The role of advection in phase-separating binary liquids

Using the advective Cahn-Hilliard equation as a model, we illuminate the role of advection in phase-separating binary liquids. The advecting velocity is either prescribed, or is determined by an evolution equation that accounts for the feedback of concentration gradients into the flow. Here, we focus on passive advection by a chaotic flow, and coupled Navier-Stokes Cahn-Hilliard flow in a thin geometry. Our approach is based on a combination of functional-analytic techniques, and numerical analysis. Additionally, we compare and contrast the Cahn-Hilliard equation with other models of aggregation; this leads us to investigate the orientational Holm-Putkaradze model. We demonstrate the emergence of singular solutions in this system, which we interpret as the formation of magnetic particles. Using elementary dynamical systems arguments, we classify the interactions of these particles.Comment: Ph.D. Thesis, Imperial College London, February 200

An analytical connection between temporal and spatio-temporal growth rates in linear stability analysis

We derive an exact formula for the complex frequency in spatio-temporal stability analysis that is valid for arbitrary complex wave numbers. The usefulness of the formula lies in the fact that it depends only on purely temporal quantities, which are easily calculated. We apply the formula to two model dispersion relations: the linearized complex Ginzburg--Landau equation, and a model of wake instability. In the first case, a quadratic truncation of the exact formula applies; in the second, the same quadratic truncation yields an estimate of the parameter values at which the transition to absolute instability occurs; the error in the estimate decreases upon increasing the order of the truncation. We outline ways in which the formula can be used to characterize stability results obtained from purely numerical calculations, and point to a further application in global stability analyses.Comment: 36 pages, 16 figures; Article has been tweaked and reduced in size but essential features remain the same; Supplementary material (16 pages) is also include

Absolute linear instability in laminar and turbulent gas/liquid two-layer channel flow

We study two-phase stratified flow where the bottom layer is a thin laminar liquid and the upper layer is a fully-developed gas flow. The gas flow can be laminar or turbulent. To determine the boundary between convective and absolute instability, we use Orr--Sommerfeld stability theory, and a combination of linear modal analysis and ray analysis. For turbulent gas flow, and for the density ratio r=1000, we find large regions of parameter space that produce absolute instability. These parameter regimes involve viscosity ratios of direct relevance to oil/gas flows. If, instead, the gas layer is laminar, absolute instability persists for the density ratio r=1000, although the convective/absolute stability boundary occurs at a viscosity ratio that is an order of magnitude smaller than in the turbulent case. Two further unstable temporal modes exist in both the laminar and the turbulent cases, one of which can exclude absolute instability. We compare our results with an experimentally-determined flow-regime map, and discuss the potential application of the present method to non-linear analyses.Comment: 33 pages, 20 figure

Particle-laden viscous channel flows: Model regularization and parameter study

We characterize the flow of a viscous suspension in an inclined channel where the flow is maintained in a steady state under the competing influences of gravity and an applied pressure drop. The basic model relies on a diffusive-flux formalism. Such models are common in the literature, yet many of them possess an unphysical singularity at the channel centreline where the shear rate vanishes. We therefore present a regularization of the basic diffusive-flux model that removes this singularity. This introduces an explicit (physical) dependence on the particle size into the model equations. This approach enables us to carry out a detailed parameter study showing in particular the opposing effects of the pressure drop and gravity. Conditions for counter-current flow and complete flow reversal are obtained from numerical solutions of the model equations. These are supplemented by an analytic lower bound on the ratio of the gravitational force to the applied pressure drop necessary to bring about complete flow reversal

Linear and nonlinear stability analysis in microfluidic systems

In this article we use analytical and numerical modeling to describe parallel viscous two-phase flows in microchannels. The focus is on idealized two-dimensional geometries, with a view to validating the various methodologies for future work in three dimensions. In the first instance, we use analytical Orr-Sommerfeld theory to describe the linear instability which governs the formation of small-amplitude waves in such systems. We then compare the results of this analysis with an in-house Computational Fluid Dynamics (CFD) solver called TPLS. Excellent agreement between the theoretical analysis and TPLS is obtained in the regime of small-amplitude waves. We continue the numerical simulations beyond the point of validity of the Orr-Sommerfeld theory. In this way, we illustrate the generation of nonlinear interfacial waves and reverse entrainment of one fluid phase into the other. We justify our simulations further by comparing the numerical results with corresponding results from a commercial CFD code. This comparison is again extremely favourable—this rigorous validation paves the way for future work using TPLS or commercial codes to perform extremely detailed three-dimensional simulations of flow in microchannels.http://www.techscience.com/journal/fdmphj2021Graduate School of Technology Management (GSTM

Mass conservation and reduction of parasitic interfacial waves in level-set methods for the numerical simulation of two-phase flows: A comparative study

International audienceA commonly used class of methods for the numerical simulation of two-phase flows is level set. It is often reported though that this method does not accurately conserve mass of each fluid, unlike other interface capturing techniques such as volume-of-fluid. A further concern besides mass conservation is the formation of any parasitic currents. Since the initial formulation of level-set methods, however, numerous modifications have been proposed, and it does not seem clear whether mass conservation errors and parasitic currents are problematic for all of these and, if not, what key steps could be taken to avoid them. Furthermore, results reported in the literature are often for benchmark tests in two dimensions, and it is not clear whether a good performance there holds up in three dimensions. We undertake here a comparative study, reporting test results in two and three dimensions for various level-set methods on a variety of problems. Kinematical tests are first performed for prescribed velocity fields, followed by benchmark tests including the solution of the Navier–Stokes equations. It is shown that higher-order schemes for spatial and temporal discretization may improve mass conservation and avoid interface distortion. In particular, two reinitialization methods that are straightforward to implement perform very well at all these tests. It is demonstrated that some schemes introduce parasitic oscillations in the simulation of Rayleigh–Taylor instability