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

Growth of non-modal transient structures during the spreading of surfactant coated films

The spreading of surfactant coated thin liquid films is often accompanied by an instability producing significant film corrugation, fingering and branching. Marangoni stresses, responsible for the rapid and spontaneous spreading, are suspected as the main cause of unstable flow. Traditional eigenvalue analysis of a self-similar solution describing Marangoni driven spreading has predicted only stable modes. We present results of a transient growth study which reveals enormous amplification of initially infinitesimal disturbances in the film thickness. This analysis provides, for the first time, evidence of an instability resembling experimental patterns

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

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

Spreading characteristics of an insoluble surfactant film on a thin liquid layer: comparison between theory and experiment

We describe measurements of the surface slope and reconstruction of the interface shape during the spreading of an oleic acid film on the surface of a thin aqueous glycerol mixture. This experimental system closely mimics the behaviour of an insoluble surfactant film driven to spread on a thin viscous layer under the action of a tangential (Marangoni) surface stress. Refracted image Moiré topography is used to monitor the evolution of the surface slope over macroscopic distances, from which the time variant interface shape and advancing speed of the surfactant film are inferred. The interfacial profile exhibits a strong surface depression ahead of the surfactant source capped by an elevated rim at the surfactant leading edge. The surface slope and shape as well as the propagation characteristics of the advancing rim can be compared directly with theoretical predictions. The agreement is quite strong when the model allows for a small level of pre-existing surface contamination of the initial liquid layer. Comparison between theoretical and experimental profiles reveals the importance of the initial shear stress in determining the evolution in the film thickness and surfactant distribution. This initial stress appears to thin the underlying liquid support so drastically that the surfactant droplet behaves as a finite and not an infinite source, even though there is always an excess of surfactant present at the origin

Three-dimensional linear instability in pressure-driven two-layer channel flow of a Newtonian and a Herschel-Bulkley fluid

The three-dimensional linear stability characteristics of pressure-driven two-layer channel flow are considered, wherein a Newtonian fluid layer overlies a layer of a Herschel-Bulkley fluid. We focus on the parameter ranges for which Squire's theorem for the two-layer Newtonian problem does not exist. The modified Orr-Sommerfeld and Squire equations in each layer are derived and solved using an efficient spectral collocation method. Our results demonstrate the presence of three-dimensional instabilities for situations where the square root of the viscosity ratio is larger than the thickness ratio of the two layers; these "interfacial" mode instabilities are also present when density stratification is destabilizing. These results may be of particular interest to researchers studying the transient growth and nonlinear stability of two-fluid non-Newtonian flows. We also show that the "shear" modes, which are present at sufficiently large Reynolds numbers, are most unstable to two-dimensional disturbances