Michael J. Miksis
Professor, Department of Mechanical Engineering
McCormick School of Engineering and Applied Science
Northwestern University, Evanston, IL 60208
E-mail
address: miksis@northwestern.edu
Editorships:
SIAM
Journal on Applied Mathematics, Editor-in-Chief
Journal of
Fluid Mechanics, Associate Editor: 2000-2003
Classes:
Fall
2011 GEN_ENG
205-4, Engineering Analysis 4, Sec 22 & 23
Current Graduate Students:
Lane McConnell
Christopher Gruber
Christopher Vogl
Previous Graduate Students:
Memberships:
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.
References:
``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.
References:
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.
References:
``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.
References:
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.