October 12-16, 2009
A seamless algorithm for multiscale simulationsOctober 16, 2009 2:15 pm - 2:55 pm
I will present a seamless algorithm for the study of multiscale problems. The multiscale method aims at
capturing the macroscale behavior of a given system which is modeled by an (incomplete) macroscale model. This is done by
coupling the macro model with a micro model: The macro model provides the necessary constraint for the micro model and the
micro model supplies the missing data (e.g. the constitutive relation or the boundary conditions) needed in the macro model.
In the multiscale method, the macro and micro models evolve simultaneously using different time steps, and they exchange
data at every step. The micro model uses its own appropriate (micro) time step. The macro model uses a macro time step but
runs at a slower pace than required by accuracy and stability considerations in order for the micro dynamics to have sufficient
time to adapt to the environment provided by the macro state. The method has the advantage that it does not require the
reinitialization of the micro model at each macro time step or each macro iteration step. The data computed from the micro model
is implicitly averaged over time.
Second chancesOctober 12, 2009 4:00 pm - 4:30 pm
Second chancesOctober 13, 2009 4:15 pm - 4:45 pm
Second chancesOctober 14, 2009 4:15 pm - 4:45 pm
Second chancesOctober 15, 2009 4:15 pm - 4:45 pm
Second chances and closing remarksOctober 16, 2009 3:00 pm - 3:30 pm
Microscale shear flow of focal conic defects in layered liquids
October 12, 2009 4:30 pm - 6:30 pm
Intermolecular interactions in liquid crystals and concentrated surfactant solutions lead to unique microstructures including lamellae, in which parallel layers are incompressible but bend easily. In such systems, planar layers are easily destabilized via external fields and nearby surfaces to produce topological defects of the order of tens of microns in size. These microscopic defects play a leading role in the flow behavior of such materials, and therefore impact numerous industrial applications including optoelectronic devices and displays and the processing of coatings, adhesives, and biomaterials to encapsulate drugs. To examine the interaction between such microscale defects and flow, we have developed a shear cell to impose a linear Couette flow in a microscale thin gap, while allowing for real time microscopic visualization. We use the shear cell to visualize the dynamics of defect formation in initially defect-free samples of a common small-molecule thermotropic liquid crystal, 8CB. We observe that the formation of focal conic defects, a specific topological defect typically found in thermotropic smectic liquid crystals, is triggered by edge effects and occurs in a series of phases, marked by distinct changes in the birefringence intensity. The defects are seen to annihilate partially or completely on reverse shear. The effect of shear rate and strain amplitude on defect formation and annihilation is studied.
The effects of polymer molecular weight on filament thinning
& drop breakup in microchannels October 13, 2009 9:00 am - 9:40 am
In this talk, the effects of fluid elasticity on the dynamics
of filament thinning and drop breakup processes are
investigated in a cross-slot microchannel. Elasticity effects
are examined using dilute aqueous polymeric solutions of
molecular weight (MW) ranging from 1.5 x 10(^3) to 1.8 ×
10(^7). Results for polymeric fluids are compared to those for
a viscous Newtonian fluid. The shearing or continuous phase
that induces breakup is mineral oil. All fluids possess similar
shear-viscosity (~0.2 Pa s) so that the viscosity ratio between
the oil and aqueous phases is close to unity. Measurements of
filament thickness as a function of time show different
thinning behavior for the different aqueous fluids. For
Newtonian fluids, the thinning process shows a single
exponential decay of the filament thickness. For low MW fluids
(103, 104, and 105), the thinning process also shows a single
exponential decay, but with a decay rate that is slower than
for the Newtonian fluid. The decay time increases with polymer
MW. For high MW (106 and 107) fluids, the initial
exponential decay crosses over to a second exponential decay in which
elastic stresses are important. We show that the decay rate of
the filament thickness in this exponential decay regime can be
used to measure the steady extensional viscosity of the fluids.
At late times, all fluids cross over to an algebraic decay
which is driven mainly by surface tension.
Stochastic Eulerian-Lagrangian methods for
fluid-structure interactions with thermal fluctuations
October 16, 2009 10:15 am - 10:55 am
Keywords: Statistical Mechanics, Soft Condensed Materials, Stochastic Eulerian Lagrangian Methods, Fluid Dynamics.
Abstract: A modeling and simulation formalism is presented for the study of soft materials. The formalism
takes into account microstructure elasticity, hydrodynamic interactions, and thermal fluctuations. As a specific motivation we consider lipid bilayer membranes and polymeric fluids. The
approach couples a Lagrangian description of the microstructures (lipid molecules / polymers)
with an Eulerian description of the hydrodynamics. Thermal fluctuations are incorporated in
the formalism by an appropriate stochastic forcing of the resulting equations in accordance
with the principles of statistical mechanics. The overall approach extends previous work on
the Stochastic Immersed Boundary Method. Simulation studies are presenting showing
applications of the methodology in the study of lipid flow in bilayer membranes, the shear
viscosity of polymer fluids and lipid structures, and the diffusivity of particles in complex fluids.
Instabilities due to microstructure growth at fluid interfacesOctober 13, 2009 1:30 pm - 2:10 pm
Keywords: wormlike micelles, hydrodynamic instability, micellar aggregation reactions
Abstract: The viscoelasticity of a wormlike micellar fluid derives from the presence at the microscale of long tubelike surfactant aggregations (micelles), which occur at low concentrations due to the presence of an organic salt/cosurfactant. Without this salt the micelles are spherical, and the fluid is completely Newtonian. I will present recent experiments on hydrodynamic instabilities caused or modified by the formation of a viscoelastic micellar layer between two Newtonian liquids containing either the surfactant or the organic salt. The micellar "reaction" produces a fragile material which grows in competition with local stretching or advection at the interface, leading to new effects in Hele-Shaw fingering, in droplet sedimentation, and in microfluidic emulsions.
Data reduction in viscoelastic turbulent channel flowsOctober 12, 2009 11:00 am - 11:40 am
Direct Numerical Simulations (DNS) of turbulent viscoelastic channel flows
typically generate a tremendous volume of information (terabytes per run.
Data reduction is therefore essential in order to allow for an efficient
processing of the data, let alone its preservation for future studies.
However, previous attempts, using a projection of the velocity to the top
Karhunen-Loeve (K-L) modes, failed to produce velocity fields that could
generate the DNS conformation field adequately. In an effort to rectify
this deficiency we investigate here three different approaches that attempt
to introduce small scale information. First, we extended the K-L analysis
that allowed us to use a hybrid measure, based on a weight of the
pseudodissipation and the fluctuating kinetic energy, as a new objective
function. Second, we used a K-L decomposition of the vorticity field using
the enstrophy (average of the square of vorticity fluctuations) as our new
objective function. As a third attempt, we used again the standard velocity
K-L approach, but in the reconstruction stage of the conformation tensor we
compensated by suitably rescaling the Weissenberg number. It is shown here
that, whereas the first two methods fail to give any improvement over the
classical K-L approach, we were able to reconstruct fairly accurately the
conformation field using the third approach, even with a relatively small
set of 1714 K-L modes. The rescaling factor in that method was calculated
objectively, based on the ratio of DNS vs. K-L-reconstructed based estimates
of the extensional deformation rate in the buffer layer. Given that fact,
we hope that this approach can also provide the starting point for future
investigations into low-dimensional modeling of viscoelastic turbulence as
well as other multiscale applications.
Single particle motion in colloids: Microrheology and
microdiffusivityOctober 15, 2009 9:00 am - 9:40 am
Keywords: Colloidal dispersions, Brownian motion, rheology
Abstract: The motion of a single individual particle in a complex
material is fundamental to understanding the dynamical
properties of the material. Monitoring such motion has given
rise to a suite of experimental techniques collectively known
as ‘microrheology,’ with the ability to probe the viscoelastic
properties of soft heterogeneous materials (e.g. polymer
solutions, colloidal dispersions, biomaterials, etc.) at the
micrometer (and smaller) scale. In microrheology, elastic and
viscous moduli are obtained from measurements of the
fluctuating thermal motion of embedded colloidal probes. In
such experiments, the probe motion is passive and reflects the
near-equilibrium (linear response) properties of the
surrounding medium. By actively pulling the probe through the
material one can gain information about the nonlinear response,
analogous to large-amplitude measurements in macrorheology.
But what exactly is measured in a microrheological experiment?
And how does the micro-rheological response compare with
conventional macrorheology? To answer these questions, we
consider a simple model – a colloidal probe pulled through a
suspension of neutrally buoyant bath colloids – for which both
micro- and macro-results can be obtained exactly. The moving
probe distorts the dispersion’s microstructure resulting in a
reactive entropic or osmotic force that resists the probe’s
motion, which can be calculated analytically and via Brownian
Dynamics simulations and used to infer the dispersion's
'effective microviscosity.' By studying the fluctuations in
the probe’s motion we can also determine the force-induced
'micro-diffusivity.' Connections between micro and macro
behavior will be explored.
Dissipative particle dynamics: Algorithms and recent
October 16, 2009 11:00 am - 11:40 am
In its original version the governing equations of Dissipative Particle Dynamics (DPD) contain three forces which need to be specified in any application, namely: i. a conservative soft repulsion, ii. a random force, iii. a dissipative force. The required thermostat is enforced by balance of ii. and iii. through the Fluctuation-Dissipation theorem . With about 3 to 4 thousand particles (number densities of 3 to 4) these forces simulate a nearly incompressible fluid whose compressibility is close to water’s, and whose viscosity is constant. When the latter is combined with the self-diffusion coefficient a characteristic radius of the DPD particle can be calculated from the Stokes-Einstein relation. Drag force calculations on single DPD particles imersed in a streaming flow show consistency with Stokes law as the Schmidt numbe increases from one. Thus DPD particles are mesoscopic entities, and are hydrodynamically similar to the beads of Brownian Dynamics (BD). However, the hydrodynamic forces between DPD particles are implicit. Complex fluids such as polymers are modeled by connecting DPD particles with spring forces. Examples include dilute, concentrated and undiluted bead-spring chains in plane Couette and in Poiseulle flow which are simulated with periodic boundary conditions. Real boundaries require carefull treatment to avoid unphysical density fluctuations near a wall. Another application of DPD is in the ‘tripple decker’ which attemps to match regions described by continuum, DPD, and Molecular Dynamics (MD) respectively.
In the original DPD all forces on a particle are central which obviates the need to deal with angular momentum. However, the calculated rotational drag on a single DPD particle was found to deveate subtantially from the Stokes value. The remedy adds a non-central term to the dissipative force, and includes angular momentum explicitly. The new formulation has been used to simulate a colloidal suspension of large hard DPD-particles in a solvent of soft DPD particles. The results show the model to be economical, and to exhibit the same features as those obtained by the Stokesian dynamics method.
A more complex case is the red blood cell (RBC) model with a membrane constructed from DPD particles connected by nonlinear springs and with an extra dissipative force to describe the known viscolelastic properties of the RBC membrane. This model succeeds in describing quantitatively a number of static and dynamic experiments without adjustment of parameters.
The models described above have been developed empirically by intuition. A more rigorous and difficult approach is to attempt to derive DPD from analysis at the molecular level. To this end MD simulations of Lennard-Jonesium (LJ) are interpreted statistically for clusters of O(10) LJ molecules to derive the soft potentials and also the dissipative forces which are found generally to be non-central.
Modeling and simulation of complex fluids via field theory October 16, 2009 9:15 am - 9:55 am
Keywords: Field theoretic polymer models, flow-structure interaction, mesoscale models, phase field models.
Abstract: We will present examples of field theoretic models of multi-component and complex fluids and discuss their main computational challenges and recent advances. We will start with simple phase field based models and progress to a class of field-theoretic models that incorporate exact thermodynamics. In particular, we will
present a model for an inhomogeneous melt of elastic dumbbell polymers which incorporates thermodynamic forces acting on the polymers into the hydrodynamic equations. The resulting equations are composed of a system of fourth order PDEs coupled with a nonlinear optimization problem to determine the conjugate fields. We develop a semi-implicit numerical method for the resulting system of PDE's in addition to a parallel nonlinear optimization solver for the conjugate mean-fields. The semi-implicit method effectively removes the fourth order stability constraint associated with explicit methods and we observe a first order time-step restriction. The algorithm for solving the nonlinear optimization problem, which takes advantage of the form of the operators being optimized, reduces the overall computational cost of simulations by several orders of magnitude.
Complex Fluids: an abstract framework, some analysis, many open
October 14, 2009 1:30 pm - 2:10 pm
Keywords: optimal transportation theory, nonlinear Fokker-Planck
equations, gradient systems in metric space, Onsager equation
Abstract: I will describe a framework for the study of the time evolution
probability distributions of complex systems based on ideas of
optimal transportation theory and gradient systems in metric
Modeling the inhomogeneous response in transient
shearing and extensional flows of entangled/micellar solutionsOctober 13, 2009 2:15 pm - 2:55 pm
Surfactant molecules (micelles) can self-assemble in solution
flexible structures known as wormlike micelles. These
forming a dense network and thus exhibit viscoelastic effects,
entangled polymer melts. In contrast to 'inert' polymeric
wormlike micelles continuously break and reform leading to an
relaxation mechanism and the name 'living polymers.'
show that, in shearing flows, wormlike micellar solutions
inhomogeneities, or shear bands. The VCM model, a two-species
network model was formulated to capture, in a self-consistent
micellar breakage and reforming. This model consists of a
coupled set of
partial differential equations describing the breakage and
two micellar species (a long species 'A' and a shorter species
‘B’) - in
addition to reptative and Rousian stress-relaxation mechanisms.
and steady-state calculations of the full inhomogeneous flow
localized shear bands that grow linearly in spatial extent
across the gap
as the apparent shear rate is incremented.
This model also captures the non-monotonic variation in the
state elongational viscosity that has been reported
the marked differences between the response of micellar
biaxial and uniaxial extensional flows. The non-monotonic
the extensional viscosity has important dynamical consequences
transient elongational flows; In filament stretching
designed to measure the extensional rheology of wormlike
solutions, it has been observed that the elongating filaments
suddenly rupture near the axial mid-plane at high strain rates
[Rothstein]. This newly-observed failure mechanism is not
the visco-capillary thinning observed in viscous Newtonian
Results of time-dependent simulations with the model carried
out in a
slender filament formulation appropriate for elongational flows
complex fluids are presented. The simulations show that
filaments described by the VCM model exhibit a dramatic and
rupture event similar to that observed in experiments. This
instability is purely elastic in nature (i.e. it is not driven
capillarity) but arises from coupling between the evolution in
tensile stress and the number density of the entangled species.
dynamics of this localized necking are contrasted with
other nonlinear viscoelastic models.
Polymeric threads: formation and instabilityOctober 15, 2009 3:30 pm - 4:10 pm
We report on recent experimental and theoretical work on
pinch-off of dilute solutions of flexible polymers. Owing to
strong extensional hardening of such solutions, pinch-off of a
liquid drop is delayed. Instead, long threads of uniform
form, whose radius decreases exponentially in time. We derive a
relationship between the thread radius and the extensional
When the thread radius has decreased to about 10 microns, the
becomes unstable to the rapid growth of small ``blisters''.
are in strong disagreement with conventional models for dilute
The ensuing dynamics are very rich, and include iterated
periodic ``breathing''. For sufficiently high polymer
the fluid never breaks, and a solid nanometer-sized thread is
Simulation and experiments on selective withdrawal of polymer solutions
October 12, 2009 4:30 pm - 6:30 pm
Selective withdrawal refers to the removal of stratified fluids by a suction tube placed near the interface. We view this as an interesting complex fluid flow problem since the interface is disturbed by the nearby sink flow, and the interfacial morphology depends on the bulk rheology of the fluids. The poster will show recent numerical and experimental results for the selective withdrawal of polymer solutions. The most notable result is a transition from a smooth continuous interface to one with a thin air jet emanating from the tip of the interface, reminiscent of the Taylor cone.
Moving contact lines and enhanced slip on textured substratesOctober 15, 2009 2:15 pm - 2:55 pm
I will discuss the use of a diffuse-interface model for simulating moving contact lines. The Cahn-Hilliard diffusion is known to regularize the singularity and makes possible a continuum-level computation. But relating the results to physical reality is subtle. I will show numerical results that suggest a well-defined sharp-interface limit, with a finite contact line speed that can be related to measurements. Furthermore, I will discuss applications of this model to simulate enhanced slip on textured substrates due to contact line depinning, with viscous or viscoelastic liquids.
Multiscale modeling and simulation of fluid flows in
deformable porous mediaOctober 12, 2009 4:30 pm - 6:30 pm
The main focus of the current poster presentation is on fluid flows in
deformable elastic media and associated multiscale problems. Many upscaling
methods are developed for flows in rigid porous media or deformable elastic
media assuming infinitely small fluid-solid interface displacements relative
to the pore size. Much research is needed for the most general and least
studied problem of flow in deformable porous media when the fluid-solid
interface deforms considerably at the pore level. We introduce a general
framework for numerical upscaling of the deformable porous media in the
context of a multiscale finite element method. This method allows for large
interface displacements and significant changes in pore geometry and volume.
For linear elastic solids we present some analysis of the proposed method.
Mesoscopic simulation of the dynamics of confined complex fluids October 16, 2009 8:30 am - 9:10 am
Keywords: Microfluidics, polymer solutions, Brownian dynamics, blood flow
Abstract: Many interesting and important phenomena in flowing complex fluids arise when the length scale of the microstructure becomes comparable with the length scale of the flow geometry. This is the case, for example, for solutions of genomic DNA in a microfluidic device or blood in the microcirculation. We describe here an efficient computational framework, based on a new real-space particle-particle/particle-mesh approach to solution of Stokes equations, for performing Brownian dynamics simulations of polymer solutions in micron-scale geometries and use them to illustrate and understand hydrodynamic migration phenomena of DNA in these geometries. The methodology will also be combined with a novel immersed boundary method for elastic capsules in Stokes flow. With this approach, we take some initial steps toward understanding the observed beneficial effects that addition of drag-reducing polymers have on blood flow.
Effective viscosity and
dynamics of dilute bacterial suspensions: A three-dimensional model
October 12, 2009 4:30 pm - 6:30 pm
We present a PDE model for dilute suspensions of bacteria
in a three-dimensional Stokesian fluid.
This model is used to calculate the statistically-stationary bulk
deviatoric stress and effective viscosity of the suspension
from the microscopic details of the interaction of an elongated body
with the background flow.
A bacterium is modeled as a prolate spheroid with self-propulsion
provided by a point force, which shows up in the model as an
inhomogeneous delta function in the PDE.
The bacterium is also subject to a stochastic torque in order to model
tumbling (random reorientation).
Due to a bacterium's asymmetric shape, interactions with a prescribed
generic background flow, such as
pure shear or planar shear, cause the bacterium to preferentially align in
certain directions. Due to the stochastic torque, the steady-state
distribution of orientations is unique for a given
background flow. Under this distribution of orientations,
self-propulsion produces a reduction in the effective viscosity.
For sufficiently weak background flows, the effect of self-propulsion
on the effective viscosity dominates all other contributions, leading
to an effective viscosity of the suspension that is lower than the viscosity
of the ambient fluid. This is in agreement with recent experiments
on suspensions of Bacillus subtilis.
An O(N) iterative scheme for viscoelastic flow
simulations with DEVSS
October 12, 2009 4:30 pm - 6:30 pm
In largescale finite element simulations of time-dependent viscoelastic flows the major computational difficulty is the solution of the linear system derived from the momentum and continuity equations. For Newtonian fluids highly efficient iterative solvers have been developed (Elman, Silvester & Wathen "Finite Elements and Fast Iterative Solvers") using block preconditioned Krylov space methods. These methods converge within a fixed number of outer iterations so that both the computational time and memory requirements are proportional to the number of unknowns. Based on these ideas we have developed an iterative scheme for viscoelastic computations discretised using the popular DEVSS (Discrete Elastic-Viscous Stress Splitting) algorithm. We show that this scheme also converges within a fixed number of outer iterations for both two and three dimensional calculations, allowing large three dimensional calculations to be performed efficiently.
Jet break-up of polymer solutions in inkjet printingOctober 15, 2009 1:30 pm - 2:10 pm
Keywords: dilute polymer solutions, FENE model, free surface flows, Finite
Abstract: The effects of polymer additives on the break-up of jets in continuous (CIJ)
and drop on demand (DOD) inkjet printing are considered. In continuous
inkjet printing the fluid jet is modulated at close to the Rayleigh
frequency to produce a steady stream of uniform drops, while in drop on
demand printing individual drops are generated by applying an impulse to the
fluid. Even at very low concentrations the presence of high molecular weight
polymers significantly affects how jets break-up into drops, due to the high
extension-rates involved. By simulating these flows with the FENE dumbbell
constitutive equation we are able to establish the parameter values
controlling the break-up length and character of jet break-up, such as the
production of small satellite droplets (which are detrimental to the
inkjetting process). For the case of drop on demand printing, we compare our
predictions to experimental measurements on dilute solutions of monodisperse
polystyrene. By using Zimm theory to predict the parameter values in the
FENE model, we are able to demonstrate quantitative agreement between
simulations and experiments.
Numerical prediction of the dynamics of nanoparticles embedded in a liquid crystalline solvent
October 12, 2009 4:30 pm - 6:30 pm
A hierarchical modeling approach has been adopted to examine the structure and dynamics of nanoparticle suspensions in confined liquid crystals. A molecular model and a combination of Monte Carlo and molecular dynamics simulations are used to investigate the defects that arise around the nanoparticles, both at rest and other imposed flow fields, and to explore how such defects influence the aggregation behavior of the particles. The continuum molecular model is solved by resorting to a unsymmetric radial basis function based technique. The validity of the model and our numerical results are established by direct comparison to results of molecular simulations and to experimental mobility data in both the isotropic and nematic phases. The model is then used to examine the response of different types of confinement, surface treatment, and flow field on the aggregation pathways of nanoparticles in liquid crystals.
Simulation of particle migration in viscoelastic fluids using the extended finite element method
October 12, 2009 4:30 pm - 6:30 pm
We present an eXtended Finite Element Method (XFEM) combined with a DEVSS-G/SUPG formulation for the direct numerical simulation of the flow of viscoelastic fluids with suspended rigid particles. For the whole computational domain including both the fluid and particles, we use a regular mesh which is not boundary-fitted. Then, the fluid domain and the particle domain are fully decoupled by using XFEM enrichment procedures. For moving particle problems, we incorporate a temporary arbitrary Lagrangian-Eulerian (ALE) scheme without the need of any re-meshing. We show the motion of a freely moving particle suspended in a Giesekus fluid between two rotating cylinders. The particle migrates to a stabilized radial position near the outer cylinder regardless of its initial position. As the Deborah number increases, the stabilized radial position of the particle shifts toward the outer cylinder.
A maximum entropy principle based closure method and
hysteresis for macro-micro models of polymeric materials
October 12, 2009 4:30 pm - 6:30 pm
We consider the finite extensible nonlinear elasticity (FENE) dumbbell
model in viscoelastic polymeric fluids. The maximum
entropy principle for FENE model is employed to obtain the solution which
maximizes the entropy of FENE model in stationary situations. Then
the maximum entropy solution is approximated using the second order
terms in microscopic configuration field to get an probability
density function (PDF). The approximated PDF gives a solution to
avoid the difficulties caused by the nonlinearity of FENE model. The moment-closure
system satisfies the energy dissipation law. The moment-closure system can also show the hysteresis which is a nonlinear
behavior of viscoelastic dilute polymeric fluids. The hysteresis of FENE model can be seen during a relaxation in simple
extensional flow employing the normal stress/the elongational viscosity versus the mean-square extension. The hysteretic
behaviors of viscoelastic dilute polymeric fluids with moment-closure approximation models, FENE-L, FENE-P, FENE-D, are
presented in extensional/enlongational flows.
Foam structure and rheology: The shape and feel of random soap frothOctober 15, 2009 9:45 am - 10:25 am
Soap froth—the quintessential foam—is composed of polyhedral gas bubbles separated by thin liquid films. Why do foams have a shear modulus and yield stress, which we usually associate with solids? How are the bubbles shaped and how are they packed? These and other questions have been explored through simulations with the Surface Evolver, a computer program developed by Brakke. The calculations are in excellent agreement with seminal experiments by Matzke (1946) on the foam structure and shear modulus measurements by Princen and Kiss (1986). The connection between elastic-plastic rheology and foam structure involves intermittent cascades of topological transitions; this cell-neighbor switching is a fundamental mechanism of foam flow. The structure and rheology of wet foams, which have finite liquid content, will also be discussed.
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.
Non-modal amplification of disturbances in channel
flows of viscoelastic fluids: A possible route to elastic
turbulence?October 12, 2009 2:00 pm - 2:40 pm
This talk will provide an overview of our recent work on
amplification of disturbances in channel flows of
viscoelastic fluids. Even if a standard linear stability
(i.e., modal) analysis predicts that a particular flow is
stable, the question of the sensitivity of the flow to
various disturbances remains. If disturbances to the
linearized governing equations are sufficiently amplified
over a finite time interval, then nonlinearities may become
important and cause transition to a more complex flow state.
This can happen if the underlying linear operator is
non-normal, and represents a non-modal mechanism of
disturbance amplification. We address this issue by adopting
an input-output point of view borrowed from the systems- and
control-theory communities. The inputs to the linearized
equations consist of spatially distributed
and temporally varying body forces that are harmonic in the
streamwise and spanwise directions and stochastic in the
direction and in time. Such inputs enable the use of
from linear systems theory that have recently been applied to
analyze Newtonian fluid flows. We find that the most
amplified disturbances are three-dimensional in nature, and
that large amplification can occur under conditions of weak
inertia and strong elasticity. The underlying physical
mechanism involves polymer stretching that introduces an
effective lift-up of flow fluctuations similar to
vortex-tilting in inertia-dominated flows. The mechanism
examined here provides a possible route for a bypass
transition to elastic turbulence and might be exploited to
enhance mixing in microfluidic devices. (Joint work with
Mihailo Jovanovic, University of Minnesota.)
Lattice-Boltzmann methods for polymer solutions - A comparison with Brownian
October 14, 2009 9:45 am - 10:25 am
I will outline the application of the fluctuating lattice-Boltzmann equation
to the simulation of polymer solutions. Then I will describe a numerical
assessment of the accuracy of lattice-Boltzmann methods for polymers, by
comparison with Brownian dynamics simulations on a similar model system.
We will examine the relaxation spectrum of an isolated chain and the
migration of individual chains in shear and pressure-driven flows.
The role of free energy in the mathematical and numerical analysis
of complex fluids modelsOctober 14, 2009 2:15 pm - 2:55 pm
The talk will overview three recent works that make use of the notion of
free energy to establish mathematical properties of some complex fluid
models. The first work (in collaboration with B. Jourdain, T. Lelievre and
F. Otto) studies the long-time behaviour of the solution to some multiscale
models. The second work (by D. Hu and T. Lelievre) introduces a notion of
free energy for purely macroscopic models. The third work (by S. Boyaval,
T. Lelievre and C. Mangoubi) makes uses of the free energy to derive better
Teaching rheology using product design
October 12, 2009 4:30 pm - 6:30 pm
The poster describes two
courses on experimental rheology offered over the past several years to
seniors and first year graduate students at our institutions (KU Leuven
Belgium and U of Minnesota). These laboratory courses use complex materials
available from the shelves of our local retailers. We have found that
measuring the rheology of face cream, shampoo, paint, chewing gum or plastic
bags provides great motivation for students to learn rheology fundamentals.
Human tear film dynamics on an eye-shaped domain
October 12, 2009 4:30 pm - 6:30 pm
We present recent progress in understanding the dynamics of human tear film on an eye-shaped domain. Using lubrication theory, we model the evolution of the tear film over a blink cycle. The highly nonlinear governing equation is solved on an overset grid by a method of lines coupled with finite difference in the Overture framework. Comparisons with experimental observations show qualitative agreement.
Continuum-microscopic computational modeling of non-equilibrium
viscoelastic flowOctober 14, 2009 3:30 pm - 4:10 pm
The problem of coupling microscopic and continuum-level descriptions of
when the microscopic system exhibits slow relaxation times is
considered. This type of
problem arises whenever the fluid exhibits significant memory effects.
The main difficulty
in this type of multiscale computation is the initialization of
microscopic configurations and
establishing the duration of microscopic evolution that has to be
computed before a continuum
time step can be taken. Density estimation theory is applied to
determine the distribution
of random variables characterizing the microscopic system. Additional
for the probability density functions required to characterize
microscopic states are determined
from successive bursts of microscopic simulation. The time evolution of
the mesoscale equations
is computed using high-order Adams-Bashforth-Moulton predictor-corrector
overall computational model is exemplified on a Rolie-Poly fluid. The
main benefit of the approach
considered here is that the complication of deriving an algorithm for
complicated constitutive laws
is sidestepped without the need for prohibitively expensive computation
at the microscale.
Particle pressure-induced phenomena in suspensions:
from osmosis to granular dilation
October 15, 2009 11:00 am - 11:40 am
The coupling of microstructure to bulk flow behavior is a hallmark of non-Newtonian flows in general. When the non-Newtonian fluid is a two-phase material where the dispersed and continuous phases may readily segregate, this coupling is found to result in quite striking migration of particles and fluid, with significant impact on the flow structure as a consequence. In this talk, we first describe a general approach to understanding the migration phenomena based on particle pressure, the nonequilibrium continuation of osmotic pressure to sheared dispersions of solids (here in Newtonian liquids only), considering the microstructural origins of the behavior, its the macroscopic consequences, and how particle pressure may be measured. This will be followed by a consideration of the flow through a channel contraction of a very concentrated suspension – near and at the jamming limit of about up to 58% solids for the system studied experimentally, so that hydrodynamic and contact forces both play a role. In the contraction geometry, the effluent generally has a lower solid fraction than the upstream suspension, a phenomenon known as self-filtration, and we will show that self-filtration may be described by a mechanism where liquid flow is driven by variation of the particle pressure within the geometry.
Effects of elasticity on high Reynolds number instabilities
in Taylor-Couette flowOctober 12, 2009 9:00 am - 9:40 am
Keywords: Elasticity, viscoelastic instability, nonlinear transitions,
Abstract: Taylor-Couette flow (i.e., flow between concentric, rotating
cylinders) has long served as a paradigm for studies of
hydrodynamic stability. For Newtonian fluids, the rich cascade
of transitions from laminar, Couette flow to turbulent flow
occurs through a set of well-characterized flow states that
depend on the Reynolds numbers of both the inner and outer
cylinders (Rei and Reo). While extensive work has been done on
(a) the effects of weak viscoelasticity on the first few
transitions for Reo = 0 and (b) the effects of strong
viscoelasticity in the limit of vanishing inertia
(Rei and Reo
both vanishing), the viscoelastic Taylor-Couette problem
presents an enormous parameter space, much of which remains
completely unexplored. Here we describe our recent experimental
efforts to examine the effects of drag reducing polymers on the
complete range of flow states observed in the Taylor-Couette
problem. Of particular importance in the present work is 1)
the rheological characterization of the test solutions via both
shear and extensional (CaBER) rheometry, 2) the wide range of
parameters examined, including Rei, Reo, and Elasticity number
El, and 3) the use of a consistent, conservative protocol for
accessing flow states. We hope to gain insights into the roles
of weak elasticity and of co- and counter-rotation on nonlinear
transitions in this flow.
Computing complex flows of complex fluidsOctober 12, 2009 2:45 pm - 3:25 pm
Flows with free surfaces and free boundaries arise in many industrial and biological applications. Examples are coating, polymer processing, ink-jet printing, DNA arrays, spraying, deformation of blood cells, blood flow in arteries and capillaries, and flow in the deep pulmonary alveoli. Most of these flows have two distinguishing features: (1) the fluid is complex (microstructured ); thus, the stress includes a visco-elastic term which is important and sometimes dominant, and (2) the surface forces are comparable to the viscous and elastic ones. Inertia is usually immaterial in these flows, because the relevant length scales are well below a millimeter.
The surface and viscoelastic forces give rise to large non-diagonal contributions in the momentum equation. Other non-diagonal terms come from the coupling of the shape of the free boundaries to the velocity field, and strong dependence of the microstructure evolution on the velocity gradient. Thus, fully-coupled algorithms for solving the steady as well as time-dependent equations of the flow are desirable.
I will discuss developments in applying mesoscopic models of microstructured liquids to three-dimensional free surface flows. In such models, the liquid microstructure is captured by tensors obeying convection-diffusion-generation equations—e.g., the gyration tensor of ensembles of polymer molecules, or the shape tensor of droplets or blood cells. Mesoscopic non-equilibrium thermodynamics ties the elastic stress to velocity-gradient-dependent terms in the microstructure evolution. This dependence yields general theories accounting for disparate microstructural models that are compatible with macroscopic transport phenomena and thermodynamics. Such theories can be incorporated into general three-dimensional finite element codes based on fully coupled formulations. Combining Newton’s method with GMRES and a Sparse Approximate Inverse Preconditioner yields a robust and efficient method for computing three-dimensional flows on low-cost parallel clusters.
I will show results on model flows of polymer solutions, and discuss developments and connections to fine-grain, microscopic models of complex fluids where microstructure is tracked by using stochastic differential equations.
Structural instability in sedimentation through
October 12, 2009 4:30 pm - 6:30 pm
A theory has been developed to describe a structural instability that is observed during the sedimentation of particulate suspensions through viscoelastic fluids. The theory is based on the assumption that the influence of hydrodynamic interactions in viscoelastic fluids, which tend to cause particles to aggregate, is in competition with hydrodynamic dispersion, which acts to maintain a homogeneous microstructure. In keeping with the experimental observations, it predicts that the suspension structure will stratify into vertical columns when a dimensionless stability parameter exceeds a critical value. The column-to-column separation, measured in particle radii, is predicted to be proportional to the square root of the ratio of the dimensionless dispersion coefficient to the product of the particle volume fraction and the Deborah number. The time for the formation of the columns is predicted to scale with the inverse of the average volume fraction. These predictions are in agreement with experimental data reported in the literature.
Spherical bubble collapse in viscoelastic fluidsOctober 12, 2009 4:30 pm - 6:30 pm
The collapse of a spherical bubble in an infinite expanse of viscoelastic fluid is considered. For a range of viscoelastic models, the problem is formulated in terms of a generalized Bernoulli equation for a velocity potential, under the assumptions of incompressibility and irrotationality. The boundary element method is used to determine the velocity potential and viscoelastic effects are incorporated into the model through the normal stress balance across the surface of the bubble. In the case of the Maxwell constitutive equation, the model predicts phenomena such as the damped oscillation of the bubble radius in time, the almost elastic oscillations in the large Deborah number limit and the rebound limit at large values of the Deborah number. A rebound condition in terms of $ReDe$ is derived theoretically for the Maxwell model by solving the Rayleigh-Plesset equation. A range of other viscoelastic models such as the Jeffreys model, the Rouse model and the Doi-Edwards model are amenable to solution using the same technique. Increasing the solvent viscosity in the Jeffreys model is shown to lead to increasingly damped oscillations of the bubble radius.
Purely-elastic instabilities in extensional flows
October 12, 2009 4:30 pm - 6:30 pm
Using a finite-volume numerical technique we demonstrate that viscoelastic flow in a range of symmetric geometries - with symmetric inlet flow conditions - containing a region of strong extensional flow goes through a bifurcation to a steady asymmetric state. We show that this asymmetry is purely elastic in nature and that the effect of inertia is a stabilizing one. Our results in one such geometry - the so called “cross-slot” - are in excellent qualitative agreement with recent experimental visualizations of a similar flow in a micro-fluidic apparatus [Arratia et al. Phys. Rev. Lett., 2006 96(14)]. We investigate effects of constitutive equation (UCM, Oldroyd-B, PTT and FENE-CR models), model parameters and effects due to three dimensionality.
Mathematical issues in stability of viscoelastic flows
October 14, 2009 9:00 am - 9:40 am
"Traditional" hydrodynamic stability studies infer stability of a flow from
a computation of eigenvalues of the linearized system. While this is well
justified for the Navier-Stokes equations, no rigorous result along these lines
is known for general systems of partial differential equations; indeed there are
counterexamples for lower order perturbations of the wave equations. This lecture
will discuss how recent results on "advective" equations can be applied to creeping
flows of viscoelastic fluids of Maxwell or Oldroyd type. For spatially periodic
flows, stability can be reduced to the study of a) the eigenvalues, and b) a system
of non-autonomous ordinary differential equations that arises from a geometric
optics approximation for short waves. A more complete result for the upper convected
Maxwell model will also be discussed.
The response of a hydrophobic superparamagnetic
ferrofluid droplet suspended in a viscous fluid in a uniform
magnetic field: the influence of microstructure on interfacial tension
October 12, 2009 4:30 pm - 6:30 pm
The microstructure of a ferrofluid influences its motion under
applied magnetic fields.
A ferrofluid typically consists of magnetite nanoparticles
suspended in a solvent. Here, we consider a
ferrofluid that has no solvent, with the advantage that the
particles do not migrate under externally applied
magnetic fields, and therefore the physical properties of the
ferrofluid can be more easily characterized.
The deformation of a biocompatible hydrophobic ferrofluid
drop suspended in a viscous medium is investigated numerically
and compared with experimental
data. At high magnetic fields, experimental drop shapes deviate
from numerical results when a
constant surface tension value is used. One hypothesis for the
difference is the dependence of
interfacial tension on the magnetic field in the experimental
data. This idea is investigated
with direct numerical simulations.
Numerical investigation of drop deformation in shearOctober 14, 2009 11:00 am - 11:40 am
The dynamics and stability of viscoelastic wormlike
micelle solutions in strong extensional flowsOctober 13, 2009 11:00 am - 11:40 am
Under the proper conditions, surfactant molecules can self-assemble into
wormlike micelles, resembling slender rods, can entangle and impart
viscoelasticity to the fluid. The behavior of wormlike micelles solutions
is similar to that of polymer solutions. The primary difference being that,
unlike covalently bound polymers, micelles are continuously breaking and
reforming under Brownian fluctuations and the imposed shear or extensional
flow field. In this talk, we will discuss the behavior of a series of
viscoelastic wormlike micelle solutions in extensional flows and describe
several newly observed instabilities and flow phenomena unique to these
fluids. In the first part of the talk, we will describe the behavior of
these fluids in the homogeneous uniaxial extensional flow produced by a
filament stretching rheometer. Like polymer solutions, wormlike micelle
solutions demonstrate significant strain hardening and a failure of the
stress-optical. At a critical stress, the wormlike micelle solutions
filaments were found to fail through a dramatic rupture near the axial
midplane. This filament failure likely stems from the local scission of
individual wormlike micelle chains. We will discuss the effect that
pre-conditioning can have on the response of these materials and demonstrate
that the presence of branching in wormlike micelle solutions can significant
reduce the strain hardening of the extensional viscosity. In the second
part of the talk, we will describe how the extensional rheology of these
wormlike micelle solutions can affect more complex flows by presenting a
series of interesting new flow phenomena unique to wormlike micelle
solutions. The experiments will include the observation of a new
instability in the flow past a falling sphere, through a periodic array of
cylinders and past a single cylinder. The flows are investigated through a
variety of experimental techniques including the use of high speed imaging,
particle image velocimetry and flow induced birefringence measurements.
Mixing and instability in two complex fluid flowsOctober 13, 2009 9:45 am - 10:25 am
Keywords: Mixing, Instability, Complex Fluids
Abstract: I will discuss two problems where flow instability drives a complex fluid -- or at least its mathematical model -- into intrinsic oscillations and unsteadiness. Both are in the Stokesian regiime where inertial effects are negligible. In the first, a visco-elastic fluid described by the Oldroyd-B model is driven by a background force that creates a local extensional flow. Beyond a critical Weissenberg number, stress accumulates rapidly there, and a symmetry breaking instability leads to coherent structures and multiple frequencies of oscillation. In the second, the complex fluid is a self-driven suspension of active swimmers. Analysis and simulation show the existence of long-wave instabilities that drive the system from isotropy to strongly mixing flows with system-size correlations.
Efficient numerical computation of fluid interfaces with soluble
surfactant: a viscous drop
October 12, 2009 4:30 pm - 6:30 pm
We address a significant difficulty in the numerical computation of fluid interfaces with soluble surfactant. At large values of bulk Peclet number
for representative fluid-surfactant systems, a
transition layer forms adjacent to the
interface in which the surfactant concentration varies rapidly.
Accurate calculation of the concentration gradient at the interface
is essential to determine
bulk-interface exchange of surfactant and the drop's dynamics.
We present a fast and accurate `hybrid' numerical method that
incorporates a separate singular perturbation reduction of the
transition layer into
a full numerical solution of the interfacial free boundary problem.
Results are presented for a drop of arbitrary viscosity in the
Stokes flow limit, where the underlying flow solver for insoluble
surfactant uses a direct (primitive variable) boundary integral method.
Planar extensional motion of an inertially-driven liquid
October 12, 2009 4:30 pm - 6:30 pm
We derive a time-dependent exact solution of the free surface problem for the Navier-Stokes equations that describes the planar extensional motion of a viscous sheet driven by inertia. The linear stability of the exact solution to one- and two-dimensional symmetric perturbations is examined in the inviscid and viscous limits within the framework of the long-wave or slender body approximation. Both transient growth and long-time asymptotic stability are considered. For one-dimensional perturbations in the axial direction, viscous and inviscid sheets are asymptotically marginally stable, though depending on the Reynolds and Weber numbers transient growth can have an important effect. For one-dimensional perturbations in the transverse direction, inviscid sheets are asymptotically unstable to perturbations of all wavelengths. For two-dimensional perturbations, inviscid sheets are unstable to perturbations of all wavelengths with the transient dynamics controlled by axial perturbations and the long-time dynamics controlled by transverse perturbations. The asymptotic stability of viscous sheets to one-dimensional transverse perturbations and to two-dimensional perturbations depends on the capillary number (Ca); in both cases, the sheet is unstable to longwave transverse perturbations for any finite Ca. This work is in collaboration with Thomas P. Witelski.
Hydrodynamic pattern formation in ultrathin metal films: Robust
route to plasmonic nanomaterials
October 12, 2009 4:30 pm - 6:30 pm
Sustainable production, storage and transportation of renewable energy
is one of the greatest challenges of the 21st century. Harnessing Sun's
energy for powering our planet has long been a dream of scientists and
engineers. Despite the universal appeal and growing usage of solar
energy systems across the globe, notably in developing economies, the
efficiency of energy conversion has remained well below desirable levels
for commercial installations. This is especially a major concern for new
generation photovoltaics, which utilize a thin film (~ 1 micron thick)
of the photoactive material. In this case, traditional light trapping
techniques such as optical gratings (~ several microns) employed for
cells based on bulk photoconductors are not applicable. Metallic
nanocomposites offer much promise in efficient and cost-effective solar
energy harvesting especially for thin film photocells. The central idea
is to exploit the plasmonic interaction between electromagnetic waves
and the localized oscillations of the free electron gas density at the
From a renewable energy perspective, plasmonics principles can be used
to tailor the spectral response of a material to fit applications such
as broadband solar absorption and photo-bioreactor design. This is
accomplished by manipulating the particle size, aspect ratio and volume
fraction as well as utilizing hybridization techniques (e.g. core-shell
materials, multi-metal composites). In this talk, a robust manufacturing
route for such materials, namely laser-induced melting, dewetting and
self-organization of ultrathin (~ nm) metal films deposited on a
suitable substrate, will be discussed [APL 91, 043105 (2007); Phys. Rev.
B, 75, 235439 (2007); Nanotechnology, 17, 4229 (2006); Phys. Rev. Lett.,
101, 017802 (2008)]. Specifically, it will be shown that the knowledge
of thin film hydrodynamic instabilities can be utilized to predict
nanoparticle size and spacing observed in such experiments. The
mechanisms of pattern formation will be illustrated using experimental
visualizations of the dewetting process.
Nonlinear pattern formation and coherent structure dynamics in
viscoelastic FlowsOctober 12, 2009 9:45 am - 10:25 am
Ability to manipulate equilibrium self-assembly and dynamical self-organization in nonlinear systems is of central importance to the success of many emerging technologies. This seminar will focus on flow instability and pattern formation in complex fluids, i.e., fluids with internal microstructure such as solutions/melts of polymers, surfactant/colloidal gels and suspensions. Specific examples discussed will include coherence and chaos in turbulent flows of “viscoelastic” dilute polymer solutions (PRL, 100, 134504 (2008)), solitary vortex solutions that manifest as a result of elastic stress-mediated self organization in complex fluids (PRL, 97, 054501 (2006)) and purely flow-induced phase transitions in surfactant micelles (J. Rheol. . 52, 527-50 (2008)).
A thermodynamically compatible rate type
fluid to describe the response of asphalt
October 12, 2009 4:30 pm - 6:30 pm
We consider a class of viscoelastic rate type models that in particular includes: (i) Oldroyd-B fluid model with three parameters, (ii) nonlinear fluid model derived Rajagopal and Srinivasa  with three parameters, and (iii) nonlinear model with five parameters. We are interested in observing how well are these models capable to capture the experimental data for asphalt performed by J. Murali Krishnan, Indian Institute of Technology, Madras using dynamic shear rheometer. We find out that the model (i) is not able to capture the experimentally observed overshoot for the torque, while we obtain overshoots for the models (ii) and (iii).
Complex motions of vesicles and red blood cells in flow
October 12, 2009 4:30 pm - 6:30 pm
Blood flow in the microcirculation is an extensively studied problem, yet the behavior of red blood cells (RBCs) continues to surprise researchers. For example, recently it was discovered that in steady shear flow RBCs not only tank-tread or tumble, but can also "swing" (tank-treading accompanied by oscillations in the inclination angle) [Abkarian et al. PRL 2007]. I will present our analytical work that quantitatively explains this behavior and other features in the RBCs dynamics.
In steady shear flows, the theory shows that a closed lipid membrane (vesicle or RBC) deforms into a prolate ellipsoid, which tumbles at low shear rates, and swings at higher shear rates. The amplitude of the oscillations decreases with shear rate. The viscosity of a dilute suspension of vesicles or RBCs exhibits a minimum at the tank-treading to tumbling transition. In quadratic flows, the theory predicts a peculiar coexistence of parachute- and bullet-like vesicle shapes at the flow centerline. Vesicles and RBCs always migrate towards the flow centerline unlike drops, whose direction of migration depends on the viscosity ratio. In time-dependent flows, vesicles can exhibit chaotic dynamics.
Shear alignment and mechanical properties of
October 13, 2009 3:30 pm - 4:10 pm
Keywords: Block copolymer solutions, hydrogels, shear aligned, soft crystals
Abstract: Self-assembled block copolymer templates can be used to control the nanoscale structure of materials that would not otherwise order in solution. In this work, we have developed a technique to use close-packed cubic and cylindrical mesophases of a thermoreversible block copolymer (PEO-PPO-PEO) to impart spatial order on dispersed nanoparticles. The thermoreversible nature of the template allows for the dispersion of particles synthesized outside the template. This feature extends the applicability of this templating method to many particle-polymer systems and also permits a systematic evaluation of the impact of design parameters on the structure and mechanical properties of the nanocomposites. The criteria for forming co-crystals has been fully characterized using contrast-matching small-angle neutron scatting (SANS) and the mechanical properties of these soft crystals determined. SANS experiments also demonstrate that shear can be used to align the nanocomposites into single-crystal macro-domains; the first demonstration of the formation of single-crystal nanoparticle superlattices. We are currently utilizing SANS to understand the flow mechanisms of both the neat block copolymer solutions and several types of these co-crystals.
Shape optimization of peristaltic pumpingOctober 12, 2009 4:30 pm - 6:30 pm
We present a variational method for optimizing peristaltic pumping in a two dimensional periodic channel with moving walls to pump fluid (peristalsis is common in biology). No a priori assumption is made on the wall motion, except that the shape is static in a moving wave frame. Thus, we pose an infinite dimensional optimization problem and solve it with finite elements. L2-type projections are used to compute quantities such as curvature and boundary stresses.
Kinetic theories for complex fluids October 16, 2009 1:30 pm - 2:10 pm
Keywords: kinetic theory, polymer-particulate nanocomposites, biaxial liquid crystal polymers.
Abstract: In this talk, I will discuss some latest development in the modeling of polymer-particulate nanocomposites (PNC) and biaxial liquid crystal polymers (BLCP) using kinetic theories. Kinetic theory formulation allows one to integrate the microscopic dynamics to the background macroscopic flow field to yield a two-level or even multi-level model for various complex fluids. Equilibrium phases and dynamical states of the PNC and BLCP will be discussed and their rheological responses to shear investigated.
Validity and limitations of the statistical scaling
hypothesis for a nematic liquid crystal flowOctober 12, 2009 4:30 pm - 6:30 pm
The transition from the isotropic into the nematic state occurs, in a
thermotropic liquid crystals, through the creation of nematically ordered
islands in the overall isotropic fluid. It was argued in the physics
literature that the domain growth of the nematic state is a scaling
phenomenon: the pattern of domains at a later time looks statistically
similar to that at an earlier time, up to a time-dependent change of scale.
The statistical scaling hypothesis states that at a large enough time the
equal time scalar correlation function C(r,t) will assume a scaling form
f(r/L(t)) where L(t) is the time-dependent length scale of nematic domains.
The precise asymptotics of L(t) for large t have been the subject of a
significant debate in the physics literature.
We present a mathematically rigorous analysis of the equations that shows
under what conditions the scaling hypothesis holds and what are the correct
asymptotics of L(t) for large t.
This is joint work with Eduard Kirr (University of Illinois at