Andreas Münch

Undercompressive shocks in thin film flows

Abstract Equations of the type ht+(h2−h3)x=−ϵ3(h3hxxx)x arise in the context of thin liquid films driven by the competing effects of a thermally induced surface tension gradient and gravity. In this paper, we focus on the interaction between the fourth order regularization and the nonconvex flux. 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 ϵ→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. The conclusions are based primarily on numerical results for the PDE, and on numerical investigations of the ODE describing traveling waves. Fourth order linear regularization is observed to produce a similar bifurcation structure of traveling waves.

Stability of compressive and undercompressive thin film travelling waves

Recent studies of liquid lms driven by competing forces due to surface tension gradients and gravity reveal that undercompressive travelling waves play an important role in the dynamics when the competing forces are comparable. In this paper, we provide a theoretical framework for assessing the spectral stability of compressive and undercompressive travelling waves in thin lm models. Associated with the linear stability problem is an Evans function which vanishes precisely at eigenvalues of the linearized operator. The structure of an index related to the Evans function explains computational results for stability of compressive waves. A new formula for the index in the undercompressive case yields results consistent with stability. In considering stability of undercompressive waves to transverse perturbations, there is an apparent inconsistency between long-wave asymptotics of the largest eigenvalue and its actual behaviour. We show that this paradox is due to the unusual structure of the eigenfunctions and we construct a revised long-wave asymptotics. We conclude with numerical computations of the largest eigenvalue, comparisons with the asymptotic results, and several open problems associated with our ndings.

Slip-controlled thin-film dynamics

In this study, we present a novel method to assess the slip length and the viscosity of thin films of highly viscous Newtonian liquids. We quantitatively analyse dewetting fronts of low-molecular-weight polystyrene melts on octadecyl- (OTS) and dodecyltrichlorosilane (DTS) polymer brushes. Using a thin-film (lubrication) model derived in the limit of large slip lengths, we can extract slip length and viscosity. We study polymer films with thicknesses between 50 nm and 230 nm and various temperatures above the glass transition. We find slip lengths from 100 nm up to 1 μm on OTS- and between 300 nm and 10 μm on DTS-covered silicon wafers. The slip length decreases with temperature. The obtained values for the viscosity are consistent with independent measurements.

Spin coating of an evaporating polymer solution

We consider a mathematical model of spin coating of a single polymer blended in a solvent. The model describes the one-dimensional development of a thin layer of the mixture as the layer thins due to flow created by a balance of viscous forces and centrifugal forces and evaporation of the solvent. In the model both the diffusivity of the solvent in the polymer and the viscosity of the mixture are very rapidly varying functions of the solvent mass fraction. Guided by numerical solutions an asymptotic analysis reveals a number of different possible behaviours of the thinning layer dependent on the nondimensional parameters describing the system. The main practical interest is in controlling the appearance and development of a “skin” on the polymer where the solvent concentration reduces rapidly on the outer surface leaving the bulk of the layer still with high concentrations of solvent. In practice, a fast and uniform drying of the film is required. The critical parameters controlling this behaviour are fou...

Shock transitions in Marangoni gravity-driven thin-film flow

Thin films of silicon oil driven up an inclined silicon wafer by a thermally induced Marangoni force develop unusual shock profiles involving a non-classical undercompressive shock, if the counteracting parallel component of gravity is sufficiently large (Bertozzi et al 1998 Phys. Rev. Lett. 81 5169-72). They arise as a result of the interaction of a non-convex flux with the fourth-order diffusion generated by surface tension. In this work, we investigate how the dynamical behaviour of the solution is affected by including second-order diffusion resulting from the normal component of gravity; this component was neglected in the previous study. Then the governing equation for the film profile h (x , t ) becomes ht + ( h 2 - h 3 )x = - ( h 3 hxxx )x + D ( h 3 hx )x D 0. The numerical simulations in this paper confirm that neglecting second-order diffusion is justified for small D , but find that for larger D , the structure of the solution changes dramatically. We give a detailed account of the transitions that occur while increasing D and make predictions for future experiments carried out at small inclination angles, corresponding to moderately large D .

The drag-out problem in film coating

The classical coating flow problem of determining the asymptotic film thickness (and hence the load) on a flat plate being withdrawn vertically from an infinitely deep bath is examined via a numerical solution of the steady-state Navier-Stokes equations. Under creeping flow conditions, the dimensionless load q is computed as a function of the capillary number Ca and, for Ca<0.4, is found to agree with Wilson’s extension [J. Eng. Math. 16, 209 (1982)] of Levich’s well-known expression. On the other hand, for Ca→∞, q asymptotes to 0.582, well below the value of 2∕3 postulated by Deryagin and Levi [Film Coating Theory (Focal, London, 1964)]. For finite Reynolds numbers Re≡mCa3∕2, where m is a dimensionless number involving only the gravitational acceleration g and the properties of the fluid, q is found to remain essentially independent of m at a given Ca, but only up to a critical capillary number Ca*, dependent on m, beyond which our numerical scheme failed. Analogous results, but only for creeping flows, ...

Dewetting rates of thin liquid films

We investigate the dewetting rates of thin liquid films using a lubrication model that describes the dewetting process of polymer melts on hydrophobized substrates. We study the effect of different boundary conditions at the liquid/solid interface, in particular, of the no-slip and the Navier slip boundary condition, and compare our numerical solutions for the no-slip and the slip-dominated cases to available results that originate from scaling arguments, simplified flow assumptions and energy balances. We furthermore consider these issues for an extended lubrication model that includes nonlinear curvature.

Stationary solutions of driven fourth- and sixth-order Cahn-Hilliard type equations

New types of stationary solutions of a one-dimensional driven sixth-order Cahn–Hilliard-type equation that arises as a model for epitaxially growing nanostructures, such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the humps of these spatially nonmonotone solutions in the limit of small driving force strength, which is the deposition rate in case of epitaxial growth. These solutions extend the family of the monotone kink and antikink solutions. The hump spacing is related to solutions of the Lambert W function. Using phase-space analysis for the corresponding fifth-order dynamical system, we use a numerical technique that enables the efficient and accurate tracking of the solution branches, where the asymptotic solutions are used as initial input. Additionally, our approach is first demonstrated for the related but simpler driven f...

Numerical and asymptotic results on the linear stability of a thin film spreading down a slope of small inclination

We model a thin liquid film moving down a slope using the lubrication approximation with a slip condition. The travelling-wave solution is derived for small inclination angle α, using singular perturbation methods, and compared to the numerical solution. For the linear stability analysis we combine numerical methods with the long-wave approximation and find a small but finite critical α∗ below which the flow remains linearly stable to spanwise perturbations. This is contrasted with the vanishing of the hump of the travelling-wave solution. Finally, the prevailing linear stability of the travelling-wave at small inclination angles is compared with recent related results using a precursor model. Here, though, a strong dependence on the magnitude of the contact angle is found, which we think has not been observed before.

Slip vs. viscoelasticity in dewetting thin films.

Abstract.Ultrathin polymer films on non-wettable substrates display dynamic features which have been attributed to either viscoelastic or slip effects. Here we show that in the weak- and strong-slip regime, effects of viscoelastic relaxation are either absent or essentially indistinguishable from slip effects. Strong slip modifies the fastest unstable mode in a rupturing thin film, which questions the standard approach to reconstruct the effective interface potential from dewetting experiments.