During the years of my graduate study, I collaborated with Professor Jon Higdon in the area of Computational Fluid Mechanics studying viscous free-surface flows. The goal of my research was to develop rigorous theoretical models to improve our understanding of physical phenomena in fluid mechanics. Specifically, I was interested in the problem of displacement of fluid droplets and bridges from solid substrates under the influence of low Re flows and gravity. This problem has application in the enhanced oil recovery, coating operations and condensation of vapor. These processes are strongly dependent on the interaction of the immiscible two-phase mixtures and the success of such operations depends on the displacement of small drops attached to solid surfaces. 

The study of fluid droplets and bridges in restricted geometries is further motivated by its similarity to the blood flow in microvessels. Blood flow in vivo is a complicated flow of numerous biological cells and inorganic ions and, for this reason, it is still not well understood. Several diseases are associated with abnormal microcirculation, including cardiovascular diseases, sickle-cell anemia and cerebral malaria. The cardiovascular disorders are the cause of heart attacks and strokes which are responsible for more than 50% of deaths in the Western world. These disasters are often associated with a blood clot, thrombus or embolus, adhering to the microvessel's inner surface and blocking a key artery that supplies blood to the heart or the brain.

In order to determine equilibrium fluid interfaces in Stokes flow, I developed a novel Newton's method which is based on the Spectral Boundary Element method. This method proved to be a robust algorithm, of high accuracy and extreme efficiency, valid to study both two and three dimensional problems. This is the first implementation of a Newton's method based on a Boundary Integral method. It is worth mentioning that the Newton method is more suitable for addressing this problem than time dependent computations which most authors have used in prior boundary element studies, and can be used to study a great variety of other problems.

During my master's research, a comprehensive study was conducted on the displacement of two dimensional droplets (J. Fluid Mech., 1997, 336, 351-378). The study covered a wide range of all the parameters which affect the problem, namely the capillary number Ca (the ratio of viscous to surface tension forces), the ratio of the fluid viscosities , the Bond number B_d (the ratio of gravitational to surface tension forces) and advancing and receding contact angles  and . We found that the viscosity ratio plays an important role for viscous droplets, especially at high contact angles, while the inviscid droplets show a dramatic and contradictory behavior compared with the viscous droplets. It was shown that the gravitational forces reduce the ability of viscous droplets to withstand fluid motion, but dramatically increase the resistance of inviscid droplets. We also demonstrated that the range of validity of asymptotic results based on lubrication theory is quite small.

In my Ph.D. research, I focused on the more challenging problem of the displacement of three-dimensional droplets. For this problem, for specific values of the advancing and the receding angles, there are many possible drop shapes which correspond to different profiles of the contact line. To determine the yield conditions for drop displacement, I sought the equilibrium shape of the fluid-fluid interface which can sustain the greatest flow rate subject to the constraint that the contact angles  remain in the interval . In this problem, the position of the contact line on the solid surface is not specified a priori, but must be determined as a part of the solution. In particular, we must solve a non-linear optimization problem with both equality (Stokes equations, continuity, boundary conditions) and inequality (contact angle) constraints. This means that the Newton's method must be combined with a successive linear optimization algorithm to solve the non-linear optimization problem.

During my first Ph.D. project, I studied the influence of low Re shear flows on droplets attached to horizontal solid surfaces (J. Fluid Mech., 1998, 377, 189-222) as well as the influence of gravity on droplets on inclined surfaces (J. Fluid Mech., 1999, 395, 181-209). The contact line contours show fore and aft asymmetry with a distorted shape not well represented by the simple circle/elliptical planforms assumed by previous authors. The distorted profiles allow sharp jumps in contact angle which increases the ability of a droplet to stick to a surface. Our predicted contours show good agreement with experimental observations. The influence of the viscosity ratio is qualitatively similar with that of the two-dimensional problem. We also show that the range of validity of asymptotic results based on lubrication theory is quite small. The normal component of the gravitational force is found to affect only weakly the displacement of sessile droplets but to have a strong effect on the displacement of pendant droplets. In addition, we show that the value of the advancing contact angle  has a significant effect on the displacement process. For a given hysteresis, increasing this angle increases the critical flow rate for values up to . Increasing  beyond this point decreases the critical flow rate.

Next, I considered the displacement of three-dimensional fluid droplets in pressure-driven flows between two parallel plates. I identified two cases: a droplet attached to the lower plane and a fluid bridge spanning the gap between the two plates. These problems have as an additional parameter the dimensionless distance between the two plates. While I considered the effect of all parameters on the problems, I was most interested in the influence of the plate separation on the critical flow rate. We note that while the previous problems revealed a number of new physical phenomena associated with the drop displacement process, these two last projects are closer to the aforementioned applications and reveal the physical mechanisms associated with them.

For the problem of a droplet attached to a single plate (J. Fluid Mech., 2001, 435, 327-350), we showed that an increase in the viscosity ratio of the droplet reduces the critical flow rate, making it easier to displace the droplet. This is consistent with my previous results for shear flows, which represent the limit of infinite plate spacing. As the plate spacing H is reduced, the critical flow rate increases until a maximum value is reached. Further reduction in the plate spacing decreases the critical flow rate. The effects of both viscosity ratio and plate separation are much more pronounced for high contact angles. Inviscid droplets (or bubbles) show behavior dramatically different from that of viscous droplets. For these droplets, a significantly high flow rate is required for drop displacement, but this critical flow rate (or capillary number Ca) decreases monotonically as the distance between the plates decreases. The displacement of inviscid droplets is strongly affected by the change in the character of the pressure force as the plate spacing is reduced.

For the case of fluid bridges spanning the gap between two parallel plates (submitted to J. Fluid Mech.), the first step in the analysis is to determine the equilibrium surfaces under quiescent conditions. (For a droplet on a single surface, the quiescent surfaces are simply sections of a sphere.) Next, I calculated the optimal equilibrium shapes for increasing flow rate, employing the optimization algorithm as described above. We showed that the influence of the viscosity ratio is consistent with my previous results for droplet displacement; however, the effect is not as strong for fluid bridges. The advancing contact angle  has a significant effect on the displacement process. For a given hysteresis, increasing this angle increases the critical flow rate for values up to  . Increasing  beyond this point decreases the critical flow rate. This behavior is valid for both viscous and inviscid bridges, in distinct contrast with the droplet displacement results. The critical flow rate is strongly affected by the plate spacing. For a given hysteresis, the critical flow rate increases when the plate separation increases from small values or when the plate separation decreases from high values, and there is a value of the plate separation for which the critical flow rate achieves a maximum value. Thus, very short or very tall bridges can be displaced more easily compared to bridges with moderate height.

My graduate research was supported by the H.G. Drickamer graduate fellowship and by the National Science Foundation (CTS-9522724, 1996-1998). The computations were performed on multiprocessor computers provided by the National Center for Supercomputing Applications.

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