Michael J. Miksis

Professor, Department of Engineering Sciences and Applied Mathematics

Professor, Department of Mechanical Engineering

McCormick School of Engineering and Applied Science

Northwestern University, Evanston, IL 60208

E-mail address: miksis@northwestern.edu



SIAM Journal on Applied Mathematics, Editor-in-Chief

Journal of Fluid Mechanics, Associate Editor: 2000-2003



Fall 2011  GEN_ENG 205-4, Engineering Analysis 4, Sec 22 & 23


Current Graduate Students:

Previous Graduate Students:


American Physical Society: Fellow

Society for Industrial and Applied Mathematics: Fellow


Research Interests: Theoretical and Computational Fluid Mechanics, Biofluids and Materials Scicence; Free Boundary Problems, Multiphase Flows, Stability Theory, Wave Propogation; Asymptotic, Perturbation Methods & Computational Methods. 



Research Projects:

Interface problems are the primary theme of my current research. Interfaces are the boundary between two phases.  Examples are the gas/liquid interface of a rising gas bubble, the gas/liquid interface of a spreading drop, the solid/vapor interface of a quantum dot, and the arterial wall of an artery within the body.  These interfaces are free boundaries which must be determined as part of the solution.  In general, there are boundary conditions along the interface that connects the dynamics on each side of the interface.  Research here has been concerned with using analytical and computational methods to solve for the dynamics.


Interface Problems in Biology


Recently my research interest has moved into the area of biology.  For example, two problems of considerable interest have been the dynamics of  lipid bilayer vesicles and  biopreservation by way of desiccation and vitrification (conversation to a glassy state), a technique known as anhydrobiosis.


Lipid bilayers are the basic component of cell membranes. Our aim is to develop solution methods and to investigate the behavior of the mathematical models governing the dynamics of these biological interfaces.  Besides being used to enhance our basic understanding of cell membranes, this investigation may have applications to enhanced drug delivery.  Our investigations have been directed to developing models which allow for the dynamics of the lipids along the membranes in flows, and we have been interested in understanding the effect of a DC electric pulse on the stability of the membrane.  A controlled application of an DC electric pulse can induce transient pores in the cell or vesicle membrane, which can reseal after the pulse is turned off but may allow the delivery of exogenous molecules.  Both small amplitude perturbation analysis and numerical methods have been used in our investigations.




``Monolayer slip effects on the dynamics of a lipid bilayer vesicle in a viscous flow'', (with J.T. Schwalbe and P.M. Vlahovska), J. Fluid Mech, 647, 403-419, 2010

``Vesicle electrohydrodynamics’’, (with J.T. Schwalbe and P.M. Vlahovska),  Phys Review E, 83, 046309, 2011

``Lipid membrane instability driven by capacitive charging’’, (with J.T. Schwalbe and P.M. Vlahovska), Phys. Fluids, 23, 041701, 2011

``A level set projection model of lipid vesicles in general flows’’, (with D. Salac),  preprint

A number of organisms (e.g., yeasts, nematodes, etc.), when encountering harsh environmental conditions, undergo anhydrobiotic preservation, emerging as viable organisms when more favorable conditions arise. Some organisms, such as the seeds of certain plants, have been known to survive decades in the preserved state. The vitrification of biological samples is induced through the injection of special sugars (e.g. trehalose) into the cell and the evaporation of water through the cell membrane. These sugar-water solution become a glass when the concentration of water decreases below a critical value. This glassy material has been shown to have an essential role in dramatically slowing intracellular transport processes, stabilizing cell membranes, and preventing cell membrane phase changes. The virtues of anhydrobiosis as a preservation technique (as compared to cryogenesis, for example) are that the process occurs at ambient temperature and that the preserved tissues are much lighter than the original material, thus reducing storage and transport costs. Our aim is to better understand this process with the hope of applying it as a general preservation technique. Our investigation has been concerned with modelling the motion and stability of a glass formation front as it advances during this diffusion process. The diffusion of water in a glassy region is anomalous and needs to be modeled by a fractional diffusion equation. Our interest is in developing asymptotic and numerical methods to determine the dynamics of the subdiffusive interface.



Nanotechnology and Problem with Multiple Scales

A whole new class of mathematical problems occurs in nanotechnology.  Part of the reason for this is that the traditional approach of using continuum models of the materials is no longer sufficient.  In this traditional approach, the fields (e.g., stress and strain) are determined by solving known partial differential equations with known coefficients. Boundary conditions are understood and many computational methods have been developed to solve these difficult, but standard, problems. However, for many problems in nanotechnology, the scales are so small (sometimes just a few nm), that continuum models are no longer applicable.  Hence the challenge is to bridge the information occurring at the atomic (microscopic) scale with the behavior on the macroscopic scale. This macroscopic scale can be as small as 10's of nanometers for structures of current interest.  Our research has involved using information from either  ab-initio or molecular-dynamics calculations into new continuum theories valid at this macroscopic scale.   For example, my graduate student Christopher Retford, developed a molecular dynamics code to study edge energies along quantum wires.  In the figure below, an example is given of a Ge quantum wire and wetting layer resting along a Si substrate.  The individual atoms are shown and the reconstruction of the interface can be observed.  As another example consider the problem of determining the shape of a nano-scale Ge island resting on a Si substrate, i.e., a quantum dot. Because of the lattice misfit between the Ge and Si crystal lattice, a misfit strain is developed which governs the shape of the island.   At these small scales, the surface energy of the interfaces is strain dependent and it is necessary to determine the shape and evolution of the island.   A multi-scale computational approach is necessary which puts information from the microscopic scale into a macroscopic scale model, i.e., the PDE's of classical elasticity modified to account for the proper surface energy. 



``Evolution of material voids for highly anisotropic surface energy'', (with M. Siegel and P.Voorhees),   J. Mech. Phys. Solids,  52, 1319-1353, 2004.

``Role of Strain-Dependent Surface Energies in Ge/Si(100) Island Formatio'', (with O.E. Shklyaev, M.J. Beck, M. Asta, and P.W. Voorhees), PRL 94, 176102, 2005

``The Effect of Contact Lines on the Rayleigh Instability with Anisotropic Surface Energy'',  (with K.F. Gurski and G.B. McFadden), SIAM J. Applied Math. 66(4), 1163-1187,  2006.

``Equilibrium Shapes of Strained Islands with Finite Contact Angle'', (with O.  Shklyaev and P.W. Voorhees), J. Mech. Phys. Solids 54, 2111-2135, 2006.

``Orientation Dependence of Strained-Ge Surface Energies Near (001): Role of Dimer-Vacancy-Lines and Their Interactions with Step'', (with C.J. Moore, C.M. Retford, M.J. Beck, M. Asta and P.W. Voorhees), PRL 96 (12), 126101, 2006.

``Energetics of {105}-faceted Ge nanowires on Si(001): An atomistic calculation of edge contribution'', (with C.M. Retford, M. Asta, P.W. Voorhees, and E.B. Webb), Physical Review B 75, 075311, 2007

``Universality and self-similarity in pinch-off of rods by bulk diffusion'', (with L. K. Aagesen, A. E. Johnson, J. L. Fife, P. W. Voorhees,  S. O. Poulsen, E. M. Lauridsen, F. Marone and M. Stampanoni),  Nature Physics, vol. 6(10), pp. 796-800, October 2010.

``Pinch-off of rods by bulk diffusion'', (with L. K. Aagesen, A. E. Johnson, J. L. Fife, P. W. Voorhees, S. O. Poulsen, E. M. Lauridsen, F. Marone and M. Stampanon, Acta Materialia, 59, 4922-4932, 2011.



Computation of Moving Boundary Problems in Fluid Dynamics

Rising gas bubbles play an important role in many physical and biological processes, such as the dynamics of multiphase flows, cavitation processes, and the flow of bubbles in the bloodstream.  The rise of gas bubbles and the observation of a path instability has been documented since the time of Leonardo Da Vinci, but questions related to the origin of this instability still exist.  Catherine Norman thesis topic was concerned with the development of a level-set numerical method to study the dynamics of rising bubbles. She considered both bubbles rising under an inclined plane and bubble free rising.  She considered both cases were there was a film of liquid between the bubble and the plane and classes where there was a three-phase contact line.  Her code allowed for adaptive meshing and she developed a full second-order method.

Below is a numerical calculation using the level-set code of C. Normann.  Here we see a three-dimensional gas bubble rising from rest.  The bubble initially rising along the center-line and flattens in the direction it is moving.  With increasing distance, a spiraling path instability occurs.  Because the bubble is rising inside of a finite channel, the spiraling instability eventually becomes a zig-zag instability where the bubble move back and forth in a center-plane normal to two of the faces.


C. Norman and M.J. Miksis, ``Dynamics of a Gas Bubble in an Inclined Channel at Finite Reynolds Number'',  (with C. Norman), Phys. Fluids, 17(2), 022102, 2005

C. Norman and M.J. Miksis, ``Gas Bubble with a Moving Contact Line Rising in an Inclined Channel at Finite Reynolds Number'', Physica D, 209, 191-204, 2005.

C. E. Norman, ``A level-set numerical method to determine the dynamics of gas bubbles in inclined channels'', Ph.D. Thesis, Northwestern University, June 2005.