Dynamics of Phase Boundaries
and Nonclassical Waves
Kirchzarten, May 7-9, 2001

Abstracts of Talks

Undercompressive shocks in driven films I:Experiments, mathematical model, and numerical computations

Andrea L. Bertozzi
Duke University, Durham

We consider viscous films driven by a thermal gradient with an opposing gravitational force. When the gravitational effect becomes non-negligible, the advancing front produces a very large capillary ridge which shows a remarkable tendency to remain stable. This phenomenon can be explained by a lubrication model of the experiment.
The advancing front evolves into an `undercompressive' capillary shock structure which is stable to contact line perturbations, unlike typical capillary ridges in driven film flows. We discuss experiments conducted by Ludvikkson/Lightfoot in 1971 and more experiments by A. M. Cazabat and colleagues (PRL 1998, JCIS 2000). These results are compared with numerical simulations and linear stability computations for traveling wave solutions.


M. Schneemilch, A. M. Cazabat, JCIS 2000.
A. Muench, A. L. Bertozzi, Phys. Fluids, 11(10) pp. 2818-2814, 1999.
A. L. Bertozzi, A. Muench, X. Fanton, A. M. Cazabat, Phys. Rev. Lett., 81(23), p. 5169-5172, 1998.
M. Bowen, A. L. Bertozzi, in preparation.

Undercompressive shocks in driven films II:Existence, bifurcations, and stability of traveling wave

Andrea L. Bertozzi
Duke University, Durham

Equations of the type

$$h_t + (h^2-h^3)_x = -\mathbf{e}^3(h^3h_{xxx})_x$$

arise in the context of thin liquid films driven by the competing effects of a thermally induced surface tension gradient and gravity. Jump initial data, from a moderately thick film to a thin precurser layer, is shown to give rise to a double wave structure that includes an undercompressive wave.
This wave, which approaches an undercompressive shock as $\mathbf{e}\to 0,$ is an accumulation point for a countable family of compressive waves having the same speed. The family appears through a series of bifurcations parameterized by the shock speed. At each bifurcation, a pair of traveling waves is produced, one being stable for the PDE, the other unstable.
We discuss an existence theorem for the undercompressive wave and Evans function theory for one-dimensional stability of the compressive and undercompressive waves.


A. L. Bertozzi, A. Muench, M Shearer, K. Zumbrun, EJAM to appear 2001
A. L. Bertozzi, M. Shearer, SIAM J. Math. Anal., 32(1), pp. 194-213, 2000
A. L. Bertozzi, A. Muench, M. Shearer, Physica D 134(4), 431-464, 1999

Sonic Phase Transitions and Chapman-Jouguet Detonations

Andrea Corli
Universita di Baria

We show that the initial-value problem for an $n\times n$ system of strictly hyperbolic conservation laws in one space dimension admits a weak global solution also in presence of sonic phase boundaries. Applications to Chapman-Jouguet detonations and elastodynamics are considered.

Linear stability of subsonic phase transitions in a van der Waals fluid

Jean-Francois Coulombel
École Normale Supérieure de Lyon

Van der Waals type fluids are known to exhibit subsonic phase transitions that can be viewed as non classical shock waves. Following Majda's approach, Sylvie Benzoni-Gavage proved that a certain class of multidimensional planar phase transitions, namely those verifying Maxwell's area rule, are weakly stable (the weakness refering to the existence of surface waves). The purpose of the talk is to obtain an energy estimate on the free boundary problem derived in this previous work. This objective will be achieved by a suitable modification in the usual construction of Kreiss' symmetrizer. The existence of surface waves will give rise to some "loss of derivatives" on the solution to the free boundary problem. This phenomenom has already been observed in anterior works of W. Domanski or M. Sable-Tougeron in elastodynamics. However, the case of phase transitions seems quite pathological compared to these results since the losses of derivatives turn out to be optimal.

Simplified asymptotic models for weakly nonlinear nonclassical waves with applications to elastodynamics

Wlodzimierz Domanski
Polish Academy of Sciences

The appearence of nonclassical waves in hyperbolic systems of conservation laws is connected with a loss of strict hyperbolicity and/or a loss of convexity. The point at which eigenvalues become multiple is called umbilic. We are interested in the analysis of the local structure of hyperbolic equations near umbilic points with the help of the method of weakly nonlinear geometric optics. This asymptotic method is used to derive simplified canonical equations for waves' amplitudes. The derivation is illustrated on the example of the equations of nonlinear elastodynamics. We consider 6x6 first order quasilinear system for plane elastic waves. Both geometric and physical nonlinearities are included in the model. We analyse an isotropic as well as an anisotropic (cubic crystal) medium. We have chosen three principal directions of wave front propagation in a cubic crystal for which, in the unstrained constant state, splitting into pure longitudinal and pure shear waves takes place. These examples illustrate different types of degeneracies occurring for quasi-shear waves. The characteristic feature of these waves is a local loss of genuine nonlinearity which is typically accompanied by a local loss of strict hyperbolicity at the zero constant state. The derived asymptotic equations in these cases confirm the fact that a cubic type nonlinearity is associated with quasi-transverse elastic waves as opposed to a quadratic nonlinearity typical for quasi-longitudinal waves. Interesting phenomena occur for $[1,1,1]$ direction where in each pair of quasi-shear waves one is locally genuinely nonlinear and the other is locally linearly degenerate. In this case new asymptotic equations for pairs of coupled quasi-transverse waves are derived.

References

W. Domanski "Weakly Nonlinear Elastic Plane Waves in a Cubic Crystal", pp. 45 - 61, Contemporary Mathematics, vol. 255, in Nonlinear PDE's, Dynamics and Continuum Physics, 1998 AMS-IMS-SIAM Joint Summer Research Conference on Nonlinear PDE's, Dynamics and Continuum Physics, edited by J. Bona, K. Saxton, R. Saxton, American Mathematical Society, Rhode Island, 2000.

W. Domanski "Propagation and Interaction of Finite Amplitude Elastic Waves in a Cubic Crystal", pp. 249 - 252, In: Nonlinear Acoustics at the Turn of the Millennium, ISNA 15, 15th International Symposium on Nonlinear Acoustics, Goettingen , September 1-4, 1999, AIP Conference Proceedings, vol. 524, edited by W. Lauterborn and T. Kurz, American Institute of Physics, Melville, New York, 2000.

Stability of undercompressive and overcompressive shocks

Heinrich Freistühler
Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig

TBA

Mechanics of Fluids with Inertia Effects

S.L. Gavrilyuk
Université d'Aix-Marseille

The aim of this lecture is to discuss the main properties for a class of novel fluid mechanics describing, in particular, bubbly fluids. Important notions of generalized vorticity and potential flows are introduced which permits us to prove the analogues of classical theorems of Fluid Mechanics for this class of models. A non-local Hamiltonian formulation of generalized potential flows is proposed and the analogues of classical explicit solutions are found.

On the Rayleigh - Taylor Instability and transition from Nucleate to Film Boiling

Lidia Klassen
Universität Freiburg

Experimental studies of boiling on a heated wall show that transition from nucleate boilng to film boiling is strongly affected by wall effects, such as thickness, material, contamination etc. At the present, no theory exists which could describe these effects quantitatively.

In the present work a linear stability analysis of a very thin static vapor layer located on a heated wall and below its liquid phase is performed. The wall is either homogeneous or consists of two layers of different materials. The mathematical model consists of incompressible Navier - Stokes equations for fluids and haet transfer equaitons for both fluids and the wall. Under the pertubation of the static state, a phase transition occurs at the liquid-vapor boundary. At the wall boundary, a constant heat flux $Q$ is kept. The stability analysis shows existence of a critical value $Q_{cr},$ such that for $Q>Q_{cr}$ the vapor layer is stable because presence of the wall enhances effects of dissipation which dominate in comparison with the Rayleigh - Taylor gravity mechanism of instability. The value $Q_{cr}$ depends on the thickness and material of the wall. This dependance is described in terms of dimensionless similarity criteria. A similarity criterion for $Q_{cr}$ coincides with the criterion by Kutateladze for the critical heat flux determining transition from nucleate to film boiling. Experimental validification is neseccary to verify whether the similarity criteria for the wall effects can be used to describe quantitativly the critical heat flux of nucleate boiling on wall of finite thickness.

Continuum-mechanical models of phase-transforming solids and related issues

James K. Knowles
California Institute of Technology

Lecture 1


This lecture presents the fundamentals of a one-dimensional, continuum-mechanical theory of stress-induced phase transitions in nonlinearly elastic solids. Although the emphasis will ultimately be on dynamic response in contexts such as impact, the lecture will begin with background material concerning equilibrium phase mixtures and quasi-static processes of the kind often arising in experiments on hysteresis in materials such as shape-memory alloys. This setting provides the natural motivation for the continuum-mechanical notion of driving force, and it exhibits the connection between this notion and the corresponding concept in materials science. Quasi-static processes also furnish the simplest framework in which to exhibit the ill-posed character of the basic problems of continuum mechanics when formulated for phase-transforming materials, making clear the need for a nucleation criterion and a kinetic relation. Next, we turn to the energetics of the fundamental dynamical system of partial differential equations of mixed type arising in this model, and the effect of inertia on the definition of driving force. We then formulate the impact problem for phase-transforming solids and discuss the global structure of solutions in the especially simple case of the two-phase trilinear material model. The roles of the nuclear criterion and the kinetic relation are discussed in detail.



Lecture 2


We begin by applying the theory of the impact problem developed in the first lecture to a series of experiments by D.J. Erskine and W.J. Nellis involving a phase transition from graphite to diamond induced by compressive impact. In particular, we describe

1. the determination of the kinetic relation from the experimental results,
2. the prediction by the model of the onset of the so-called "overdriven" response observed in the experiments, and
3. the relationship of this phenomenon to "supersonic" phase boundaries for which kinetics cannot be prescribed.

We turn next to the incorporation of thermal effects into the theory by sketching the modifications needed to model phase-transforming thermoelastic solids, and we briefly describe the impact problem in this context. Finally, we discuss the sense in which the notions of driving force and kinetic relation may be mathematically applicable in a model for impact-induced tensile waves in a rubberlike material, for which the system of partial differential equations, while always hyperbolic, fails to be genuinely nonlinear.

Compaction Waves in a Granular Bed

Ralph Menikoff
Los Alamos National Laboratory

Continuum mechanics typically treats materials as homogeneous. This is valid for gases and liquids but not for solids. A granular bed provides an extreme example of a heterogeneous material. Behind a moderate strength wave, the shock compression in a granular material is due to squeezing out pore space. The increase in the density of individual grains is negligible. This type of shock is known as a compaction wave. The key properties of compaction waves are displayed in meso-mechanics simulations (continuum mechanics calculations in which individual grains are resolved). Behind the wave front, fluctuations in hydrodynamic quantities arise from material heterogeneities. Nevertheless, average wave profiles have the appearance of a dispersed shock wave. Consequently, on a coarse grain scale heterogeneous materials can be described by the usual continuum mechanics models. However, the average constitutive properties of a granular material need to account for visco-elastic or visco-plastic response not present in the individual grains. Determining the effective constitutive properties of a composite from those of its components has important engineering applications. Numerical experiments can provide insight and guidance for developing an analytical theory for heterogeneous materials.

LA-UR-01-1274
Workshop: Dynamics of Phase Boundaries and Nonclassical Waves
Frieburg, Germany
May 7-9, 2001

The Thickness of a Marangoni-Driven Thin Liquid Film Emerging from a Meniscus

Andreas Münch
TU München

Recent experimental and theoretical investigations have uncovered the rich and unusual wave structure that occurs in thin liquid films that are driven up an inclined wafer by a thermally induced Marangoni force. Depending on the thickness of the precursor ahead of the rising front and on the trailing film thickness $h_\infty$, we have either a simple compressive capillary wave or a double wave involving a leading undercompressive shock profile. In this talk we focus on how $h_\infty$ is fixed by the meniscus that connects the film to the liquid reservoir.

We develop a new model which describes the thin film as well as the meniscus. Stationary solutions for the meniscus (and $h_\infty$) are locally unique iff

$$h_\infty < \frac{2\text{Ca}^{1/2}}{\cos\alpha}$$

(Ca Capillary number), which is satisfied for small inclination angles. Our numerical and matched asymptotic solutions for $h_\infty$ show good agreement with experimental measurements. For larger inclination angles, however, the uniqueness property is lost. This is related to negative speed of the trailing Lax shock, and the fact that the undercompressive shock height is determined by the precursor thickness $b$.

Dynamic morphology evolution in twinning and phase transitions

Phoebus Rosakis
Cornell University

A two-dimensional, continuum theory of twinning and phase transition dynamics is presented, aimed at describing complex morphological evolution of curved interfaces. Twinning is modelled using a nonconvex elastic stored energy with multiple wells, determined by lattice symmetry. Equilibrium twinning deformations are briefly considered; they involve discontinuous shear localized within narrow twin zones. Their shape is restricted by exclusion of instability: twin boundaries, which may be curved, must be closely alligned with special planes and terminate into cusps. This prediction is in agreement with observed twin morphology.

Various dynamic twin growth problems will be discussed. A kinetic relation governing twin boundary evolution is imposed; it involves boundary orientation dependence, whose form is obtained by appealing to the dislocation model of twinning.

Steady growth of a semi-infinite twin lamella under remotely applied stress is considered. The steady-state tip velocity may be subsonic, sonic or supersonic. The steady dynamic problem is solved explicitly. Under a broad class of kinetic laws we find that subsonic growth cannot be steady, while supersonic motion is possible under high enough applied stress, in agreement with recent developments of dislocation dynamics.

The problem of quasi-steady growth is addressed, concerning the case when shape changes are slow compared with the average constant growth speed. Interface evolution is governed by a nonlocal evolution equation. The stability and large-time dynamics of the evolving interface curve are determined.

In order to describe more complex morphologies, an adaptation of the level-set method, ordinarily used to describe curvature driven motion, is employed. This method allows imposition of an independent kinetic relation, which governs interface motion through a Hamilton-Jacobi equation, coupled with momentum balance. Finite-difference computations exhibit complex morphological dynamics, including cusp formation, needle growth, tip splitting and topological changes resulting in microstructure refinement, in qualitative agreement with experimental observations.

Phase domains and phase solitons in nonlinear optical systems

Kestitutis Staliunas
Physikalisch-Technische Bundesanstalt Braunschweig

Recent studies have lead to prediction and experimental observation of phase domains and phase solitons in some nonlinear optical systems, such as in broad aperture degenerate optical parametric oscillators, and degenerate four wave mixers. The phase domains and phase solitons can be applied in parallel information processing by nonlinear optical systems. There is a ESF network, where the phase domains and phase solitons in nonlinear optics are studied:
http://www.esf.org/physical/pn/phase/phaseb.htm
During the lecture I will speak about phase domains in nonlinear optics, theory and experiment, and about research directions in frame of the above ESF network. Also I will illustrate and demonstrate main features of dynamics of phase domains and phase solitons using interactive simulators.

Discrete BGK Approximations for Dynamic Phase Transitions

Shaoqiang Tang
Universität Konstanz

In this talk, we shall discuss low-order regularizations in the form of discrete BGK approximations to dynamic phase transitions. In 1-D, numerical results show that such approximations are stable and robust. Moreover, it may yield a catagory of kinetic relations and nucleation criteria, by choosing different velocity fields and local maxwellian functions in the BGK models. In particular, it contains Suliciu's model and Jin-Xin's relaxation model as specific cases. For a tri-linear structural relation in these two models, we may work out explicitly the corresponding Riemann solutions for the Suliciu's model and relaxation model.

Numerically, the low order regularization enjoys many nice features, including the clarity and modular form of the coding, no Riemann solver, no high-order derivatives, etc. Moreover, low order approximations are easier to cope with in higher dimensions. Some numerical simulations in 2 space-dimensions will be presented.

Drop Simulations of Newtonian and Non-Newtonian Fluids.

Kerstin Wielage
Paderborn Center for Parallel Computing, Paderborn University

In the industry the production of powder coating particles is done by energy-costly milling of polymers. Particles obtained from this process are of sharp-edged irregular shape. With a novel technique some disadvantages of actual used production process of polymer powder can be avoided. Particles are created by disintegration of polymer melt in an ultrasonic standing wave field.

For the more exact investigation of the disintegration process it is important to examine the behavior of spherical and deformed droplets in a gas flow before.

In the talk we will present some numerical simulations calculated by a 3-D program which has been developted by our cooperation partners in Stuttgart and it's emphasize in our project supported by the Ministerium für Schule, Wissenschaft und Forschung (MSWF).

The numerical program solves the Navier-Stokes equations for incompressible two-phase flows by using a Finite Volume method. The conservation equation of the liquid mass is solved using an highly evolved VOF (Volume of Fluid) - Method.

The emhasis of the talk should be the actual investigations about Non-Newtonian Fluids.

References:

[1] Hirt, C. W. and Nichols, B. D.: Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries, J. Comp. Phys., Vol. 39, No. 1, pp. 201-225, 1981.

[2] Rieber, M.; Graf, F.; Hase, M.; Roth, N.; Weigand, B.: Numerical simulations of moving spherical and strongly deformated droplets. Proc of the ILASS-Europe 2000. Darmstadt, Germany (2000).

The Cahn-Hilliard-Equation with Elasticity.

Ulrich Weikard
Universität Bonn

Phase separation and the evolution of phases in binary alloys can be described by the Cahn-Hilliard model. This fourth-order parabolic equation can be extended to include elastic effects. The elasticity has a profound effect on the shape of the interfacial regions.

After introducing the model a discretization based on finite elements is presented. Several properties of the modeland its discretization are discussed and numerical results are shown.

On some three-dimensional nonlinear thermoviscoelastic models for phase transitions in shape memory alloys

Johannes Zimmer
TU München

As a first model, we will study a coupled system of nonlinear evolution equations derived from conservation of linear momentum and conservation of energy. These equations of nonlinear thermoviscoelasticity are used as a phenomenological model of the thermodynamic behaviour of martensitic transformations in two or three space dimensions. To be able to describe phase transitions in martensitic crystals, the constitutive assumptions do not require the stored energy potential $\Phi$ to be convex, but $\Phi$ has to satisfy certain growth conditions and bounds. Unlike in other work, we do not introduce a regularizing higher-order term. The heat equation is not linearized; this leads to a parabolic equation with a right hand side in $L^1\left(0,T;L^1(\Omega)\right)$.
To deal with this, we use the theory of renormalized solutions. To complete the initial-value boundary problem, Dirichlet data is assumed both for displacement and temperature. We prove the existence of a solution that is global in time.

If time permits, we would like to address another model which includes a capillarity term and has been recently studied by Pawlow and Zochowski. Extensions of their results to boundary conditions which are motivated from medical applications are discussed. We will give a concrete example of a stored energy potential which satisfies the symmetry requirements for a cubic to tetragonal phase transition and the growths conditions made in the existence theorems mentioned above.