Research

Acoustics

This figure describes the pressure field obtained from a simulation of sound from a jet interacting with an array of flexible aircraft type panels (C. C. Fenno, Jr., Alvin Bayliss and L. Maestrello). In a real aircraft vibrations of the panels can lead to enhanced interior noise levels and enhanced structural fatigue of the panels. In this simulation the lower domain, which includes the jet exiting from a contracting nozzle, models the aircraft exterior, while the upper domain models the aircraft interior. Sources in the jet radiate sound which serves to excite the panels causing them to vibrate and radiate sound into the upper domain.

Picture Credits: C. C. Fenno, Jr., Alvin Bayliss, and L. Maestrello

Faculty studying in this topic: Alvin Bayliss; Michael J Miksis;

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Asymptotic Analysis

The majority of mathematical problems describing realistic systems cannot be solved exactly. They are therefore often solved by numerical methods. Numerical solutions have, however, a number of draw-backs. For one, it is usually quite difficult to get an overall impression of the dependence of the behavior of a system on its parameters. This type of information is readily available from analytical solutions. It is therefore often of great interest to obtain analytical solutions even if they solve the problem only approximately. In addition, the complete numerical solution often does not reveal which mechanisms are important for the understanding of a certain phenomenon and which are irrelevant. Here again analytical, approximate approaches can give valuable insight; by considering limiting cases in which only certain aspects of the problem are kept one can distill out which mechanisms are essential for a phenomonen that is obtained numerically or observed experimentally. There are a number of standard asymptotic approaches that are used to consider certain limiting cases. In each of them some quantity is assumed to be small or large. The most common ones involve expansions in the small amplitude of a solution, the assumption of a slow spatial (or temporal) variation of the variables, or the assumption of a fast variation in these quantities. The assumption of a slow temporal evolution of some modes in a system, while the remaining modes evolve fast (i.e. the separation of time scales), is a key ingredient of bifurcation theory and - related to it - of pattern formation. In cases in which a quantity depends on one spatial variable much more slowly than on the other long-wave equations can be derived, which are important in studies of thin films and many fluid dynamical problems (e.g. lubrication). Such assumptions lead also to insight in many wave phenomena (WKB-method) and in spatial patterns (phase dynamics). In boundary layers progress is often made by assuming that one spatial variable varies much faster than all the others. The course Asymptotic and Perturbation Methods provides a powerful "tool-box" for the study of a vast range of problems in applied mathematics.

Faculty studying in this topic: Stephen H Davis; William L. Kath; Moshe Matalon; Bernard J. Matkowsky; Michael J Miksis; Hermann Riecke; Mary Silber; Vladimir A Volpert;

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Bifurcation Theory

In many systems a smooth change in a control parameter can lead to a change in the behavior of the system that is not smooth. A simple example is the buckling of a rod. If a straight rod is compressed by a small load its length shrinks somewhat but it remains straight. For larger load, however, it starts to buckle. Mathematically, the solution corresponding to a straight rod still exists, but it is unstable for the large load applied and very small transversal perturbations make the rod buckle. The transition from the unbuckled to the buckled state occurs via a bifurcation, i.e. at the onset of the instability a new solution corresponding to the buckled rod comes into existence. A typical dependence of the buckling amplitude on the load is shown below. Bifurcations arise in many systems. Extensively studied are transitions in fluid dynamics, which often lead to flows that exhibit regular or complex spatial or spatio-temporal patterns. An example is shown in the picture below. It shows the top view of a thin layer of a liquid crystal that is exposed to an AC-voltage. For sufficently large voltage a bifurcation to a flow in the form of traveling convection rolls appears. Due to the anisotropy of the system these waves travel in one of four directions (north-east, north-west, south-west, south-west). If the applied AC-voltage is modulated in time in resonance with the frequency of the waves, standing waves arise instead of the traveling waves (H. Riecke, J.D. Crawford, and E. Knobloch, Phys. Rev. Lett. 61 (1988) 1942). In the liquid-crystal system more complicated standing wave patterns can arise due to the competition between the waves traveling in different directions. The bifurcation diagram below shows a bifurcation sequence from standing rectangles (PSRe) to standing rolls (PSRo) via standing cross rolls (PSCR).

Picture Credits: Fig.2. M. Dennin, G. Ahlers, and D. Cannell, Science 272, 388 (1996), Fig.3. H. Riecke, M. Silber, and L. Kramer, Phys. Rev. E 49 (1994) 4100

Faculty studying in this topic: Stephen H Davis; Bernard J. Matkowsky; Hermann Riecke; Mary Silber; Vladimir A Volpert;

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Combustion Theory

Cellular Flames

Each of the color figures depicts a snapshot of a top view of a cellular flame at a fixed instant of time. The figures were obtained from computations of mathematical models of flames (cf. A. Bayliss, B.J. Matkowsky, H. Riecke). Each figure depicts a flame with four cells. The figures are color coded according to the colors of the spectrum, i.e., red corresponds to the highest temperatures, yellow the next highest temperatures, etc. Figure 1A depicts a stationary (time independent) cellular solution, as can be seen from figure 1B which exhibits the temperature as a function of the angle at a fixed distance r from the center, at successive instants of time. Figure 2A corresponds to a traveling (rotating) wave solution, as can be seen from figure 2B. The branch of traveling wave solutions arises from the stationary solution branch via an infinite period bifurcation. Note that the four cells in figure 1A are identical and are reflection symmetric about a line (theta = constant) through the middle of the cell, while in the figure 2A, though the four cells are still identical, this symmetry is destroyed. Thus, we are led to believe that symmetry is associated with the lack of motion, while breaking the symmetry leads to motion. Figure 3A exhibits four cells which are no longer identical. Two of them are very nearly symmetric, characteristic of stationary cellular solutions, while the other two are decidedly nonsymmetric, characteristic of traveling wave cellular solutions. We are therefore led to believe that the first two cells are stationary while the latter two are in motion. This is confirmed figure 3B. As the forward crest (temperature maximum) of a given cell is jerked clockwise (to the left in the figure), the cell expands, becomes asymmetric and moves clockwise. This, in turn, pulls the cell behind it so that it too expands, becomes asymmetric and moves clockwise. At the same time the next cell pushes the cell ahead of it, compressing and causing it to become nearly symmetric and nearly stationary. This process then repeats with the next set of cells, etc. and can be viewed as a localized wave of asymmetry propagating counter-clockwise through the stationary symmetric cellular array. This type of cellular flame, which we have termed a "pushme-pullyu" cellular flame, is analogous to the behavior of the hopping modes of cellular flames observed experimentally by M. Gorman at Univ. Houston.

Picture Credits: A. Bayliss, B.J. Matkowsky, H. Riecke

Faculty studying in this topic: Alvin Bayliss; Moshe Matalon; Bernard J. Matkowsky; Vladimir A Volpert; Hermann Riecke;

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Fluid Dynamics

Fluid Chains Produced by Obliquely Intersecting Viscous Jets Connected by a Thin Free Liquid Film

A rather unique fluid dynamic configuration was observed in the course of studying rivulet formation from a contact line moving down an inclined plate about 1 m2. The exiting liquid was funneled via an inclined flat trough with narrowing sidewalls into an open vessel below. The liquid flowed principally along the funnel sidewalls. but also as a thin sheet in between. This formed two obliquely colliding falling jets with a thin liquid film connecting them. To our surprise the jets retained their character for long distances downstream, colliding and recolliding to form stable liquid rings, always with the surface tension of the liquid sheet pulling the outwards-pointing rebound jets back into the ring configuration. In each case, the rebound jets conserved momentum normal to the plane of entering jets by rotating the plane of exiting jets 90°. The jet and sheet structure is made fluorescent by illuminating the liquid, consisting of a 50% by volume mixture of glycerin and water with small amounts of fluorescein, by strong ultraviolet light. The entering jets are about the same diameter, and their centerlines appear close to intersecting. Thus, a mixing region is observed from which the exiting jets emerge.

Picture Credits: M. F. G. Johnson, M. J. Miksis, R. A. Schluter, and S. G. Bankoff. Phys. Fluids, Vol.8, No.9, September 1996

Faculty studying in this topic: Alvin Bayliss; Stephen H Davis; Moshe Matalon; Michael J Miksis; Hermann Riecke; Mary Silber;

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Information Technology

Analysis of Dispersion Managed Solitons for Optical Fiber Communications Systems

Increases in internet use have generated exponential growth in demand for communications services. Unprecedented amounts of information are now being transferred over the the internet. All this information is moved from place to place by encoding it in the form of light waves that propagate inside of optical fibers. As these light waves are guided toward their destinations they are deformed by a number of physical effects that tend to degrade the signal by introducing errors. To counteract these effects scientists and engineers have introduced a number of techniques that help to maintain the fidelity of the information. Dispersion management is one of the most important techniques now under development. In the figure a dispersion-managed-soliton is shown as it propagates through a section of an optical fiber communications link. A soliton is a localized wave that maintains its shape as it propagates and therefore can act as a single bit in a digital signal. Using dispersion management, this soliton becomes more resistant to errors than a soliton in the absence of such a control technique. Professors Kath and Luther use modern mathematical techniques to model and analyze nonlinear waves in a variety of communications systems. Much of their recent work has been devoted to improving the dispersion management of solitons.

Picture Credits: W.L. Kath and G.G. Luther

Faculty studying in this topic: William L. Kath;

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Math Biology

Today, mathematicians are finding more and more opportunities and a greater variety of interesting biological problems to explore than ever before. This emphasis is driven by the desire of the biological community to use mathematical modeling to explain biological phenomena, as well as by the increased availability of quantitative data to compare to the models. Elias Zerhouni, director of the NIH, on the future of biomedical research, stated (The NIH Roadmap, Science, 302(3), 2003), "To devise and use the state-of-the-art technologies developed from the roadmap effort, we will need the expertise of nontraditional teams of biological scientists, engineers, mathematicians, physical scientists, computer scientists, and others." There are many diverse biological applications that are being studied by members of the department. These applications include the study of bacterial biofilms (see Figs. 1, 2), neurophysiology (see Fig. 3), and cryo-preservation of cells to name a few. Each of these applications require the use of modeling, asymptotic analysis, and numerical methods to try to help understand the underlying biological phenomena.

Picture Credits: Fig. 1: Matt Parsek, U. Iowa; Fig. 2: David Chopp

Faculty studying in this topic: David Chopp; William L. Kath; Hermann Riecke;

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Microfluidics

A droplet of liquid (water in Fig.1) is frozen by cooling the substrate yielding the frozen drop depicted in Fig.2. It can be shown that the cusp is present whenever the liquid does not perfectly wet the solid. These experiments were performed by S.H. Davis et al.

Picture Credits: S.H. Davis et al

Faculty studying in this topic: Stephen H Davis; Michael J Miksis; Sasha Golovin

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Moving Boundary Problems

Motion by Gauss Curvature, Collapse of a Cube

Gauss curvature flow means that the velocity of a point on the surface is equal to the local Gauss curvature in the direction of the inward normal. For example, a sphere of radius r has constant Gauss curvature on its surface, so each point on the surface will travel inwards with speed 1/r^2 and the sphere collapses to a point in finite time. This flow is a model for the rounding of objects undergoing blunt impacts for random directions, e.g. stones tumbling in a stream.  This flow has the peculiar property that flat regions remain flat for finite non-zero time before the entire cube becomes strictly convex. This is in contrast to mean curvature flow where the cube will become strictly convex instantly (i.e. no flat spots after time zero).  This is one example of a very large class of problems involving moving interfaces and free boundaries.

Faculty studying in this topic: David Chopp; Stephen H Davis; Moshe Matalon; Bernard J. Matkowsky; Michael J Miksis;

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Nonlinear Dynamics

Movie of a numerical simulation of parametrically driven waves by G. Granzow and H. Riecke. In this regime all parallel waves are unstable. They perpetually break up and merge via the creation and destruction of defect pairs.

Picture Credits: G. Granzow and H. Riecke

Faculty studying in this topic: Stephen H Davis; William L. Kath; Bernard J. Matkowsky; Hermann Riecke; Mary Silber; Vladimir A Volpert;

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Pattern Formation

This "superlattice" pattern was obtained by numerically integrating a system of two coupled reaction-diffusion equations in the vicinity of a Turing bifurcation. The investigation of such Turing patterns was part of the thesis work of our recent applied math Ph.D. Stephen Judd. A striking feature of the superlattice pattern is that it has structure on two disparate lengthscales; the pattern is periodic on a large scale, and has small scale structure in each of the periodic "tiles". Its formation is surprising because the Turing instability has just a single characteristic lengthscale associated with it. To find out more about this research, see "Simple and Superlattice Turing Patterns in Reaction-Diffusion Systems: Bifurcation, Bistability, and Parameter Collapse", preprint, by S.L. Judd and M. Silber. This, as well as related pattern formation papers, may be downloaded from Prof. Silber's homepage.

Picture Credits: Fig.1. Kudroli, Pier, and Gollub 1998, Fig. 2. Silber and Proctor 1998

Faculty studying in this topic: Alvin Bayliss; Stephen H Davis; Moshe Matalon; Bernard J. Matkowsky; Hermann Riecke; Mary Silber; Vladimir A Volpert;

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Waves

Water wave simulation

This is a simulation of water waves in shallow water moving past a given location. The lines represent fluid levels with constant average depth, while the circles represent specific particles. Note that on the surface, particles move in circular paths, and as the depth increases they become more and more elliptical.

Faculty studying in this topic: Alvin Bayliss; William L. Kath; Michael J Miksis; Hermann Riecke;

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