Session CY: Instability: Interfacial and Thin Film II

The development of nonlinear instability waves in two-fluid shear flows with complex interfacial topology

We present a nonlinear stability formulation for laminar, two-fluid shear flows which undergo changes in interface topology. The formulation combines the nonlinear Parabolized Stability Equations with a scalar-based interface capturing scheme. In doing so, this approach retains the flexibility and insight of instability-wave based methods, and accurately accounts for the nonlinear physical mechanisms responsible for complex deformations to the interface. This new approach is used to simulate the nonlinear evolution of instability waves in spatial, two-phase mixing layers with density and viscosity stratification. We demonstrate that the dynamics of the disturbance waves are well-predicted, by comparing our results to direct Navier-Stokes simulations. The new method accurately captures the formation of large-scale structures such as Kelvin-Helmholtz vortex rolls and liquid ligaments in two and three dimensions. Using this formulation, we also illustrate the importance of the mean flow distortion, nonlinear interactions, and finite amplitude effects to the development of the two-fluid structures. In addition to accuracy, the computational efficiency of our method is compared to direct Navier-Stokes simulations.   

Non-linear evolution of vortical disturbances in two-fluid boundary layers

The stability characteristics of boundary layers are altered by the presence of a wall-film of different density or viscosity. The effects of stratification on modal stability of this flow has previously been studied using linear theory. The transient amplification of disturbances is also altered. We demonstrate that an optimal choice of viscosity and density ratios can result in a reduction of transient energy growth up to $30\%$. The predictions of linear, locally-parallel analyses do not, however, take into account the spreading of the mean flow. The contribution of non-parallelism is quantified in the case of spatially developing boundary layers. Furthermore, beyond the early linear stage, the amplitude of the instability waves becomes appreciable and non-linear effects can no longer be neglected. As a result, an accurate description of the evolution of disturbances in two-fluid boundary layers must account for non-linear interactions and mean flow modification. We apply the framework of the non-linear Parabolized Stability Equations (PSE) in order to study the evolution of the most amplifying disturbances in the two-fluid flow. Our methodology takes into account non-linear modal interactions, the mean flow correction and finite deformation of the interface.   

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

We investigate the three-dimensional linear characteristics of pressure-driven two-layer channel flow, focussing on the range of parameters for which Squire's theorem does not exist, wherein a Newtonian fluid layer overlies a layer of a Herschel-Bulkley fluid. The modified Orr-Sommerfeld and Squire equations in each layers 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 destabilising. These results may be of particular interest to researchers studying the transient growth and nonlinear stability of two-fluid flows. We also show that the ``shear'' modes, which are present at sufficiently large Reynolds numbers, are most unstable to two-dimensional disturbances.   

Three dimensional spatio-temporal instabilities in two-layer flows at high Reynolds numbers

Interfacial instabilities in Newtonian two-layer flows are investigated via three-dimensional direct numerical simulations using the diffuse-interface method to capture the interface. The simulations study the effect of waves, generated by a random 3D noise, at the inlet on the spatio-temporal behaviour of the instabilities. Of specific interest are the conditions of growth/decay of the spanwise interfacial perturbation. Preliminary results show a sustained growth of the spanwise mode, irrespective of the primary streamwise mode, at various streamwise locations in the domain. At positions close to the inlet the spanwise wave grows linearly until a non-linear distortion which eventually saturates the amplitude. This work extends our recently reported two-dimensional studies on spatiotemporal interfacial instabilities (J. Fluid Mech. (2010), vol. 656, pp. 458--480) to i) three dimensions and ii) higher Reynolds numbers.   

Numerical simulations of two-fluid channel flow with wall deposition and ageing effects

We study the dynamics of two immiscible fluids with a high viscosity contrast in pressure-driven channel flow using direct numerical simulations at moderate Reynolds numbers. The equations of mass, momentum and energy conservation in both fluids are solved using a procedure based on the diffuse interface method. A Cahn-Hilliard equation is also solved for the volume fraction. Numerical solutions are obtained subject to no-slip and no-penetration conditions at the walls, and constant flow rate conditions at the channel inlet; outflow conditions are imposed at the outlet. This model accounts for a thermal instability in the bulk, through a chemical equilibrium model based on the Gibbs free energy, which leads to the formation of the highly viscous phase and its deposition at the channel walls. We also account for the evolution of the rheology of the deposited phase through ``ageing.'' We present results showing typical flow dynamics and the effect of system parameters on the average deposit thickness.   

Dynamics of surfactant-laden lenses

We consider the dynamics of lenses of immiscible fluids laden with surfactant. We use lubrication theory to derive equations for the positions of the interfaces and the surfactant concentrations. The surfactant is allowed to exist in the form of monomers as well as micelles and the model accounts for the effects of surfactant on the moving contact line. We use a finite-element formulation to obtain numerical solutions of the evolution equations and carry out a full parametric study. Our results catalogue the various types of behaviour encountered, which range from complete spreading of the lens, to spreading followed by retraction, to sustained oscillations.   

Surfactant-induced superspreading of liquid drops on solid substrates

The mechanisms driving surfactant-enhanced superspreading of droplets on solid substrates are investigated. Lubrication theory for the droplet motion, and advection-diffusion equations as well as chemical kinetic fluxes for the surfactant transport, lead to coupled evolution equations for the drop thickness, interfacial concentrations of surfactant monomers and bulk concentrations of monomers and micellar aggregates. The surfactant is allowed to adsorb on the substrate either directly from the bulk monomer concentrations or from the liquid-air interface through the contact line. The evolution equations are solved numerically using the finite element method. The results show that basal adsorption of surfactant plays a crucial role in the spreading process: the continuous removal of surfactant from the liquid-air interface, due to the adsorption at the solid surface, is capable of inducing high Marangoni stresses, close to the droplet edge, driving very fast spreading. In many of the cases studied, the droplet radius, R, grows with time, t, with power laws of unity or even higher, close to the reported experimental values for superspreading. The spreading rates depend non-monotonically on the initial surfactant concentration also in accordance with experimental observations. In certain cases, the spreading droplet forms a rim at its leading edge, or creates a ``secondary'' front separated from its main body.   

Dynamics of surfactants spreading on gel layers: cracking and pattern formation

The spreading of surfactant droplets on gel layers is observed to be accompanied by ``starbursts'' resembling cracking patterns on the gel surface. Marangoni stresses induced by surface tension gradients between the spreading surfactant and the underlying gel layer are identified to be the driving force behind these phenomena. A parametric study that involves different surfactants on various gels aims to investigate the ways that system parameters such as the surfactant chemistry and concentration and the gel type and strength can affect the morphology and dynamics of the cracking patterns. The surfactants used in this study include SDS (Sodium Dodecyl Sulphate) and the ``super-spreader'' Silwet L-77 (a Trisiloxane ethoxylate); the different gel substrates are made of agar and gelatine. The instability asscociated with the cracking on the surface of the gels is characterised in terms of the number of ``arms'' that forms and of their mean length as a function of time. Qualitative information on the surfactant distribution and gel topography, where the patterns are formed, is also obtained.   

Gravitational stabilization of the interfacial surfactant-induced instability of shear flows

The linear stability of a two-layer plane Couette-Poiseuille flow with an insoluble surfactant on the interface in the presence of gravity is considered. Previous work has shown that when gravity is absent, the interfacial surfactant in the incompressible inertialess shear flow implies its instability. Considering now the case when gravity is included and the denser fluid is at the bottom, only the normal modes whose wavenumbers $\alpha$ are smaller than some marginal value $\alpha_{0}$ are expected to be unstable. Also, $\alpha_{0}$ should decrease as the Bond number $Bo$ (proportional to the acceleration of gravity) increases. A natural question is, as $Bo$ increases, does $\alpha_{0}\rightarrow0$ as $Bo\rightarrow Bo_{0},$ some finite threshold value? The answer is ``no'' for both the infinite and finite thickness ratios, but in differing ways. By the standard normal mode approach, the dispersion equations found to be quadratic in $\gamma$, the complex ``growth rate.'' It yields the dispersion relation Re$\gamma=$ Re$\gamma(\alpha;Bo,M,s,m),$ where $M$ is the Marangoni number, $m$ is the viscosity ratio, and $s$ is the bottom-side interfacial shear rate. The theory goes without the lubrication approximation: it accounts for the normal modes of all wavelengths.   

Long-wavelength Marangoni convection in liquid layer with insoluble surfactant in modulated thermal field

Marangoni convection in a horizontal liquid layer with an insoluble surfactant distributed on the flat nondeformable free surface is considered. The temperature flux applied to the rigid bottom boundary of the horizontal layer is modulated periodically near a fixed mean value. On the free surface the Biot number is assumed to be small (poorly conducting surface). The surface tension varies linearly with temperature and surface concentration. It has been found formerly that in the absence of temperature flux modulations there exist two longwave instability modes, monotonic and oscillatory ones, which determine the instability threshold. The linear analysis shows that in the case of the monotonic mode, the periodic modulation of the heat flux increase the critical Marangoni number. This effect is especially strong in the region of a small modulation frequency. In the case of the oscillatory mode the influence of modulation depends on the parameters of the problem. A weakly nonlinear analysis near the stability threshold is performed. The reseach was supported by the grant \#3-5799 of the Israeli Ministry of Science and partically supported by the EU via the FP7 Marie Curie scheme (grant \#PITN-GA-2008-214919 (MULTIFLOW).