Unstable van der Waals driven line rupture in Marangoni driven thin viscous films

Abstract

An intriguing, dramatic and, at present, not fully understood instability often accompanies surfactant driven flows on thin films. This paper investigates a candidate mechanism that could create and drive this instability, van der Waals rupture, via numerical simulations coupled with analytical techniques. The spreading process itself is modelled with a pair of coupled evolution equations for the fluid film thickness and surfactant concentration that are derived in the lubrication approximation. These equations are then linearized about a base state that corresponds to the one-dimensional rupturing solution, and equations for the evolution of the transverse disturbances are derived. These linearized equations are investigated in several ways: numerical simulations where the perturbations are driven by the time evolving base state, or where the base state is frozen at a time tf close to the rupture event. The quasistatic initial value problem is also investigated as an eigenvalue problem, where the eigenvalue represents the quasistatic growth rate. We also take advantage of recent similarity scalings and results deduced for rupture, in the absence of surfactant, to motivate some of our numerical investigations. Additionally, we investigate the fully nonlinear equations including the transverse components. Perhaps interestingly, three-dimensional reconstructions of the film profile using the most dangerous mode from linear theory, as well as profiles from direct numerical simulations of the full nonlinear governing equations, that is, including interactions in the transverse direction, assume the form of finger-like patterns.

Flow of surfactant-laden thin films down an inclined plane

“…This subject has been covered extensively by previous investigators who have examined the steady-state [17], dynamic spreading of insoluble, [18][19][20], and soluble surfactants [21,22], and the process of surfactant-enhanced drug delivery [23,24]. The stability of the spreading process, which is often accompanied by fingering phenomena has also received considerable attention [2][3][4], [25][26][27][28].…”

“…Evolution equations for the film thickness and surfactant monolayer concentration are derived using lubrication theory and applied to a flow with constant flux; a precursor layer model is used to relieve the contact line singularity. The effects of gravity, Marangoni stress, inclination angle, precursor layer thickness, capillarity, and surface diffusion on the flow are considered, while van der Waals, and inertial forces are neglected [25]. In addition to the base state height and surfactant concentration profiles, the stability of the flow is examined using suitable transient growth measures over a wide range of conditions.…”

Instability of nanometric fluid films on a thermally conductive substrate

“…The studies are often carried out using the long-wave approach; within this framework, a significant body of work has been established in the recent years, including extensive research on linear and weakly nonlinear instability mechanisms [4][5][6], as well as discussion of monotone and oscillatory type of Marangoni effect governed instabilities [3,[7][8][9] (only a subset of relevant works is listed here). While most of the works have focused on the regime where gravitational effects are relevant, there is also an increasing body of work considering the interplay between the instabilities caused by Marangoni effect and by liquid-solid interaction that becomes important for the films on nanoscale, see, e.g., [10][11][12][13]. Understanding the influence of Marangoni effect on film stability is simplified in the settings where temperature of the film surface could be related in some simple way to its thickness; however it is not always clear that a simple functional relation can be accurately established, particularly in the setups such that the temperature field and the film thickness evolve on the comparable time scales so that the temperature of the fluid may be history dependent.…”

Marangoni effect from density variations in apolar ultrathin films

“…Assuming that the film density ρ is a function of h only [3,6], the surface tension gradient can be decomposed using the chain rule, so that ∂ x γ = ∂ ρ γ∂ h ρ∂ x h. When the film is not uniformly flat, so that ∂ x h = 0, a film density gradient ∂ h ρ = 0 will produce a gradient in surface tension, which in turn leads to a Marangoni flow [9,10,12,13].…”

“…From the continuity equation and the kinematic boundary condition, assuming that the film is so thin that density gradients along the z-direction are negligible [12], an advection equation can be derived:…”