We investigate two-dimensional and axisymmetric van-der-Waals rupture of a thin liquid film on a solid substrate in the long-wavelength limit. Numerical solutions to the lubrication equations show that the dynamics close to rupture is determined by both surface tension and van-der-Waals disjoining pressure. The equations possess a countably infinite number of similarity solutions, in each of which the horizontal lengthscale decreases like t^2/5 and the film thickness decreases like t^1/5, where t is the time remaining to rupture. Only one of these solutions is thought to be dynamically stable and realized in practice.
[Cd.02] Rupture by Van der Waals Forces
Dmitri Vaynblat, Michael Brenner (Dept. of Mathematics, MIT), John Lister (DAMPT, University of Cambridge), Thomas Witelski (Dept. of Mathematics, MIT)
We construct scaling theories for the rupture of thin bodies of fluid in three geometries: (i) a 2D layer with planar symmetry, (ii) with central symmetry, and (iii) a cylindrical thread. The model equations contain the effects of surface tension, viscosity, and van der Waals forces, as originally derived by Enreux and Davis~(T. Erneux and S. H. Davis, Nonlinear rupture of free films, Physics of Fluids A, 5 (1993) 1117--1122.). \underlineAt high Reynolds numbers, there is (in each geometry) a countable set of solutions with the same scaling exponents. In 2D, the solutions are symmetric, whereas for the thread the solutions are 0. For each geometry, only one solution is dynamically selected for generic initial conditions. In all geometries, at high Reynolds number the solutions violate the long wavelength assumption, under which the equations were derived. \underlineAt low Reynolds number, the situation is very different: There is a continuous family of similarity solutions with different scaling exponents. The dynamics picks special members of this family. These special solutions all obey the long wavelength assumption and, therefore, should be physically realizable. The selection mechanism determining which solution occurs will be remarked upon.
[Cd.03] A Positivity Preserving Finite Difference Scheme for Thin Film Equations
Liya Zhornitskaya, Andrea Bertozzi (Duke University)
Lubrication equations are commonly used to describe thin films or liquid layers driven by surface tension. Recent studies of singularities, describing rupture of the fluid layer, show that such equations exhibit complex dynamics which can be difficult to simulate accurately. In particular, one must ensure that the numerical approximation of the interface does not show a false premature rupture. Generic finite difference schemes have the potential to manifest such instabilities especially when underresolved. We design a new numerical method that is guaranteed to have a positivity preserving property, regardless of the spatial resolution, whenever the PDE has such a property.
[Cd.04] Long-wave intabilities and saturation in thin film equations: scaling theory, Lyapunov functions, and weak solutions.
Mary Pugh (Department of Mathematics, University of Pennsylvania), Andrea Bertozzi (Department of Mathematics, Duke University)
Long-wave unstable thin film models form a subclass of Cahn-Hilliard type equations with degenerate mobility. Hocherman and Rosenau (Physica D, 67):113-125, 1993 conjectured that long-wave unstable Cahn-Hilliard type interface models develop finite time singularities when the nonlinearity in the destabilizing term grows faster at larger amplitudes than nonlinear effects in the stabilizing term. For the thin-film subclass, dimensional analysis with a constrained volume suggests a different blowup regime than conjectured in [1]. We present this analysis and discuss how, via Lyapunov function arguments, to prove this result. We also discuss, in a physical context, new weak solution theory for moving contact lines. Specific problems include gravitationally destabilized films and the gravity driven Hele-Shaw cell.
[Cd.05] Long-wave instabilities and blowup in thin film equations: computational results.
Andrea Bertozzi (Department of Mathematics, Duke University), Mary Pugh (Department of Mathematics, University of Pennsylvania)
Hocherman and Rosenau (Physica D, 67):113-125, 1993 conjectured that long-wave unstable Cahn-Hilliard type interface models develop finite time singularities when the nonlinearity in the destabilizing term grows faster at larger amplitudes than nonlinear effects in the stabilizing term. We revise their conjecture for a subclass of models, corresponding to a family of equations often used to model thin films in a lubrication context, to say that the destabilizing term can be stronger (by up to a power of two in the nonlinear diffusion coefficient) yet the solution still remains globally bounded. This alternate scaling is motivated by a conservation of volume constraint, and can be rigorously proved for the bounded solutions. We present numerical computations of singularities near the borderline case of this conjecture, confirming both the blowup and global boundedness regimes. We also discuss relationship with recent results (A. J. Bernoff and A. L. Bertozzi, Physica D, 85):375--404, 1995 for a Modified Kuramoto Sivashinsky equation.
[Cd.06] Mathematical Modeling and Numerical Simulation of Coating Flows over Curved Surfaces
R.V. Roy, L. W. Schwartz (University of Delaware)
Certain defects in surface coatings arise from the flow of liquids. Many different physical factors are known to create flows with corresponding surface defects. Here we examine the less-understood influence of substrate curvature which can radically modify the long-term appearance of a coating. First we derive a mathematical model for the three-dimensional flow of a viscous, incompressible, Newtonian liquid layer on a curved substrate. By exploiting the thinness and slowness of the fluid layer, we arrive at a higher-order lubrication equation of the flow dynamics under the influence of viscous, surface tension and gravitational forces. Our model systematically accounts for the curvature of the substrate and that of the surface of the film. Generic features of substrate curvature effects can be demonstrated. We also show how numerical simulation of this model can be performed in an efficient manner. Time-dependent, three-dimensional numerical simulations will be shown on various complex surfaces that exhibit specific features of the dynamics of such thin fluid films.
[Cd.07] Axial Instability of Coating Flow in a Horizontal Rotating Cylinder
A.\ E.\ Hosoi (The University of Chicago), L.\ Mahadevan (Massachusetts Institute of Technology)
We investigate the axial instability of the free-surface of a viscous fluid in a horizontal cylinder rotating about its major axis. At low rotation rates, the shape of the free-surface appears to be determined by the balance between gravitational and viscous forces. Following earlier work (Benjamin, Pritchard and Tavener), we use an asymptotic expansion in the small parameter \alpha = \sqrtØmega \nu øver g R where \nu is viscosity, Ømega is angular velocity, g is gravity and R is the radius of the cylinder, to derive a simplified evolution equation for the free-surface. This equation is solved numerically to determine the base state with no axial variation, which is then perturbed to examine the onset of unstable axial modes. Various computational results will be presented for the shape of the free-surface and the wavelength of the axial instability. We show that inertia plays an important role in the onset of the instability and we derive the power law \lambda = \gamma^1/3 where \lambda is the wavelength of the axial instability and \gamma is surface tension.
[Cd.08] Two-Dimensional Flows in Langmuir Monolayers : Optical and Rheological Study.
Rajesh Ghaskadvi, Michael Dennin (UC Irvine Department of Physics and Astronomy)
Langmuir monolayers consist of amphiphilic molecules that are confined to the air-water interface and are intrinsically two-dimensional systems. In addition to isotropic gas and liquid phases, they are known to possess a large number of liquid crystal phases that are two-dimensional analogues of three dimensional smectic liquid crystals. The liquid crystal phases are composed of randomly oriented, mesoscopic domains and are known to exhibit anomalous viscous behavior(L. E. Copeland, W. D. Harkins, and G. E. Boyd, J. Chem. Phys. 10), 357 (1942).. However, the source of the anomalous viscosity is still not well understood. For example, is the measured viscosity due to the motion of molecules or the motion of domains? And if the later is the case, how does this mechanism depend on the microscopic structure? To explore these questions, we have built an apparatus to study mechanical and flow properties of Langmuir monolayers. It consists of a torsion pendulum to measure the shear modulus and the viscosity in a cylindrical geometry. The outer wall can be rotated to create a 2-d Couette flow. A Brewster angle microscope is used to directly observe domain motions, rearrangements, and extensions. We will present the results of simultaneous measurements of domain dynamics and viscoelastic coefficients.
[Cd.09] Stability of Liquid Layers with Blowing
M. A. Karabeyoglu (Stanford University), D. Altman (UTC/CSD), B. J. Cantwell (Stanford University)
The stability of a liquid layer under strong blowing and subject to large shear forces is investigated. This case is of practical importance for applications such as the regression rate estimation of cryogenic hybrid rockets. An Orr-Sommerfeld equation for the linear stability of the liquid gas interface is derived and an exact solution is found for a linear base velocity profile. The exact solution for the liquid phase is coupled to the linearized gas phase response with appropriate boundary conditions at the interface to give an eigenvalue problem for the linear stability of the layer. The results for liquid layer Reynolds numbers of practical interest (50-300) show the existance of a range of unstable wave numbers. It is observed that both the most amplified wave number and the maximum amplification increases with the Reynolds number. At these moderately high Reynolds numbers the effect of gravity is small.
[Cd.10] Capillary effects at newly formed liquid-liquid contacts
Martin Rein (DLR-Institute of Fluid Mechanics, Göttingen, Germany)
When two immiscible liquids are brought into contact with a negligible velocity capillarity is of primary importance. Recent experiments of Dooley et al.(B. S. Dooley, A. E. Warncke, M. Gharib and G. Tryggvason, Exp. in Fluids 22), 369 (1997). indicate that this may also be the case when liquids are miscible. In particular, the experimental results point at a process that may be interpreted as a self-spreading of liquids. We have investigated capillary processes occurring right after the instant of contact between arbitrary pairs of liquids. The geometry considered is that of a drop touching the plane free surface of a pool. This geometry is representative of any configuration in which a surface of finite curvature touches another one whose curvature is much greater. Based on energy arguments we first show that right after formation of contact the pool liquid should always spread up the surface of the drop. In a second step the dynamics of the fluid motion caused by capillarity are described by a Lagrangian equation of motion. The results are finally discussed in the light of vortex ring formation which is a phenomenon characteristic of drops coalescing with a free surface. In particular, spreading liquids enable a mechanism of vorticity generation that has not been considered in this context before.