Mathematics Colloquia and Seminars

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Description

A gel is a polymer network permeated with a fluid solvent. The rheology and dynamics of these complex materials can change dramatically in response to temperature, stress, and chemical stimulus. Because of their adaptivity, gels are important in many biological systems, e.g. gels make up the cytoskeleton and cytoplasm of cells and the mucus in the respiratory and digestive systems, and they are involved in the formation of blood clots. In examples such as these, gels are not adequately described as a single continuous medium because polymer and solvent move with distinct velocities, which results in relative motion between the two materials. In this talk we discuss a mathematical model for gels that takes into account this relative motion. The model treats the gel as a two-fluid system where the network is modeled as a viscoelastic fluid and the solvent is modeled as a viscous fluid. The dynamics are then governed by a coupled system of time-dependent partial differential equations which consist of transport equations for the volume fractions of the two fluids, transport equations for the viscoelastic stresses, two coupled momentum equations for the velocity fields of the two fluids, and a volume-averaged incompressibility constraint. We discuss the details of an efficient computational method for simulating this system of equations and present results illustrating its accuracy and robustness for several complicated model problems.