Te-Sheng Lin

Thin films flowing down inverted substrates: Two dimensional flow

We consider free surface instabilities of films flowing on inverted substrates within the framework of lubrication approximation. We allow for the presence of fronts and related contact lines and explore the role which they play in instability development. It is found that a contact line, modeled by a commonly used precursor film model, leads to free surface instabilities without any additional natural or excited perturbations. A single parameter D=(3 Ca)1/3cot α, where Ca is the capillary number and α is the inclination angle, is identified as a governing parameter in the problem. This parameter may be interpreted to reflect the combined effect of inclination angle, film thickness, Reynolds number, and fluid flux. Variation of D leads to change in the wavelike properties of the instabilities, allowing us to observe traveling wave behavior, mixed waves, and the waves resembling solitary ones.

Instabilities of Layers of Deposited Molecules on Chemically Stripe Patterned Substrates: Ridges versus Drops.

A mesoscopic continuum model is employed to analyze the transport mechanisms and structure formation during the redistribution stage of deposition experiments where organic molecules are deposited on a solid substrate with periodic stripe-like wettability patterns. Transversally invariant ridges located on the more wettable stripes are identified as very important transient states and their linear stability is analyzed accompanied by direct numerical simulations of the fully nonlinear evolution equation for two-dimensional substrates. It is found that there exist two different instability modes that lead to different nonlinear evolutions that result (i) at large ridge volume in the formation of bulges that spill from the more wettable stripes onto the less wettable bare substrate and (ii) at small ridge volume in the formation of small droplets located on the more wettable stripes. In addition, the influence of different transport mechanisms during redistribution is investigated focusing on the cases of convective transport with no-slip at the substrate, transport via diffusion in the film bulk and via diffusion at the film surface. In particular, it is shown that the transport process does neither influence the linear stability thresholds nor the sequence of morphologies observed in the time simulation, but only the ratio of the time scales of the different process phases.

Note on the hydrodynamic description of thin nematic films: Strong anchoring model

We discuss the long-wave hydrodynamic model for a thin film of nematic liquid crystal in the limit of strong anchoring at the free surface and at the substrate. We rigorously clarify how the elastic energy enters the evolution equation for the film thickness in order to provide a solid basis for further investigation: several conflicting models exist in the literature that predict qualitatively different behaviour. We consolidate the various approaches and show that the long-wave model derived through an asymptotic expansion of the full nemato-hydrodynamic equations with consistent boundary conditions agrees with the model one obtains by employing a thermodynamically motivated gradient dynamics formulation based on an underlying free energy functional. As a result, we find that in the case of strong anchoring the elastic distortion energy is always stabilising. To support the discussion in the main part of the paper, an appendix gives the full derivation of the evolution equation for the film thickness via asymptotic expansion.

Modelling spreading dynamics of nematic liquid crystals in three spatial dimensions

We study spreading dynamics of nematic liquid crystal droplets within the framework of the long-wave approximation. A fourth-order nonlinear parabolic partial differential equation governing the free surface evolution is derived. The influence of elastic distortion energy and of imposed anchoring variations at the substrate are explored through linear stability analysis and scaling arguments, which yield useful insight and predictions for the behaviour of spreading droplets. This behaviour is captured by fully nonlinear time-dependent simulations of three-dimensional droplets spreading in the presence of anchoring variations that model simple defects in the nematic orientation at the substrate.

Modeling and simulations of the spreading and destabilization of nematic droplets

A series of experiments [C. Poulard and A. M. Cazabat, “Spontaneous spreading of nematic liquid crystals,” Langmuir 21, 6270 (2005)] on spreading droplets of nematic liquid crystal (NLC) reveals a surprisingly rich variety of behaviors. Small droplets can either be arrested in their spreading, spread stably, destabilize without spreading (corrugated surface), or spread with a fingering instability and corrugated free surface. In this work, we discuss the problem of NLC drops spreading in a simplified two-dimensional (2D) geometry. The model that we present is based on a long-wavelength approach for NLCs by Ben Amar and Cummings [“Fingering instabilities in driven thin nematic films,” Phys. Fluids 13, 1160 (2001); L. J. Cummings, “Evolution of a thin film of nematic liquid crystal with anisotropic surface energy,” Eur. J. Appl. Math. 15, 651 (2004)]. The improvements in the model here permit fully nonlinear time-dependent simulations. These simulations, for the appropriate choice of parameter values, exhib...

Coherent Structures in Nonlocal Dispersive Active-Dissipative Systems

We analyze coherent structures in nonlocal dispersive active-dissipative nonlinear systems, using as a prototype the Kuramoto-Sivashinsky (KS) equation with an additional nonlo- cal term that contains stabilizing/destabilizing and dispersive parts. As for the local generalized Kuramoto-Sivashinsky (gKS) equation (see, e.g., (T. Kawahara and S. Toh, Phys. Fluids, 31 (1988), pp. 2103-2111)), we show that sufficiently strong dispersion regularizes the chaotic dynamics of the KS equation, and the solutions evolve into arrays of interacting pulses that can form bound states. We analyze the asymptotic characteristics of such pulses and show that their tails tend to zero al- gebraically but not exponentially, as for the local gKS equation. Since the Shilnikov-type approach is not applicable for analyzing bound states in nonlocal equations, we develop a weak-interaction theory and show that the standard first-neighbor approximation is no longer applicable. It is then essential to take into account long-range interactions due to the algebraic decay of the tails of the pulses. In addition, we find that the number of possible bound states for fixed parameter values is always finite, and we determine when there is long-range attractive or repulsive force between the pulses. Finally, we explain the regularizing effect of dispersion by showing that, as dispersion is increased, the pulses generally undergo a transition from absolute to convective instability. We also find that for some nonlocal operators, increasing the strength of the stabilizing/destabilizing term can have a regularizing/deregularizing effect on the dynamics.

Three-dimensional coating flow of nematic liquid crystal on an inclined substrate †

We consider a coating flow of nematic liquid crystal (NLC) fluid film on an inclined substrate. Exploiting the small aspect ratio in the geometry of interest, a fourth-order nonlinear partial differential equation is used to model the free surface evolution. Particular attention is paid to the interplay between the bulk elasticity and the anchoring conditions at the substrate and free surface. Previous results have shown that there exist two-dimensional travelling wave solutions that translate down the substrate. In contrast to the analogous Newtonian flow, such solutions may be unstable to streamwise perturbations. Extending well-known results for Newtonian flow, we analyse the stability of the front with respect to transverse perturbations. Using full numerical simulations, we validate the linear stability theory and present examples of downslope flow of nematic liquid crystal in the presence of both transverse and streamwise instabilities.

Two-dimensional pulse dynamics and the formation of bound states on electrified falling films

The flow of an electrified liquid film down an inclined plane wall is investigated with the focus on coherent structures in the form of travelling-wave solutions on the film surface, in particular, single-hump solitary pulses and their interactions. The flow structures are analysed first using a long-wave model derived on the basis of thin-film theory and second using the Stokes equations for zero Reynolds number flow. Bifurcation diagrams of travelling-wave solutions for the long-wave model are obtained for acute and obtuse wall inclination angles. For obtuse angles, gravity is destablising and solitary pulses exist even in the absence of an electric field. For acute angles, spatially non-uniform solutions exist only beyond a critical value of the electric field strength; moreover solitary-pulse solutions are present only at sufficiently high supercritical electric field strengths. The electric field increases the amplitude of the pulses, generate recirculation zones in the humps, and alters the far-field decay of the pulse tails from exponential to algebraic. A weak-interaction theory which incorporates long-range effects is developed to analyse attractions and repulsions and the formation of bound states of pulses. The infinite sequence of bound-state solutions found for non-electrified flow is shown to reduce to a finite set for electrified flow due to the algebraic decay of the tails. The existence of single-hump pulse solutions and two-pulse bound states is confirmed for the Stokes equations via boundary-element computations. An absolute-convective instability analysis of single-hump pulse solutions is performed both for the long-wave model and for the Stokes equations. The electric field is shown to trigger a switch from absolute instability to convective instability, thereby regularising the dynamics, and this is confirmed by time-dependent simulations of the long-wave model.

Continuation methods for time-periodic travelling-wave solutions to evolution equations

Abstract A numerical continuation method is developed to follow time-periodic travelling-wave solutions of both local and non-local evolution partial differential equations (PDEs). It is found that the equation for the speed of the moving coordinate can be derived naturally from the governing equations together with a condition that breaks the translational symmetry. The derived system of equations allows one to follow the branch of travelling-wave solutions as well as solutions that are time-periodic in a frame of reference travelling at a constant speed. Finally, we show as an example the bifurcation and stability analysis of single and double-pulse waves in long-wave models of electrified falling films.