Research

Dynamics of contact lines

Huh and Scriven noted in 1971 that liquids shouldn't slide on solid surfaces -- at least if one applies the usual rules of fluid mechanics. The motion of the contact line (where liquid, solid and vapor meet) has since been one of the most studied and controversial subjects in fluid mechanics. Research has largely focused on which additional physical mechanisms should be included. The problem is that there is not just one - many different mechanisms can regulate contact line motion. From a practical perspective, we might ask if it is important to identify the particular mechanisms at work, or is it possible that all mechanisms have the same macroscopic consequences?

The crux of the contact line "paradox" is infinite energy dissipation. In reality, energy dissipation is finite, and we could therefore characterize the manifestation of microscopic physics that regulate contact line motion in terms of the energy dissipated. This can be done in terms of a variational formulation which has been recently studied by the author.

An important dynamic limit is the quasisteady case, where the fluid is nearly at mechanical equilibrium except near the contact lines. This leads to a substantial reduction of the equations of motion, as is the basis for numerically efficient boundary integral calculations (see figure).

Flow over a surface of templated wettability (white = high)
References:

A boundary integral formulation of quasi-steady fluid wetting , J. Comp. Phys., 207, pp. 529-541 (2005).

Variational models for moving contact lines and the quasi-static approximation, Euro. J. Appl. Math, 16, pp. 1-28 (2005).

Spreading of droplets under the influence of intermolecular forces, Phys. of Fluids, Vol.15, pp. 1837-1842 (2003).

Homogenization of contact line dynamics, preprint (2006).

Grain boundary dynamics

Simulations of the regularized Cross-Newell equation, showing grain boundaries:
  • Movie #1
  • Movie #2

    Reference:

    Grain boundary motion arising from the gradient flow of the Aviles-Giga functional , Physica D,Volume 215, pp. 80-98 (2006).

    Coarsening dynamics of dewetting films

    Liquid films are useful in many everyday applications: lubricants, paints, coatings and electronics manufacturing. Mathematical models can adequately describe many events which occur in thin films, such as spreading, fingering, evaporation and other pattern-forming processes.

    A competition between long and short range forces can produce complex instabilities in otherwise uniform films, a process sometimes referred to as spinodal dewetting because of the analogy to Cahn- Hilliard dynamics. The film destabilizes and ruptures (see figure on right) and eventually forms ridge patterns and droplets. The droplets gradually coarsen over long time scales. In contrast to Ostwald ripening (the long term behavior of Cahn-Hilliard dynamics), this process is accomplished by two mechanisms: both mass exchange between "particles" (i.e. droplets) and relative motion between them. The result is an interacting particle system that exhibits dynamic scaling and quasi-self similarity (see figure below).
    Instability and droplet formation of a uniform film.



    Quasi-self-similar droplet trajectories and collisions in 1-D


    References:

    Collision vs. collapse of droplets in coarsening of dewetting thin films, with T. P. Witeslski, Physica D, 209, 80-104 (2005).

    Coarsening Dynamics of Dewetting Films, with T. P. Witelski, Phys. Rev. E, Vol. 67, 016302 (2003).

    Bioconvection

    Swimming microorganisms may be attracted upward toward oxygen sources, and simultaneously be dragged downward by negative buoyancy. In analogy to the Rayleigh-Taylor instability, this leads to a phenomenon called bioconvection. In liquid droplets, this process plays out near the contact lines, producing patterns of horizontal stripes (see Mendelson & Lega, J. Bacteriology, 180,1998). Below are some simulations of this process, showing the development of stripes and eventual "plumes". The contact line of the drop is at the bottom of each figure.

    Gradient Dynamics of Dissipative Fluid Systems

    Dissipative dynamical systems can many times be described by a gradient flow, a system which "flows" in a direction so as to minimize some energy. One example is the motion of a fluid trapped between two glass plates, the so-called Hele-Shaw cell. The bubbles evolution follows a path on the energy "landscape" which reduces the bubble's perimeter (i.e. surface energy). This amounts to an intriguing gradient flow related to optimal transport problems . In a ferrofluid, there is a competition between the attractive nature of surface tension and the repulsive nature of magnetic forces, causing the fluid bubble to evolve into complicated shapes (see figure at right).

    References:

    A Diffuse Interface Approach to Hele-Shaw Flow , Nonlinearity , Vol. 16, 1-18 (2003).

    A Diffuse Interface Model for Electrowetting Droplets in a Hele-Shaw Cell , w/ H.-W. Lu, A. Bertozzi, C.-J. Kim (2005).

    Recent Papers

    Ostwald Ripening in Thin Film Equations , preprint (2007).

    Ostwald Ripening of Droplets: The Role of Migration , w/ F. Otto, T. Rump. D. Slepcev, preprint (2007).

    The dynamics of pendant droplets on a one dimensional surface, Phys. Fluids (in press) (2007).

    A Diffuse Interface Model for Electrowetting Droplets in a Hele-Shaw Cell , w/ H.-W. Lu, A. Bertozzi, C.-J. Kim, JFM, Vol. 590, 411-435 (2007).

    Homogenization of contact line dynamics, Int. Free Bound., Volume 8, (2006).

    Grain boundary motion arising from the gradient flow of the Aviles-Giga functional , Physica D,Volume 215, pp. 80-98 (2006).

    A boundary integral formulation of quasi-steady fluid wetting, J. Comp. Phys., 207, pp. 529-541 (2005).

    Collision vs. collapse of droplets in coarsening of dewetting thin films, with T. P. Witeslski, Physica D, 209, 80-104 (2005).

    Variational models for moving contact lines and the quasi-static approximation , Euro. J. Appl. Math, 16, pp. 1-28 (2005).

    Spreading of droplets under the influence of intermolecular forces, Phys. of Fluids, Vol.15, pp. 1837-1842 (2003).

    Coarsening Dynamics of Dewetting Films, with T. P. Witelski, Phys. Rev. E, Vol. 67, 016302 (2003).

    A Diffuse Interface Approach to Hele-Shaw Flow , Nonlinearity , Vol. 16, 1-18 (2003).

    Solute Trapping and the Non-equilibrium Phase Diagram for Solidification of Binary Alloys , Physica D , 151 (1-4) (2001) pp. 253-270.

    Nonlinear Preconditioning for Diffuse Interfaces , Journal of Computational Physics, Vol. 174, No. 2, Dec 2001, pp. 695-711.

    Rapid Growth and Critical Behavior in Phase Field Models of Solidification, European J. Appl. Math, Vol. 12, pp. 39-56, (2001).

    Dual Fronts in a Phase Field Model, with R. Almgren, Physica D , 146, pp. 328-340 (2000).

    Traveling Waves in Rapid Solidification, Elec. J. Diff. Eq. , 2000, #16, pp. 1-28 (2000).