Bubble dynamics in a maxwell fluid with extensional viscosity

Abstract

Abstract In this paper we investigate the motion of a single oscillating bubble immersed in a viscoelastic fluid, undergoing non-linear pulsations. The ambient fluid is composed of a Newtonian liquid and a dilute volume fraction of additives as few ppm of polymer solution. The constitutive equation for the fluid has been based on a Maxwell model with an extensional viscosity for the viscous contribution. This approach results in a modified version of the classical Rayleigh–Plesset equation of bubble dynamics that might be integrated by using a fifth order Runge–Kutta scheme with an adaptive time step. While the extensional viscosity is related to the strong anisotropy produced in the flow by the macromolecule stretching, the elastic contribution accounts for the relaxation time of the additive. The elastic stress tensor is written in terms of a convolution or memory integral, solved by means of an extra ordinary differential equation to the bubble dynamic governing equation. We present results for the nonlinear oscillatory motion of a spherical bubble as a function of the elastic parameter (i.e. Deborah number) and the anisotropic parameters. The results show that a degree of elasticity in the ambient liquid for the linear viscoelastic model examined here, increases the degree of instability in the oscillatory motion of the bubble. This is in contrast with the bubble response in the presence of an anisotropic effect only (i.e. an extensional viscosity), representing the tendency of additives like rigid fibers aligned with the flow direction.