Numerical simulation of bubble growth in Newtonian and viscoelastic filaments undergoing stretching.

*(English)*Zbl 1143.76329Summary: Numerical results are presented concerning bubble growth in Newtonian and viscoelastic filaments undergoing stretching. In practice, such bubbles or cavities develop in materials (either in their bulk or at their interface with a substrate) such as the pressure sensitive adhesives. The problem we address here is how such an initially spherical bubble deforms inside a filament (which at the beginning has the shape of a cylinder with uniform radius) undergoing stretching with a constant pulling velocity. The lower end of the liquid filament is fixed at the substrate; stretching along the filament axis is achieved by means of an imposed force on the upper end. The governing equations consist of the momentum, continuity and constitutive equations and the free surface boundary conditions at the two interfaces (the bubble-liquid and the liquid-air interfaces). These are solved by a finite element/Galerkin method coupled with a first-order implicit Euler scheme for time integration. In the case of a viscoelastic medium (here the Phan-Thien/Tanner constitutive model is chosen), the elastic viscous stress splitting-G (EVSS-G) technique is used to separate the elastic and viscous contributions to the stress tensor together with a streamline upwind Petrov-Galerkin (SUPG) discretization of the constitutive equation. Our numerical calculations provide information on the effect of a number of important parameters on bubble and filament growth rate and deformation. These include mainly the capillary and Deborah numbers. The effect of other variables such as the relative size of the bubble, the proximity of the bubble to and liquid slippage on the substrate are also studied and discussed in detail.

##### MSC:

76T10 | Liquid-gas two-phase flows, bubbly flows |

76A10 | Viscoelastic fluids |

76M10 | Finite element methods applied to problems in fluid mechanics |