diffusion there exists unique solution of (*), and the required root ex-ists only if v² > C², where vo is the wave velocity and C is the sound speed in bubbly liquid. 0 0 The states before (x→ +∞) and behind the shock wave (x→∞) are equilib-rium ones, which is why an analysis of the asymptotic behaviour at X +∞ is necessary to get away from the critical point, The integral curve is connecting these two states and determins the structure of the steady shock wave.