Inertio-capillary cross-streamline drift of droplets in Poiseuille flow using dissipative particle dynamics simulations

Abstract

We find using dissipative particle dynamics (DPD) simulations that a deformable droplet sheared in a narrow microchannel migrates to steady-state position that depends upon the dimensionless particle capillary number , which controls the droplet deformability (with V the centerline velocity, μ the fluid viscosity, Γ the surface tension, R the droplet radius, and H the gap), the droplet (particle) Reynolds number , which controls inertia, where ρ is the fluid density, as well as on the viscosity ratio of the droplet to the suspending fluid κ = μ/μ. We find that when the Ohnesorge number is around 0.06, so that inertia is stronger than capillarity, at small capillary number Ca < 0.1, the droplet migrates to a position close to that observed for hard spheres by Segre and Silberberg, around 60% of the distance from the centerline to the wall, while for increasing Ca the droplet steady-state position moves smoothly towards the centerline, reaching around 20% of the distance from centerline to the wall when Ca reaches 0.5 or so. For higher Oh, the droplet position is much less sensitive to Ca, and remains at around 30% of the distance from centerline to the wall over the whole accessible range of Ca. The results are insensitive to viscosity ratios from unity to the highest value studied here, around 13, and the drift towards the centerline for increasing Ca is observed for ratios of droplet diameter to gap size ranging from 0.1 to 0.3. We also find consistency between our predictions and existing perturbation theory for small droplet or particle size, as well as with experimental data. Additionally, we assess the accuracy of the DPD method and conclude that with current computer resources and methods DPD is not readily able to predict cross-stream-line drift for small particle Reynolds number (much less than unity), or for droplets that are less than one tenth the gap size, owing to excessive noise and inadequate numbers of DPD particles per droplet.

Cross-stream migration of a Brownian droplet in a polymer solution under Poiseuille flow

“…Without polymer, the droplet initially migrated outward as f x increased, but the peak of this distribution moved inward with additional increases in f x (Fig. 7a), consistent with the simulations of Marson et al 35 At φ p = 5.0% (Fig. 7b), there was an initial trend to focus when f x 0.003 ε/d.…”

“…In this article, we test and confirm this hypothesis using particle-based computer simulations of the crossstream migration of a Brownian fluid droplet in a polymer solution under Poiseuille flow. Although the droplet migrated outward in a Newtonian solvent (in agreement with prior simulations 35 ), we found that it focused onto the channel centerline in solutions of sufficiently long polymers at modest concentrations. The flow-rate dependence of the focusing was nontrivial due to a combination of effects from droplet deformation and the elastic force exerted by the polymers.…”

“…We analyzed the absolute value |z c | based on the symmetry of the microchannel and to improve sampling. Previous studies 34,35 have reported the average center-of-mass position, |z c | , which is the first moment of the distribution of |z c |. However, a Brownian droplet can adopt a variety of distributions in the channel depending on the conditions, and we found that |z c | was not sufficiently discriminating between these.…”

“…We define a channel Reynolds number, Re = 2ρU H/µ s , and a droplet Reynolds number Re d = Re(R/H) 2 . 21,35 When the channel Reynolds number is sufficiently small, the flow is expected to be laminar. When Re d is small, inertial forces on the droplet are not significant and results from the Stokes flow limit are expected to apply.…”

“…26,27 At finite fluid inertia, Legendre and Magnaudet demonstrated that there is lift on a droplet 28 analogous to the Saffman lift on a rigid particle 29,30 but with a magnitude that depends on the viscosity ratio between the droplet and the fluid. Experiments 4,31 and simulations [32][33][34][35] have shown that droplets undergo Segré-Silberberg-type migration in Poiseuille flow, and that the preferred lateral position depends on several dimensionless parameters, as recently discussed in detail by Marson et al 35 High-throughput applications like filtration or sorting may require focusing particles onto the channel centerline, 16,21 which is not always achieved by inertial or deformation-induced migration in simple channel geometries. Considerable efforts have been dedicated to design various microfluidic device geometries that can manipulate particles in this way, 21 but finding such geometries can be difficult and highly problem specific.…”