Stuart Kent
Program in Applied Mathematics
University of Arizona
Program in Applied Mathematics
University of Arizona
Welcome to my website. I'm a fifth year graduate student in the
Program in Applied Mathematics at the University of Arizona. A summary of my research is given below. Please use the
links above to view details of my other academic work, to check homework assignments, or to find out how to contact me.
Since Spring 2009 I have been working with Dr. Venkataramani, analyzing the topological transition observed in selective withdrawal experiments. A brief summary of our work to date (Nov. 2010) is given below.
A poppy seed of diameter 0.8mm is entrained in a thin water spout
Our problem is motivated by a series of experiments that investigate polymer coatings of small particles [1]. A tube is placed just above the boundary between two immiscible fluid layers, and fluid is withdrawn through this tube at a constant rate. For low withdrawal rates, only fluid from the upper layer is withdrawn once the system reaches a steady state. As the flow rate is increased, the steady state of the system transitions from this `selective withdrawal' regime to a new regime in which a thin spout of the lower fluid is entrained within the upper fluid and withdrawn through the tube.
In the second state described, a small particle that is initially placed in the lower fluid will be entrained in the spout and drawn towards the tube. The presence of the particle causes the spout interface to break, leaving the particle coated with a thin film of the lower fluid. The experiments of Cohen et al [1] show that the film thickness is controllable to some degree, but it is unclear whether such a film can be made as thin as desired. Precise control of coating thickness and uniformity is important from an applications point-of-view - for example, transplanted biological cells often require finely-tuned coatings in order to prevent unwanted immune system responses. The overall goal of our current work is to investigate coating control by first analyzing the hump to spout transition. We hope that mathematical analysis of the transition between the two steady states described will allow any restrictions on the minimal film width to be determined in terms of the fluid properties.
However, the full free-boundary problem (including both viscosity and surface tension) is difficult to approach analytically. Numerical simulations of the transition have been performed by Berkenbusch et al [3], and these indicate a logarithmic coupling between hump height and hump tip curvature near the transition. This contradicts the suggestion of Cohen & Nagel [2] that the steady-state humps continuously approach a power-law cusp as the flow rate is increased.
Our initial approach to the problem has been to convert the non-variational fluid free-boundary problem to a variational electrostatic problem (where the fluid interface is modeled by a conducting elastic sheet) whose boundary behavior is altered to match that of the fluid problem qualitatively. We are therefore able to apply ideas and results from the study of electrostatically-motivated problems to our reformulated model as a precursor to attempting to analyze the full fluid problem.
[1] I. Cohen, H. Li, J. L. Hougland, M. Mrksich, S. R. Nagel - `Using selective withdrawal to coat microparticles' - Science 292:265-267 (2001)
[2] M. Berkenbusch, I. Cohen, W. Zhang - `Liquid interfaces in viscous straining flows: numerical studies of the selective withdrawal transition' - J. Fluid Mech. 613:171203 (2008)
[3] I. Cohen, S. R. Nagel - `Scaling at the selective withdrawal transition through a tube suspended above the fluid surface' - Phys. Rev. Lett. 88, 074501 (2002)
A poppy seed of diameter 0.8mm is entrained in a thin water spout
Our problem is motivated by a series of experiments that investigate polymer coatings of small particles [1]. A tube is placed just above the boundary between two immiscible fluid layers, and fluid is withdrawn through this tube at a constant rate. For low withdrawal rates, only fluid from the upper layer is withdrawn once the system reaches a steady state. As the flow rate is increased, the steady state of the system transitions from this `selective withdrawal' regime to a new regime in which a thin spout of the lower fluid is entrained within the upper fluid and withdrawn through the tube.
In the second state described, a small particle that is initially placed in the lower fluid will be entrained in the spout and drawn towards the tube. The presence of the particle causes the spout interface to break, leaving the particle coated with a thin film of the lower fluid. The experiments of Cohen et al [1] show that the film thickness is controllable to some degree, but it is unclear whether such a film can be made as thin as desired. Precise control of coating thickness and uniformity is important from an applications point-of-view - for example, transplanted biological cells often require finely-tuned coatings in order to prevent unwanted immune system responses. The overall goal of our current work is to investigate coating control by first analyzing the hump to spout transition. We hope that mathematical analysis of the transition between the two steady states described will allow any restrictions on the minimal film width to be determined in terms of the fluid properties.
However, the full free-boundary problem (including both viscosity and surface tension) is difficult to approach analytically. Numerical simulations of the transition have been performed by Berkenbusch et al [3], and these indicate a logarithmic coupling between hump height and hump tip curvature near the transition. This contradicts the suggestion of Cohen & Nagel [2] that the steady-state humps continuously approach a power-law cusp as the flow rate is increased.
Our initial approach to the problem has been to convert the non-variational fluid free-boundary problem to a variational electrostatic problem (where the fluid interface is modeled by a conducting elastic sheet) whose boundary behavior is altered to match that of the fluid problem qualitatively. We are therefore able to apply ideas and results from the study of electrostatically-motivated problems to our reformulated model as a precursor to attempting to analyze the full fluid problem.
[1] I. Cohen, H. Li, J. L. Hougland, M. Mrksich, S. R. Nagel - `Using selective withdrawal to coat microparticles' - Science 292:265-267 (2001)
[2] M. Berkenbusch, I. Cohen, W. Zhang - `Liquid interfaces in viscous straining flows: numerical studies of the selective withdrawal transition' - J. Fluid Mech. 613:171203 (2008)
[3] I. Cohen, S. R. Nagel - `Scaling at the selective withdrawal transition through a tube suspended above the fluid surface' - Phys. Rev. Lett. 88, 074501 (2002)