Influence of attractive van der Waals interactions on the optimal excitations in thermocapillary-driven spreading

Recent investigations of microfluidic flows have focused on manipulating the motion of very thin liquid films by modulating the surface tension through an applied streamwise temperature gradient. The extent to which the choice of contact line model affects the flow and stability of such thermocapillary-driven films is not completely understood. Regardless of the contact line model used, the linearized disturbance operator corresponding to the evolution of the film height is non-normal, and a generalized non-modal stability analysis is required. Surprisingly, early predictions of frontal instability that stemmed from conventional modal analysis of thermocapillary flow on a flat, infinite precursor film showed excellent agreement with experiment. Within the more rigorous framework provided by a generalized stability analysis, this work investigates the transient dynamics and amplification of optimal disturbances subject to a finite precursor film generated by attractive van der Waals forces. Convergence of the disturbance growth rates and perturbed shapes to the asymptotic solutions obtained by conventional linear stability analysis occurs early in the spreading process. In addition, the level of transient disturbance amplification is minimal. The equations governing thermocapillary-driven spreading exhibit a small degree of non-normality, which explains the source of agreement between modal theory and experiment. The more rigorous generalized stability analysis presented here, however, affords critical insight into the types of disturbances leading to maximum unstable growth and the exact influence of the contact line model used

On a generalized approach to the linear stability of spatially nonuniform thin film flows

The presence of a deformable free surface in thin films driven to spread by body or shear forces gives rise to base states that are spatially nonuniform. This nonuniformity produces linearized disturbance operators that are non-normal and an eigenvalue spectrum that does not necessarily predict stability behavior. The falling film provides a simple example for demonstrating a more generalized, rigorous nonmodal approach to linear stability for free surface flows. Calculations of the pseudospectra and maximum disturbance amplification in this system, however, reveal weak effects of non-normality and transient growth such that the modal growth rate is rapidly recovered. Subdominant modes contribute little energy to the leading eigenvector because the oscillatory behavior is rapidly damped by surface tension. Generalization of these results to numerous other lubrication flows involving surface tension suggests similarly weak non-normality and transient growth

Influence of boundary slip on the optimal excitations in thermocapillary driven spreading

Thin liquid films driven to spread on homogeneous surfaces by thermocapillarity can undergo frontal breakup and parallel rivulet formation with well-defined wavelength. Previous modal analyses have relieved the well-known divergence in stress that occurs at a moving contact line by matching the front region to a precursor film. Because the linearized disturbance operator is non-normal, a generalized, nonmodal analysis is required to probe film stability at all times. The effect of the contact line model on nonmodal stability has not been previously investigated. This work examines the influence of boundary slip on thermocapillary driven spreading using a transient stability analysis, which recovers the conventional modal results in the long-time limit. In combination with earlier work on thermocapillary driven spreading, this study verifies that the dynamics and stability of this system are rather insensitive to the choice of contact line model and that the leading eigenvalue is physically determinant, thereby assuring results that agree with the eigenspectrum. Modal results for the flat precursor film model are reproduced with appropriate choice of slip coefficient and contact line slope

Linear stability analysis of an insoluble surfactant monolayer spreading on a thin liquid film

Recent experiments by several groups have uncovered a novel fingering instability in the spreading of surface active material on a thin liquid film. The mechanism responsible for this instability is yet to be determined. In an effort to understand this phenomenon and isolate a possible mechanism, we have investigated the linear stability of a coupled set of equations describing the Marangoni spreading of a surfactant monolayer on a thin liquid support. The unperturbed flows, which exhibit simple linear behavior in the film thickness and surfactant concentration, are self-similar solutions of the first kind for spreading in a rectilinear geometry. The solution of the disturbance equations determines that the rectilinear base flows are linearly stable. An energy analysis reveals why these base flows can successfully heal perturbations of all wavenumbers. The details of this analysis suggest, however, a mechanism by which the spreading can be destabilized. We propose how the inclusion of additional forces acting on the surfactant coated spreading film might give rise to regions of adverse mobility gradients known to produce fingering instabilities in other fluid flows

Spreading of a surfactant monolayer on a thin liquid film: Onset and evolution of digitated structures

We describe the response of an insoluble surfactant monolayer spreading on the surface of a thin liquid film to small disturbances in the film thickness and surfactant concentration. The surface shear stress, which derives from variations in surfactant concentration at the air–liquid interface, rapidly drives liquid and surfactant from the source toward the distal region of higher surface tension. A previous linear stability analysis of a quasi-steady state solution describing the spreading of a finite strip of surfactant on a thin Newtonian film has predicted only stable modes. [Dynamics in Small Confining Systems III, Materials Research Society Symposium Proceedings, edited by J. M. Drake, J. Klafter, and E. R. Kopelman (Materials Research Society, Boston, 1996), Vol. 464, p. 237; Phys. Fluids A 9, 3645 (1997); O. K. Matar Ph.D. thesis, Princeton University, Princeton, NJ, 1998]. A perturbation analysis of the transient behavior, however, has revealed the possibility of significant amplification of disturbances in the film thickness within an order one shear time after the onset of flow [Phys. Fluids A 10, 1234 (1998); "Transient response of a surfactant monolayer spreading on a thin liquid film: Mechanism for amplification of disturbances," submitted to Phys. Fluids]. In this paper we describe the linearized transient behavior and interpret which physical parameters most strongly affect the disturbance amplification ratio. We show how the disturbances localize behind the moving front and how the inclusion of van der Waals forces further enhances their growth and lifetime. We also present numerical solutions to the fully nonlinear 2D governing equations. As time evolves, the nonlinear system sustains disturbances of longer and longer wavelength, consistent with the quasi-steady state and transient linearized descriptions. In addition, for the parameter set investigated, disturbances consisting of several harmonics of a fundamental wavenumber do not couple significantly. The system eventually singles out the smallest wavenumber disturbance in the chosen set. The summary of results to date seems to suggest that the fingering process may be a transient response which nonetheless has a dramatic influence on the spreading process since the digitated structures redirect the flux of liquid and surfactant to produce nonuniform surface coverage

Influence of periodic wall roughness on the slip behaviour at liquid/solid interfaces: molecular-scale simulations versus continuum predictions

The influence of surface roughness on the slip behaviour of a Newtonian liquid in steady planar shear is investigated using three different approaches, namely Stokes flow calculations, molecular dynamics (MD) simulations and a statistical mechanical model for the friction coefficient between a corrugated wall and the first liquid layer. These approaches are used to probe the behaviour of the slip length as a function of the slope parameter ka = 2πa/λ, where a and λ represent the amplitude and wavelength characterizing the periodic corrugation of the bounding surface. The molecular and continuum approaches both confirm a monotonic decay in the slip length with increasing ka but the rate of decay as well as the magnitude of the slip length obtained from the Stokes flow solutions exceed the MD predictions as the wall feature sizes approach the liquid molecular dimensions. In the limit of molecular-scale wall corrugation, a Green–Kubo analysis based on the fluctuation–dissipation theorem accurately reproduces the MD results for the behaviour of the slip length as a function of a. In combination, these three approaches provide a detailed picture of the influence of periodic roughness on the slip length which spans multiple length scales ranging from molecular to macroscopic dimensions

Growth and decay of localized disturbances on a surfactant-coated spreading film

If the surface of a quiescent thin liquid film is suddenly coated by a patch of surface active material like a surfactant monolayer, the film is set in motion and begins spreading. An insoluble surfactant will rapidly attempt to coat the entire surface of the film thereby minimizing the liquid's surface tension. The shear stress that develops during the spreading process produces a maximum in surface velocity in the region where the moving film meets the quiescent layer. This region is characterized by a shock front with large interfacial curvature and a corresponding local buildup of surfactant which creates a spike in the concentration gradient. In this paper, we investigate the sensitivity of this region to infinitesimal disturbances. Accordingly, we introduce a measure of disturbance amplification and transient growth analogous to a kinetic energy that couples variations in film thickness to the surfactant concentration. These variables undergo significant amplification during the brief period in which they are convected past the downstream tip of the monolayer, where the variation in concentration gradient and surface curvature are largest. Once they migrate past this sensitive area, the perturbations weaken considerably and the system approaches a stable configuration. It appears that the localized disturbances of the type we consider here, cannot sustain asymptotic instability. Nonetheless, our study of the dynamics leading to the large transient growth clearly illustrates how the coupling of Marangoni and capillary forces work in unison to stabilize the spreading process against localized perturbations

Molecular Origin and Dynamic Behavior of Slip in Sheared Polymer Films

The behavior of the slip length in thin polymer films subject to planar shear is investigated using molecular dynamics simulations. At low shear rates, the slip length extracted from the velocity profiles correlates well with that computed from a Green-Kubo analysis. Beyond chain lengths of about N = 10, the molecular weight dependence of the slip length is dominated strongly by the bulk viscosity. The dynamical response of the slip length with increasing shear rate is well captured by a power law up to a critical value where the momentum transfer between wall and fluid reaches its maximum

The development of transient fingering patterns during the spreading of surfactant coated films

The spontaneous spreading of an insoluble surfactant monolayer on a thin liquid film produces a complex waveform whose time variant shape is strongly influenced by the surface shear stress. This Marangoni stress produces a shocklike front at the leading edge of the spreading monolayer and significant film thinning near the source. For sufficiently thin films or large initial shear stress, digitated structures appear in the wake of the advancing monolayer. These structures funnel the oncoming flow into small arteries that continuously tip-split to produce spectacular dendritic shapes. A previous quasisteady modal analysis has predicted stable flow at asymptotically long times [Phys. Fluids A 9, 3645 (1997)]. A more recent transient analysis has revealed large amplification in the disturbance film thickness at early times [O. K. Matar and S. M. Troian, "Growth of nonmodal transient structures during the spreading of surfactant coated films," Phys. Fluids A 10, 1234 (1998)]. In this paper, we report results of an extended sensitivity analysis which probes two aspects of the flow: the time variant character of the base state and the non-normal character of the disturbance operators. The analysis clearly identifies Marangoni forces as the main source of digitation for both small and large wave number disturbances. Furthermore, initial conditions which increase the initial shear stress or which steepen the shape of the advancing front produce a larger transient response and deeper corrugations in the film. Disturbances applied just ahead of the deposited monolayer rapidly fall behind the advancing front eventually settling in the upstream region where their mobility is hampered. Recent findings confirm that additional forces which promote film thinning can further intensify disturbances [O. K. Matar and S. M. Troian, "Spreading of surfactant monolayer on a thin liquid film: Onset and evolution of digitated structures," Chaos 9, 141 (1999). The transient analysis presented here corroborates our previous results for asymptotic stability but reveals a source for digitation at early times. The energy decomposition lends useful insight into the actual mechanisms preventing efficacious distribution of surfactant

Thinning and disturbance growth in liquid films mobilized by continuous surfactant delivery

A generalized linear stability analysis is applied to the case of a thin liquid film propelled to spread by a continuous supply of surfactant. The time-dependent base states for the film thickness and surfactant concentration give rise to a nonautonomous system describing disturbance propagation. As a first approximation, the nonautonomous operator is treated as time independent, thereby reducing the system of equations to a standard eigenvalue problem. For the range of parameters investigated, this modal approximation reveals a band of unstable modes corresponding to the growth of transverse, sinusoidal corrugations. A transient growth analysis of the fully time-dependent system, which requires the solution of an initial value problem, also signals the possibility of large disturbance growth. In both cases, significant amplification of infinitesimal disturbances can be traced to the region of the film most rapidly thinned by Marangoni stresses, which is characterized by large interfacial curvature and a sharp variation in shear stress. In contrast to previous models implementing a finite surfactant source that predict asymptotic stability, large transient growth and asymptotic instability are possible for the case of sustained surfactant release