Simulation of the Deformation of Non-Newtonian Drops in a Viscous Flow

In this thesis we describe the development of a boundary integral method for the simulation of non-Newtonian drops and vesicles subjected to a viscous flow. A vesicle consists of a viscous drop encapsulated by a lipid bilayer and is modelled as a two-layer drop, of which the outer layer is viscoelastic. As the typical Reynolds number of the flow problem considered is very low, we can describe flow fields inside and outside the drop by the Stokes equations. In order to account for the non-Newtonian character of the fluids in the drop, we incorporate a non-Newtonian stress tensor in the Stokes equations. The time evolution of this stress tensor is describe by a characteristic non-Newtonian model, such as the Oldroyd-B model. By treating the contribution of the non-Newtonian stress tensor to the flow field as a source term, we can derive an integral formulation for the velocity field. However, a disadvantage of this approach is that the source term leads to a domain integral which increases the computational effort substantially compared to Newtonian fluids. The advantage of the method over a more direct (finite difference) discretization of the Stokes equations lies in the fact that only the relatively small non-Newtonian domains of the drop have to be discretized. Other advantages are the flexible and accurate modelling of the boundary shapes including the behavior at infinity and the fact that the incompressibility of the fluids is guaranteed. Besides the development of a general integral expression for the velocity field, we also considered two specific situations: a single drop in an axisymmetric flow field, and a periodic dispersion of Newtonian drops. In the first case we can transform the integral equation to cylindrical coordinates. By performing the integration over the azimuthal angle analytically we may reduce the dimension of the computational problem. For the second problem it turns out that the periodicity of the problem can be simply included in the Green's functions, whereas the boundary integral formulation remained of the same form as for single non-Newtonian drops. In this thesis numerical simulations of (non-)Newtonian drops and vesicles subjected to an axisymmetrical flow and of a two-dimensional periodic dispersion of Newtonian drops are described and analyzed. For the actual simulation several numerical techniques are used which are described in detail. The developed methods were tested thoroughly using a numerical and physical validation procedure. This validation showed that most of the numerical methods are at least second order accurate and require only a small amount of points to yield accurate results. From the validation process for a periodic dispersion of drops it was also found that only a few drops are needed for the generation of reliable macroscopic properties. This result is very promising with respect to the computational costs if one wishes to simulate a dispersion of non-Newtonian drops. The simulation results of non-Newtonian drops indicate that a fluid-like non-Newtonian material mainly influences the deformation process, and to a smaller extent the final shape of the drop. For a solid rubber-like material, however, the final shape of the drop is mainly determined by the elasticity of the material. For vesicles we concentrated on the identification of a characteristic breakup mechanism of the outer layer. From the simulations it appeared that a breakup criterion based on a critical stress in the outer rubber-like layer of the vesicle, is related to the most likely mechanism. Besides in non-Newtonian drops and vesicles, we also found strong non-Newtonian behavior in a dispersion of Newtonian drops. This behavior expresses itself in an effective viscosity which depends on the applied velocity field.