Publication Date



A good understanding of drop evolution and breakup is important in many applications. For instance, controlling the liquid droplet size is crucial in atomization processes such as fuel combustion and fertilizer application, as well as drop-on-demand technologies such as ink-jet printing and DNA arraying. In these applications, the length scales are very small relative to viscosity so that the Reynolds number is much less than unity. The aim of this work is to investigate the evolution and breakup of drops in Stokes flow. Drop evolution depends on different factors, such as the drop size, the viscosity, any applied force, or surface tension. In this dissertation, the behavior of axisymmetric viscous drops in a nonlinear strain field is investigated for various parameters. The three non-dimensional parameters that determine the flow in our case are: the capillary number Ca which measures the strength of the strain field and drop viscosity relative to surface tension, the ratio of inner to outer viscosities, and the relative nonlinearity c2 of the background flow. It is known that the drop approaches a steady state for sufficiently small values of Ca and that there exists a critical value of the capillary number, Cacr, above which no steady states exist. We examine the evolution of the drop as a function of these three parameters. Our main results are explained in three parts. (1) A full classification of the steady-state solutions in the parameter-space for Ca ≤ Cacr is presented. In particular, we describe the deformation, maximum curvature and the critical capillary number as functions of the key parameters. We find previously unobserved biconcave steady shapes. (2) The non-steady evolution for Ca > Cacr is studied and classified. With c2 = 0, the drop keeps elongating and becomes more pointed in time. With positive values of c2, the surface approaches a cusp as it increases in length. With negative values of c2, the surface collapses at two points on the axis in finite time. Thus the solution has a finite time pinch-off singularity. (3) Based on experimental observations, the drop surface is expected to break at the time of pinch-off and reconnect to form several smaller drops. We develop a numerical method to simulate the break-and-reconnection process. This enables us to compute the after pinch-off drop evolution. Our simulations indicate that this phenomenon has a linear self-similar behavior before and after pinch-off. Further pinch-offs is observed. Throughout this work the fifth-order boundary integral method presented by Nitsche et al. [1] is used. This method enables us to resolve the flow using fewer computational points compared to the commonly used second-order method. Furthermore, it is shown that the uniformly fifth-order method proposed in earlier work [1] makes a significant improvement in the results in certain cases.

Degree Name


Level of Degree


Department Name

Mathematics & Statistics

First Committee Member (Chair)

Monika Nitsche

Second Committee Member

Stephen Lau

Third Committee Member

Evangelos A. Coutsias

Fourth Committee Member

Peter Vorobieff

Project Sponsors





Drops, Stokes flow, Axial flow, Boundary element methods.

Document Type