Answer Happy

Consider a spherical bubble in an infinite incompressible liquid subject to a standing wave with a low amplitude as follows: p(z,t) = -8B sin (kz) sin(ax), 0<<<<1, B =0(1). (1) The linear solution for the bubble radius R) is given by R(t)=R. (1+ex), (2a) x= A, sin(ax +). (26) B sin(kz) 1 A (2c) PR Va-w*)* +(2700) ø=arga1 - 0-i207), (20) 2u 1 y = 0,= 3KP 20 R (2e) PR | PR where Ro is the equilibrium bubble radius, Pgo the equilibrium pressure, k the polytropic index of the bubble gas, p the liquid density, u the liquid viscosity, o the surface tension and ze the position coordinate of the bubble centre. (a) Derive the formula for the Bjerknes force on the bubble surface due to the pressure of the standing wave. (b) Calculate the average Bjerknes force over an oscillation period of the wave. (c) Derive the simplified result for the average Bjerknes force for my <<\w; -w'). (d) Using the result of (c), determine how bubbles with Ro = 3um and 4 um migrate in a standing wave with the frequency IMHz, respectively. Other parameters are p = 1000 kgm", po = 100 kPa, pı = 0, x = 1.4, and a = 0.07Nm"!