3 results
Flow of a concentrated emulsion with surfactant through a periodic porous medium
- Alexander Z. Zinchenko, Jacob R. Gissinger, Robert H. Davis
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- Journal:
- Journal of Fluid Mechanics / Volume 953 / 25 December 2022
- Published online by Cambridge University Press:
- 07 December 2022, A21
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High-resolution, long-time three-dimensional simulations are presented for slow, pressure-driven flow of a periodic emulsion of deformable drops through a dense, simple cubic array of solid spheres (one drop and one particle per periodic cell). The drops, covered with insoluble, non-diffusive surfactant, are large compared with pores, and they squeeze with high resistance, very closely coating the solids to overcome surface tension and lubrication effects. The solid volume fraction is 50 %, the emulsion concentration $c_{em}$ in the pore space is 36 % or 50 %, the drop-to-medium viscosity ratio $\lambda$ is 0.25 to 4. The contamination measure $\beta \leq 0.1$ keeps the linear surfactant model (assumed in most of the work) physically relevant. The boundary-integral solution requires extreme resolutions (tens of thousands of boundary elements per surface) achieved by multipole acceleration with special desingularizations, combined with flow-biased surfactant transport algorithms for numerical stability. The time-periodic regime is typically attained after a few squeezing cycles; the motion period is used in the extrapolation scheme to evaluate critical capillary numbers $Ca_{crit}$ demarcating squeezing from trapping. Due to Marangoni stresses, even light ($\beta =0.05$) to moderate ($\beta =0.1$) contaminations significantly reduce the average drop-phase migration velocity (up to 2.8 times, compared with clean drops), especially at small $\lambda =0.25$. In contrast, $Ca_{crit}$ is weakly sensitive to contamination and levels off completely at $\beta =0.05$. At $\lambda =0.25$ and $c_{em}=0.36$, the average drop-phase velocities are much different for lightly and moderately contaminated emulsions, except for near-critical squeezing when they become the same. Nonlinear surfactant models (Langmuir, Frumkin) are used to validate the linear model.
Drop squeezing between arbitrary smooth obstacles
- Jacob R. Gissinger, Alexander Z. Zinchenko, Robert H. Davis
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- Journal:
- Journal of Fluid Mechanics / Volume 908 / 10 February 2021
- Published online by Cambridge University Press:
- 10 December 2020, A33
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A fully three-dimensional boundary-integral method (BIM) is developed for the interaction of drops, suspended in a uniform far-field flow at small Reynolds number, with arbitrary Lyapunov surfaces. The close approach of fluid interfaces to solid surfaces poses significant challenges for numerical BIM implementations, due to the highly singular behaviour of single- and double-layer boundary integrals. Two new methods are described that generalize the accurate calculation of the highly singular surface integrals used by high-order desingularization techniques. The first method is semi-analytical, and applies to axisymmetric solid obstacles (in an arbitrary three-dimensional configuration). An axisymmetric particle can be divided into a series of characteristic disks along its axis, for which closed-form expressions for single and double layers are derived in terms of elliptic integrals. To accommodate arbitrary smooth surfaces, a multimesh desingularization method is introduced that calculates surface integrals utilizing a hierarchy of embedded mesh resolutions, together with distance-activated mesh interactions. Several particle shapes, including spherocylinders (capsules) and flat plates, are used to represent major classes characteristic of porous media. A droplet approaching a capsule will break up after forming two lobes, connected by a thin filament, on either side of the capsule. The cross-sectional shape of the filament affects lubrication behaviour. A constriction made of two parallel capsules, even of low aspect ratio, significantly retards drop passage compared to two spheres. Trends in drop squeezing between two capsules are summarized over a range of capillary number, viscosity ratio, drop size and capsule length. A constriction of two coplanar plates results in notably different lubrication and squeezing behaviour. Flow rectification is demonstrated for constrictions that are non-symmetrical with respect to flow reversal, for several non-axisymmetric particles.
Drops with insoluble surfactant squeezing through interparticle constrictions
- Jacob R. Gissinger, Alexander Z. Zinchenko, Robert H. Davis
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- Journal:
- Journal of Fluid Mechanics / Volume 878 / 10 November 2019
- Published online by Cambridge University Press:
- 10 September 2019, pp. 324-355
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The interfacial behaviour of surfactant-laden drops squeezing through tight constrictions in a uniform far-field flow is modelled with respect to capillary number, drop-to-medium viscosity ratio and surfactant contamination. The surfactant is treated as insoluble and non-diffusive, and drop surface tension is related to surfactant concentration by a linear equation of state. The constriction is formed by three solid spheres held rigidly in space. A characteristic aspect of this confined and contaminated multiphase system is the rapid development of steep surfactant-concentration gradients during the onset of drop squeezing. The interplay between two physical effects of surfactant, namely the greater interface deformability due to decreased surface tension and interface immobilization due to Marangoni stresses, results in particularly rich drop-squeezing dynamics. A three-dimensional boundary-integral algorithm is used to describe drop hydrodynamics, and accurate treatment of close squeezing and trapped states is enabled by advanced singularity subtraction techniques. Surfactant transport and hydrodynamics are coupled via the surface convection equation (or convection–diffusion equation, if artificial diffusion is included), the interfacial stress balance and a solid-particle contribution based on the Hebeker representation. For extreme conditions, such as drop-to-medium viscosity ratios significantly less than unity, it is found that upwind-biased methods are the only stable approaches for modelling surfactant transport. Two distinct schemes, upwind finite volume and flow-biased least squares, are found to provide results in close agreement, indicating negligible numerical diffusion. Surfactant transport is enhanced by low drop-to-medium viscosity ratios, at which extremely sharp concentration gradients form during various stages of the squeezing process. The presence of surfactant, even at low degrees of contamination, significantly decreases the critical capillary number for droplet trapping, due to the accumulation of surfactant at the downwind pole of the drop and its subsequent elongation. Increasing the degree of contamination significantly affects surface mobility and further decreases the critical capillary number as well as drop squeezing times, up to a threshold above which the addition of surfactant negligibly affects squeezing dynamics.