Session M21: Instability: Interfacial and Thin Films II

Instabilities of evaporating non-isothermal ultra-thin film with insoluble surfactant

The stability of an evaporating ultra-thin liquid layer with insoluble surfactant spreading over a free deformable interface is investigated within lubrication theory. The evaporation process is described by 2D one-sided model based on the assumptions of density, viscosity and thermal conductivity of the gaseous phase being small compared to the same properties of the liquid phase. It is assumed that the thermal resistance to the evaporation at the interface is an increasing linear function of surfactant concentration. The evaporation mass flux depends on the interface temperature and vapor pressure. Using the long-wave approach and assumption of slow time evolution, a system of nonlinear equations governing the nonequilibrium evaporation is obtained. The system retains main physical effects which take place in the system. A linear stability analysis is also carried out. Both monotonic instability mode and oscillatory one are found and analyzed. The analysis does not include the Born repulsion force in intermolecular interactions.   

Three-dimensional modelling of film flows over spinning disks

Film flows over spinning disks are of central importance to a wide array of industrial processes, such as the augmentation of heat and mass transfer in chemical reactors, or power production in metallurgy. As a result they have been extensively investigated experimentally. Theoretically they constitute an interesting problem due to the interplay of inertial, capillary, centrifugal and Coriolis forces. However, modelling efforts have typically been restricted to the consideration of the one-dimensional axisymmetric situation. We extend the existing models to incorporate azimuthal variations. The resultant system is solved via the use of an operator-splitting method. In addition, we have performed Direct Numerical Simulations of the system. We compare the low order model, the direct simulations and the results of experiments, to reveal a wide variety of different flow regimes in accordance with existing literature.   

Buckling instabilities in photopolymerised gels

Frontal photopolymerisation (FPP) is a process whereby solid polymer networks are created by illuminating a monomer-rich bath with collimated light. In practice, FPP can be used to rapidly fabricate intricate small-scale structures. Due to the attenuation of light as it propagates through the bath, polymerisation occurs in a wave-like fashion from the illuminated surface into the bulk. At low temperatures, the polymerisation front remains planar; however, at higher temperatures, it can undergo large deformations. We believe this is due the buckling of a thin gel layer that forms between the polymer-rich and solvent-rich phases. The gel is thought to buckle due to high compressive stresses that are generated as it absorbs solvent and swells. In this talk, we will present a mathematical model for gel formation which captures the phenomenon of buckling due to swelling. The gel is treated as a deformable porous medium and the solvent is assumed to flow according to Darcy's law. We will also examine the buckling patterns that emerge from the model and compare them with experiments.   

Elastic membranes in confinement

An elastic membrane stretched between two walls takes a shape defined by its length and the volume of fluid it encloses. Many biological structures, such as cells, mitochondria and DNA, have finer internal structure in which a membrane (or elastic member) is geometrically ``confined'' by another object. We study the shape stability of elastic membranes in a ``confining'' box and introduce repulsive van der Waals forces to prevent the membrane from intersecting the wall. We aim to define the parameter space associated with mitochondria-like deformations. We compare the confined to `unconfined' solutions and show how the structure and stability of the membrane shapes changes with the system parameters.   

Free surface dynamics of nematic liquid crystal

Spreading thin films of nematic liquid crystal (NLC) are known to behave very differently to those of isotropic fluids. The polar interactions of the rod-like molecules with each other, and the interactions with the underlying substrate, can lead to intricate patterns and instabilities that are not yet fully understood. The physics of a system even as simple as a film of NLC spreading slowly over a surface (inclined or horizontal) are remarkably complex: the outcome depends strongly on the details of the NLC's behavior at both the substrate and the free surface (so-called ``anchoring'' effects). We will present a dynamic flow model that takes careful account of such nematic-substrate and nematic-free surface interactions. We will present model simulations for several different flow scenarios that indicate the variety of behavior that can emerge. Spreading over a horizontal substrate may exhibit a range of unstable behavior. Flow down an incline also exhibits intriguing instabilities: in addition to the usual transverse fingering, instabilities can be manifested behind the flowing front in a manner reminiscent of Newtonian flow down an inverted substrate.   

Electrified film flows at moderate Reynolds number

We examine the flow of a thin, inclined film sandwiched between two parallel electrodes. We follow the Weighted Residual Integral Boundary Layer method, which has been shown via comparison with both direct numerical simulations and experiments to give good results in both the drag-gravity and drag-inertia regimes. We extend existing models to give an accurate model of electrostatic effects via a similar separation of variables approach. A disparity in material properties between the liquid and gas regions induces a Maxwell stress at the interface, which affords a significant degree of control over the behaviour of the film. In one dimension, linear stability comparisons are made with a full Orr-Sommerfeld calculation, and nonlinear comparisons are made with direct numerical simulations, both showing excellent agreement in large parts of parameter space. The model is also extended to fully two-dimensional simulations.   

Phase diagram for the onset of rolling waves and flow reversal in inclined falling films

The onset of rolling waves and the onset of flow reversal in inclined falling films is investigated in dependence of the Reynolds and the inclination number. For this, the weighted integral boundary layer model (WIBL) and direct numerical simulations (DNS) are used. Analytical criteria for the onset of rolling waves and flow reversal based on the wave celerity, the average film thickness and the maxi-mum/minimum film thickness have been approximated using self-similar parabolic velocity profiles. This approximation has been validated by second-order WIBL and DNS simulations. It is shown that the various transitions in the phase diagram for homoclinic solutions (waves of infinite wave length) are strongly dependent on the inclination, but independent on the streamwise viscous dissipation effect. Compared to the onset of flow reversal, the onset of rolling waves occurs for higher Reynolds numbers, resulting in a regime in which flow reversal and non-rolling waves coexist. Furthermore, simulation results for limit cycles (finite wave length) reveal a strong increase of the critical Reynolds number with the excitation frequency.   

Improved Measurements of the Dominant Mode Wavelength in Viscous Nanofilms Undergoing 3D Pillar Growth Via B\'enard Type Instability

Free surface viscous nanofilms exposed to an initial uniform and very large transverse thermal gradient are prone to spontaneous formation and growth of nanopillars typically separated by tens of microns or less. Linear stability analyses of various interface equations in the long wavelength limit suggest these formations can result either from fluctuations in electrostatic forces between the fluid interface and induced image charge distribution,\footnote{S. Y. Chou and L. Zhuang, J. Vac. Sci. Technol. B 17, 3197 (1999)} radiation pressure induced by acoustic phonon reflections,\footnote{E. Sch\"affer \emph{et al}., Macromolecules 36, 1645 (2003)} or thermocapillary stresses leading to B\'enard-like deformations.\footnote{M. Dietzel and S. M. Troian, Phys. Rev. Lett. 103 (7), 074501 (2009); M. Dietzel and S. M. Troian, J. Appl. Phys. 108, 074308 (2010)} Here we review improvements over previous comparison to theoretical predictions\footnote{E. McLeod, Y. Liu and S. M. Troian, Phys. Rev. Lett. 106, 175501 (2011)} which suggest even closer agreement with the thermocapillary model; however, systematic discrepancies persist. We have therefore redesigned our experimental system for more accurate thermal flux control and estimation and will discuss our newest results.   

Lyapunov Analysis of the Stability of Nanodroplet Arrays Arising From Steady State B\'enard Flow in the Long Wavelength Limit

Previous work in our group has focused on a novel B\'enard like instability leading to nanopillar arrays in ultrathin viscous films subject to a transverse thermal gradient.\footnote{M. Dietzel and S. M. Troian, Phys. Rev. Lett. 103 (7), 074501 (2009)}$^,$\footnote{M. Dietzel and S. M. Troian, J. Appl. Phys. 108, 074308 (2010)}$^,$\footnote{E. McLeod, Y. Liu and S. M. Troian, Phys. Rev. Lett. 106, 175501 (2011)} The shape and size of these formations is influenced by the relative strength of the thermocapillary to capillary stresses. In turn, this ratio is dependent on the system geometry, fluid material properties, overall magnitude of the applied thermal gradient, and whether volume is conserved. Here we examine the parameter regime corresponding to steady state solutions resembling either isolated or extended sinusoidal-like states. The linear stability of rectilinear and axisymmetric formations is investigated by a combination of Lyapunov analysis, asymptotic methods, and numerical simulations. Our findings indicate that radially symmetric arrays with small peak heights are linearly stable. The existence of stable axisymmetric states for parameter values accessible to experiment offers an intriguing route for non-contact fabrication of microlens arrays.