Computational study of multiphase viscoelastic flows

Drop dynamics in the presence of viscoelasticity is an inherently non-linear, moving boundary problem rich in mathematical complexity and fascinating phenomena. Drops in an emulsion, deform, relax, coalesce, migrate and break, modifying its microstructure and thereby changing its effective properties. Viscoelastic drop dynamics is fundamental from both theoretical and practical viewpoints, having applications in fields as diverse as emulsions, polymer processing, biological motion, filtration, and enhanced oil recovery. In this thesis, we computationally study the behavior of drops and how viscoelasticity modifies them. A front tracking finite difference method adapted for viscoelastic fluids is used to numerically study multiphase flows. Viscoelasticity is modeled using rate-type equations (Oldroyd-B and FENE-CR). We have investigated four different phenomena: deformation, retraction, sedimentation, and lateral migration.;We first investigate the effects of viscosity ratio on a viscoelastic drop in a shear flow of viscous liquid. We find that for low drop-to-matrix viscosity ratio, drop deformation decreases with increasing Deborah number, while in higher viscosity ratio systems, the drop response is non-monotonic—the steady drop deformation first decreases with increasing Deborah number but above a critical Deborah number, it increases. Next we study the effects of drop and matrix viscoelasticity on the retraction of a sheared drop. Retraction of an Oldroyd-B drop in a Newtonian matrix is initially faster and later slower with increasing drop viscoelasticity. This observed behavior is explained using an ordinary differential equation model representing the dominant balance between various forces during retraction. The initial faster relaxation of viscoelastic drops is due to viscoelastic stresses pulling the drop interface at the tips inward. The later slower retraction is due to the slowly-relaxing viscoelastic forces at the equator, where they act against the capillary force. Matrix viscoelasticity slows the relaxation of a Newtonian drop because of the increasingly slow relaxation of highly stretched polymers near the drop tip with increasing Deborah number.;We further study the deformation and sedimentation velocities of a viscoelastic drop falling through a Newtonian medium. We found that viscoelasticity deforms in a unique way, resulting in a variety of shapes. An approximate analysis is performed to model the stress development responsible for all of the shape transition (spherical-oblate-dimple-toroidal).;Finally, we investigate the viscoelastic effects on the dynamics of a drop deforming, orienting and laterally migrating in a shear flow near a wall. The matrix elasticity reduces the migration velocity, the reduction scales approximately linearly with viscoelasticity. Two mechanisms for the viscoelastic effect on drop migration are identified: the first one stems from the normal stress differences from the curved streamlines around the drop and the second mechanism results from the reduced inclination angle of the drop with increasing viscoelasticity. We conducted an approximate perturbative analysis to show that the effect is due to viscoelasticity modifying the wall induced image terms in the stresslet field. For highly viscous Newtonian drops, the lateral velocities are comparatively smaller in magnitude and the addition of viscoelasticity in the matrix drastically changes the velocities and the direction of motion. The drop migrates in the opposite direction; towards the wall depending on the amount of viscoelasticity. Finally, we observed that a viscoelastic drop in a Newtonian matrix follows a non-monotonic velocity trend, due to two competing factors: the interfacial and the non-Newtonian stresses.