Unsteady motion of a spherical bubble in a complex fluid: Mathematical modelling and simulation

The nonlinear response of an oscillatory bubble in a complex fluid is studied. The bubble is immersed in a Newtonian liquid, which may have a dilute volume fraction of anisotropic additives such as fibers or few ppm of macromolecules. The constitutive equation for the fluid is based on a Maxwell model with an extensional viscosity for the viscous contribution. The model is considered new in the study of bubble dynamics in complex fluids. The numerical computation solves a system of three first order ordinary differential equations, including the one associated with the solution of the convolution integral, using a fifth order Runge-Kutta scheme with appropriated time steps. Asymptotic solutions of governing equation are developed for small values of the pressure forcing amplitude and for small values of the elastic parameter. A study of the bubble collapse radius is also presented. We compare the results predicted by our model with other model in the literature and a good agreement is observed. The calculated asymptotic solutions are also used to test the results of the numerical simulations. In addition, the orientation of the additives is considered. The angular probability density function is assumed to be a normal distribution. The results show that the model based on the fully aligned additives with the radial direction overestimates the tendency of the additives to stabilize the bubble motion, since the effect of extensional viscosity occurs due to the particle resistance to the movement throughout its longitudinal direction.