Commonly Taught Courses:
MTH 145 --- Calculus I
MTH 146 --- Calculus II
MTH 245 --- Calculus III
MTH 260 (CSC 281) -- Discrete Mathematics
MTH 303 --- Differential Equations and Mathematical Modeling
MTH 304 --- Differential Equations for Scientists and Engineers
MTH 421 --- Numerical Methods
MTH 431 --- Introduction to Complex Analysis
Specialty: Applied analysis and partial differential equations with particular emphasis on fluid dynamics and computational issues.
Research: My work involves the theoretical and numerical study of “free boundary” fluid flow — a rich and challenging class of problems dealing with flows which have an evolving interface with another fluid (e.g., air). Since the space occupied by the fluid is constantly changing in response to the flow variables, the fluid domain is itself an unknown in free boundary problems. Modeling free boundary flow involves coupling the Navier-Stokes Equations (NSE) with free surface boundary conditions which govern the interaction of the dominant forces shaping the interface between fluids. In its simplest form, the free surface boundary condition balances viscous forces in the fluid with the external pressure being applied to the fluid’s interface. However, more general forms incorporate the often significant effects due to surface tension. Simple examples of free boundary flows include the coating of a wire as it is withdrawn from a bath of molten plastic, the breakup of a liquid jet into droplets, and the spreading of a viscous fluid (e.g., honey) as it is poured onto a rigid surface.
Of particular interest to me are the important (though not well understood) multiple-scale problems of how free boundary models governing bulk flow (e.g. the full three-dimensional NSE) scale down to the simplified models which govern flow in “thin” fluid domains. In the cases of liquid sheets and jets, these models are often referred to as thin-film and thin-filament approximations and are obtained from the NSE by using the assumption of thinness to reduce the number of spatial dimensions required to describe the flow to two and one respectively; examples include Yeow’s equations for film casting and the Matovich-Pearson equations for fiber spinning. Such models play central roles in quantitatively describing the free surface when numerical computations using the full three-dimensional Navier-Stokes equations are cost-prohibitive.
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