Interfacial flows in complex fluids are an important subject, scientifically rich and technologically important. The main scientific attraction comes from the fact that the microstructure of the bulk fluids may evolve during interfacial flow, and thereby generating non-Newtonian stresses that act on the interface. Thus, interfacial motion and conformation of the microstructure are coupled. Such flow situations arise in many industrial applications, including processing of polymer blends, foaming, and emulsification. In this thesis, I describe three projects aimed at exploring interfacial dynamics of viscoelastic polymeric liquids. The first project consists of finite-element simulations of drop deformation in converging flows in an axisymmetric conical geometry. The moving interface is captured using a diffuse-interface model and accurate interfacial resolution is ensured by adaptive refinement of the grid. The drop experiences a predominantly elongational flow. The amount of deformation sustained by the drop depends, besides the geometry and kinematics of the flow, on the rheology of both the drop and the matrix fluids. The second and third projects concern the same process of selective withdrawal, in which stratified layers of immiscible fluids are withdrawn from a tube placed a certain distance from the interface. We have chosen to work with an air-liquid system, with the suction tube embedded in the Newtonian or viscoelastic liquid. The second project is an experimental study, where we used video recording and imaging processing to analyze how the interfacial deformation is influenced by the non-Newtonian rheology of the liquid. We discover three regimes, subcritical, critical and supercritical. The third project consists of sharp-interface, moving-grid finite-element simulations of selective withdrawal for Newtonian and viscoelastic Giesekus liquids. The experiments and computations are in reasonable agreement. The work of this thesis has led to two main outcom.

?Pub Inc A computational study of pair-wise interactions between drops and the dynamics of concentrated emulsion is undertaken using a three-dimensional front-tracking finite difference method. The study is motivated by numerous industrial and domestic applications of multi-fluid flow. We first investigate the effects of inertia on pair-wise interaction between drops in a Newtonian system. It is seen that at high Re, increase in the initial cross-stream separation between the two interacting drops slightly increases the drops deformation. In addition, it is observed that if the separation position of the two interacting drops along the flow direction is so small that the drops can slide past each other even at high Re, increase in Re leads to an increase in the drops' lateral displacement after separation. However, if the separation position of the two interacting drops is more than twice the undeformed radius of the drops, the drops may not be able to slide past each other if Re is sufficiently large. The deformability of the drops also affects their lateral displacement. We observe that deformable drops are not able to slide past each other under the same conditions where drops with very low Ca can slide past each other. It appears that the flow modification caused by the drops deformation creates a downward force on the left drop. Next, we examine the effects of inertia on drop dynamics in a concentrated emulsion. We observe that the effects of inertia on weak hydrodynamic interaction at low dispersed phase volume fraction do not result in increased volume-averaged drop deformation. However, for concentrated emulsions, interaction between drops at increased Re leads to an increase in volume-averaged drop deformation especially when Ca of the drops is low. A single drop tends to rotate towards the vertical direction at increased Re. However, interactions between drops suppress this tendency. Finally, we investigate the effects of viscoelasticity on interactions between drops. We notice that the interaction between viscoelastic drops is similar to that between Newtonian drops. However, we observe that the interaction in the case of Newtonian drops in a viscoelastic matrix is different from that of Newtonian fluids. Viscoelastic stresses in the matrix-phase viscoelastic system inhibit the drops' lateral displacement and cause the drops to align more with the flow direction. Similar to what has been observed in single-drop deformation, the De of the matrix-phase viscoelasticity has non-monotonic effects on the drops' deformation when the drops are aligned with each other along the compressional quadrant of the shear flow.

This monograph is based on a series of lectures presented at the 1999 NSF-CBMS Regional Research Conference on Mathematical Analysis of Viscoelastic Flows. It begins with an introduction to phenomena observed in viscoelastic flows, the formulation of mathematical equations to model such flows, and the behavior of various models in simple flows. It also discusses the asymptotics of the high Weissenberg limit, the analysis of flow instabilities, the equations of viscoelastic flows, jets and filaments and their breakup, as well as several other topics.

In this thesis two different models and numerical methods have been developed to investigate the dynamics of bubbles in viscoelastic fluids. In the interests of gaining crucial initial insights, a simplifed system of governing equations is first considered. The ambient fluid around the bubble is considered incompressible and the flow irrotational. Viscoelastic effects are included through the normal stress balance at the bubble surface. The governing equations are then solved using a boundary element method. With regard to spherical bubble collapse, the model captures the behaviour seen in other studies, including the damped oscillation of the bubble radius with time and the existence of an elastic-limit solution. The model is extended in order to investigate multi-bubble dynamics near a rigid wall and a free surface. It is found that viscoelastic effects can prevent jet formation, produce cusped bubble shapes, and generally prevent the catastrophic collapse that is seen in the inviscid cases. The model is then used to investigate the role of viscoelasticity in the dynamics of rising gas bubbles. The dynamics of bubbles rising in a viscoelastic liquid are characterised by three phenomena: the trailing edge cusp, negative wake, and the rise velocity jump discontinuity. The model predicts the cusp at the trailing end of a rising bubble to a high resolution. However, the irrotational assumption precludes the prediction of the negative wake. The corresponding absence of the jump discontinuity supports the hypothesis that the negative wake is primarily responsible for the jump discontinuity, as mooted in previous studies. A second model is developed with the intention of gaining further insight into the role of viscoelasticity and corroborating the finndings of the first model. This second model employs the full compressible governing equations in a two dimensional domain. The equations are solved using the spectral element method, while the two phases are represented by "marker particles". The results are in qualitative agreement with the first model and confirm that the findings presented are a faithful account of bubble dynamics in viscoelastic fluids.

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