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Here are the problems presented at the 2019 Graduate Student Mathematical Modeling Camp, as well as summary reports of the work accomplished.
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Daniel Anderson, George Mason University
Abstract: Ophthalmic drugs are commonly delivered via eye drops. Drainage out of the eye during a blink and the resupply of fresh tears are among the issues that limit the residence timeand effectiveness of the drug in the eye. The use of contact lenses is an alternative means for sustained drug delivery for the treatment of chronic and acute eye disorders.Soft contact lenses are made out of hydrogels and the understanding of diffusion of chemicals through these materials along with transport mechanisms of the drugin the tear film environment are fundamental to the success of these types of treatments. In this project, we explore mathematical models for drug delivery from a contact lens during wear and aim to develop predictive models for drug delivery in contact lens and tear films.
Richard Moore, New Jersey Institute of Technology
We’ve all seen how water can cut through mountains with its erosive power and crack rock when pumped underground at high pressure. We’re perhaps less familiar with water jets used as a precision cutting tool, employed regularly in the food and textile industries. Abrasive water jets, with small particles of garnet in the stream, can cut hard materials such as glass and metal. They are competitive with other precision machine tools such as laser, plasma, and wire EDM cutters but don’t generate a heat-affected zone in the material.
There are some challenges with abrasive water jets, however. Depending on the traverse speed of the cutter, striations can form on the cut face, a blank region can be left in the sample, and the kerf (i.e., the gap left by the cut) can develop an undesired internal angle (bevel).
Our goal for the camp is to understand the basic operation of the abrasive water jet and the mechanisms underlying the problems described above. Time permitting, we’ll propose some parameter choices or procedural changes to address the problems.
Pejman Sanaei, New York University
Abstract: Membrane filters are used in various industrial engineering
processes and one of the most significant applications is water purification, where target particles,
colloids and macromolecules, are removed from the water flow by applying microfiltration. Hence
mathematical models to predict their efficacy are potentially very useful, as such models can suggest
design modifications to improve filter performance and lifetime. Many models have been proposed to
describe particle capture by membrane filters and the associated fluid dynamics, but most of such models
are based on a very simple structure in which the pores of the membrane are assumed to be simple
circularly cylindrical tubes spanning the depth of the membrane. Real membranes used in applications can
have much more complex internal structure, with interconnected pores that may branch
and bifurcate, and pore-size variation across the membrane. However, during the filtration process,
membrane fouling due to the block of large particles and deposition of small particles occur and decreases
the membrane performance. Thus, the membrane’s permeability decreases as the filtration progresses.
Two driving mechanisms can be considered an here: (i) constant pressure drop across the membrane
specified; and (ii) constant flux through the membrane specified. In the former case the flux will decrease
in time as the membrane becomes fouled; in the latter, the pressure drop required to sustain the constant
flux will rise as fouling occurs. Considering elasticity to sub-branches in constant flux scenario, in some stage of filtration process,
the radius of pores may tend to expand due to the effect of high pressure on the elastic sub-branches, which is not negligible.