Session A20: Interfacial/Thin Film Instability I

Predicting Stability of Air--Water Interface on Superhydrophobic Surfaces

In this work, two different methodologies for predicting the stability of the air-water interface on submerged superhydrophobic surfaces are presented. The first method is an analytical approach developed by balancing the hydrostatic pressure with the capillary forces over the interface, and results in a second-order partial differential equation. The solution to this equation provides the 3-D interface shape and the critical pressure beyond which the superhydrophobic surface departs from the Cassie state. The second method presented here is an approximate numerical technique based on the so called Full Morphology method in which the Young--Laplace equation is used to relate a capillary pressure to the most constricted opening of the pore space between the peaks of the surface roughness. Predictions of the methods presented in this study are compared with the available studies in the literature ({\it Applied Physics Letters} {\bf 98}:20, 203106, 2011).   

Three-dimensional convective and absolute instabilities in pressure-driven two-layer channel flow

A generalized linear stability analysis of three-dimensional disturbance in a pressure-driven two-layer channel flow, focusing on the range of parameters for which Squire's theorem does not exist is considered. Three-dimensional linear stability equations, in which both the spatial wavenumber and temporal frequency are complex, are derived and solved using an efficient spectral collocation method. A Briggs-type analysis is then carried out to delineate the boundaries between convective and absolute instabilities in m-Re space. We find that although three-dimensional disturbances are temporally more unstable than the two-dimensional disturbances, absolute modes of instability are most unstable for two-dimensional disturbances. An energy ``budget'' analysis also shows that the most dangerous modes are ``interfacial'' ones.   

Diffuse interface approach to rotating Hele-Shaw flows

When two fluids of different densities move in a rotating Hele-Shaw cell, the interface between them becomes centrifugally unstable and deforms. Depending on the viscosity contrast of the system distinct types of complex patterns arise at the fluid-fluid boundary. Deformations can also induce the emergence of interfacial singularities and topological changes such as droplet pinch-off and self-intersection. We present numerical simulations based on a diffuse interface model for this particular two-phase displacement that capture a variety of pattern forming behaviors. This is implemented by employing a Boussinesq Hele-Shaw-Cahn-Hilliard approach, considering the whole range of possible values for the viscosity contrast, and by including inertial effects due to the Coriolis force. The role played by these two physical contributions on the development of interface singularities is illustrated and discussed.   

Solitary waves running down a vertically falling film: low-dimensional models

We consider the wavy regime of vertically falling fluid film. Derivation of LDM for this flow, based on the LW expension have a long history(see review in[1]). A crucial test of such models is the correct prediction of the properties of SW as a function of the distance from the instability threshold. The latter is usually quantified in terms of the reduced $Re$, $\delta=3^\frac{4}{3}Re^\frac{11}{9}Ka^\frac{-1}{3}$. Though most models predict similar behavior in the drag-gravity regime($\delta\ll1$), they exhibit large differences from each other in the drag-inertia regime($\delta\gg1$). Characteristics of SW solutions to available LDM are shown and contrasted with DNS results. The best agreement with DNS is found with the 4-equation model derived in[2] and the convergence rate of the wave speed to the asymptotic limit $c_\infty$ at $\delta\to\infty$ is affected by viscous diffusion terms and is governed by the $Re$ as $|c-c_\infty|\propto1/Re^2$. The asymptotic behavior of the speed, amplitude, lengths of the wave-tail and front capillary ripples are discussed. References: [1] R.V.Craster, O.K.Matar. Reviews of modern physics 81 (2009). [2] C.Ruyer-Quil, P.Manneville. Eur.Phys.J. 15 357-369 (2000).   

Three-dimensional instabilities of miscible fingers in a Hele-Shaw cell

We perform three-dimensional DNS simulations of the transient, variable viscosity Boussinesq Navier-Stokes equations, coupled to a convection-diffusion equation for a concentration field, to simulate miscible viscous fingers in Hele-Shaw cells. The three-dimensional problem allows for new instabilities and patterns that cannot be captured by traditional gap-averaged modeling. For constant density displacements, we find that a streamwise vorticity quadrupole forms that induces fluid transport from the walls of the cell to its center, thereby leading to a new hydrodynamic instability, termed ``inner splitting.'' If gravity is included, the nature of the two-dimensional base flow and its subsequent instability changes dramatically. The interaction between Saffman-Taylor and Rayleigh-Taylor instabilities can lead to additional splitting events, and it can significantly enhance the mixing rates of the two fluids, thereby altering the overall displacement efficiency. This work is supported by NSF, and a CAPES/Fulbright fellowship.   

Thin film drainage of a deformable droplet moving toward a wall with finite inertia

Direct numerical simulations of a deformable droplet approaching toward a solid wall through another fluid are performed by solving the full Navier-Stokes equations using a finite volume/ moving mesh interface tracking method with high fidelity. Cases with Reynolds numbers of 25 and 50 and capillary numbers of $5\times 10^{-3}$ and $1\times 10^{-2}$ are simulated for both head-on and oblique approaching scenarios. The front head of the droplet is flattened as the droplet nearly touches the wall, and a dimpled thin film is observed. Because of the great viscous forces of the flow inside the thin film, the droplet slows down dramatically which leads to a significant increase of the drag force. An asymmetric thin film is observed for the oblique approaching. The numerical prediction on the central separation at which a dimple is formed agrees fairly well with previous analysis based on the lubrication theory. The simulated thinning rate is slower than the rate predicted by previous approximate models. The differences are mainly due to the finite Reynolds number of the simulated cases.   

Contact line induced instabilities for thin fluid films

We study contact line induced instabilities for thin film of complete and partially wetting fluids spreading down an inclined plane with inclination angle ranging from 0 to pi. It is found that a contact line may lead to free surface instability without any additional perturbations. We investigate the effect of both inclination angle and of contact angle on surface and transverse (fingering) instabilities, as well as on dewetting process.   

The dynamics of foams with mobile interfaces

Using a novel technique for resolving nearly singular integrals, we investigate the dynamics of two-dimensional foams with mobile interfaces and an incompressible, inviscid gas phase by a boundary integral method. For foams with small liquid fractions ($\le 5\%$), we observe node motion, lamellar bending, drainage, and T1 transitions. Node motion occurs on the fastest timescale and is well-described by considering only the surface forces on each Plateau border. The lamellar bending is characterized by viscida theory, but the drainage occurs at a different rate than predicted by the asymptotic theory for the examples studied. Topological transitions occur on a timescale intermediate to Plateau border rearrangement and drainage.   

Optical interference effect on Marangoni instability of irradiated thin liquid films

The Marangoni instability of thin liquid films on solid substrates induced by irradiative heating is investigated within the framework of the long-wave approximation. The energy transfer that includes the energy absorption and reflection is taken into account. In order to examine the optical interference effects on the instability, focus is placed on a transparent film/absorbable substrate system irradiated by a monochromatic wave with laterally uniform intensity distribution. In such a case, the energy reflectance varies periodically with the film thickness due to optical interference. Numerical simulation results show that the stability of the film depends on the first derivative of the energy reflectance with respect to the film thickness at a reference point, and the resultant surface patterns differ depending on the reference thickness and initial perturbation.   

Numerical simulations of two-fluid boundary layers beneath free-stream turbulence

In two-fluid boundary layers, a wall-film is sheared by an external stream with different density and viscosity. As a result, the flow becomes prone to both shear and interfacial instabilities. In this study, the evolution of two-fluid boundary layers beneath free-stream vortical forcing is investigated using DNS. The simulations employ a conservative level-set technique in conjunction with a ghost fluid approach in order to capture a sharp interface. The wall film is less viscous than the outer flow, and its thickness is $10\%$ of that of the boundary layer at the inlet. The choice of viscosity ratio influences the spatial development of disturbances within the boundary layer. The spatial growth of instabilities is examined into the non-linear regime, which includes the region of breakdown to turbulence. We demonstrate that, at moderate levels of free-stream turbulence intensities, appropriate choice of the viscosity ratio can yield considerable transition delay.