Interaction of an ultrasonic wave with a bubbly mixture

The interaction of an ultrasonic wave with a bubbly two-phase flow is studied both experimentally and theoretically. Brief theoretical reviews of acoustic wave generation by a piston and of the interaction of a plane wave with a single bubble are given. A theory relating ultrasonic wave transmission through a bubbly flow with two-phase flow parameters, notably the bubble size and the volumetric interfacial area, is derived and compared with preliminary data. The theoretical and experimental limitations concerning the application of ultrasonic transmission measurements to the study of bubbly flow are discussed in detail, and recommendations for future work in this area are made.Nomenclature A projected cross-sectional area of a bubble - a bubble radius - a_T emitter radius - c speed of sound - d equivalent bubble diameter - I intensity - I₀ incident intensity - J₀ Bessel function of zero order - J₀ Bessel function of first order - j₁ spherical Bessel function of the first kind and of order l - k wave number - n_l spherical Bessel function of the second kind (Nishi's notation) - n number of bubbles per unit volume or per unit area - P₀ pressure amplitude at the emitter; equilibrium pressure in the liquid - p pressure perturbation - r spherical coordinate, radial distance to the x-axis - S total scattering cross-sectional area, surface of the piston - S_i scattering cross-sectional area of the i-th bubble - T transmittance - t time - U complex source strength divided by the source area - W ratio of radial distance from the axis on the emitter surface to the radius of the emitter - x axis coinciding with the direction of propagation of the plane wave; distance between the transducers - void fraction - interfacial area per unit volume - spherical coordinate - ultrasonic wavelength - density - angular frequency