Pressure waves in a liquid suspension with solid particles and gas bubbles

Abstract

532.529The propagation of pressure disturbances in a suspension of a liquid and solid particles is a subject that has been studied quite extensively. The theory of multiple scattering was used in [1, 2] to obtain expressions for the velocity and attenuation factor of an acoustic wave, and the results were compared with experimental data. Good agreement between theoretical and experimental data on the speed and attenuation of sound in suspensions was obtained in [3] on the basis of the Biot model for the propagation of sound in saturated porous media. The author of [4] presented averaged equations of the mechanics of disperse media that make it possible to examine the evolution of waves in two-phase mixtures. The results of a series of theoretical and experimental studies on wave dynamics in gas-liquid media were presented in [5], and the propagation of waves in porous media saturated with a gas-liquid mixture was examined in [6].Our goal here is to experimentally study the evolution and structure of pressure waves of moderate intensity in a suspension of a liquid with solid particles and gas bubbles. We also want to generalize the empirical data on the basis of a theoretical analysis.We will examine the propagation of unidimensional pressure disturbances in a liquid containing suspended solid spheres and gas bubbles. We assume that the length of the waves associated with the disturbances is much greater on the dimensions of the spheres, the dimensions of the bubbles, and the distances between them. We represent a liquid with gas bubbles as a homogeneous medium having the mean density Pm, pressure p, and velocity v m. For a disperse medium (solid spheres + homogeneous gas-liquid medium), the system of equations describing the propagation of unidimensional pressure perturbations has the form [4] 8t~O m O(t~OmVni~ O(l --m)p 1 0(1 --re)pit.' 1 + =0, + -0, Ot ax at ax aom aol at, F, (1) ,~mp -~--(,~ -1) mpm~ = -.l -~dvm ~1 -(a --1) mPm~t + ((1 -m)p t + (a -1) rnPm) --~-+ F, Pt = const, = --(1 m) oxwhere Pm = P2( 1 -~P) + P3~ o; m is the porosity of the medium; ~o is the volumetric gas content of the liquid (thus, the quantity ~om will correspond to the volumetric gas content in the three-phase medium at low values of ~); c~ is a coefficient expressing the apparent additional mass of the liquid. The subscripts 1, 2, and 3 pertain to the solid, liquid, and gas phases, m pertains to the gas-liquid mixture, and 0 denotes the initial state of a phase. We represent the interfacial force F~ as [4] s cpm(lm) (v m-vl)' F=-4 dHere, d is the diameter of the solid spheres; c~ is the resistance coefficient, determined experimentally. At low relative velocities, the interfacial force depends linearly on relative velocity:Novosibirsk.