Quantifying transport within a two-cell microdroplet induced by circular and sharp channel bends

Abstract

A passive method for obtaining good mixing within microdroplets is to introduce curves in the boundaries of the microchannels in which they flow. This article develops a method which quantifies the role of piecewise circular or straight channel boundaries on the transport within a two-cell microdroplet. Transport between the two cells is quantified as an easily computable time-varying flux, which quantifies how lobes intrude from one cell to the other as the droplet traverses the channel. The computation requires neither numerically solving unsteady boundary value problems nor performing trajectory integration, thereby providing an efficient new method for investigating the role of channel geometry on intra-droplet transport.

Transport between two fluids across their mutual flow interface: the streakline approach

“…agitation is time-periodic in a specific way, or for more general perturbations it is possible to define an instantaneous transport [32,78]. However, the current situation is different: the original flow interface is not a heteroclinic manifold.…”

“…However, the current situation is different: the original flow interface is not a heteroclinic manifold. Nevertheless, the ideas from [3,32,78] can be adapted to this situation.…”

“…These methods all identify the flow barriers purely by examining the velocity field, i.e., they determine curves/surfaces arising from analysis of the Lagrangian flow equation ẋ = u(x, t), where u(x, t) is the (potentially unsteady) velocity field, known either explicitly or from discrete data. and improvements to existing methods continue to be reported [2,10,11,15,[31][32][33]. However, given that these are all based directly on the velocity, they ignore actual flow interfaces between two fluids.…”

Unsteadily manipulating internal flow barriers

“…This may help address broader goals of flow control such as ensuring robust flow properties (Brunton & Noack 2015;Kucala & Biringen 2014;Fish & Lauder 2006;Karnik et al 2007;Sattarzadeh & Fransson 2016;Tounsi et al 2016;Boujo et al 2015) or enhancing or suppressing mixing (Ho & Tai 1998;Park et al 2014;Cheikh & Lakkis 2016). The methods described in this paper are therefore from a different angle than more established flow control methods such as velocity modification based on flow sensing (Beebe et al 2000;Frank et al 2016;Jadhav et al 2015;Tounsi et al 2016), stabilising unstable or chaotic trajectories (Ott et al 1990;Boccaletti et al 2000;Pyragas 1992;Tamaseviciute et al 2013), controlling autonomous vehicles by sampling fluid velocities (Heckman et al 2015;Michini et al 2014;Mallory et al 2013;Senatore & Ross 2008), designing optimal geometries for microfluidic devices (Jeong et al 2016;Ionov et al 2006;Balasuriya 2015), determining control velocities for energy/enstropyconstrained mixing (Lin et al 2011;Hassanzadeh et al 2014;Cortelezzi et al 2008;Mathew et al 2007;Balasuriya & Finn 2012), and many others (Kim & Bewley 2007). Knowing the Eulerian velocities which engender a particular unsteady flow barrier can be used as a condition to build various control strategies: determining the optimal global Eulerian velocity to control flow barriers, finding a control forcing that must be applied, determining where to place flow actuators and when to invoke them, etc.…”

Local Stable and Unstable Manifolds and Their Control in Nonautonomous Finite-Time Flows

“…A frequently considered assumption in oceanographic and other flows [38,59,29], this form of model has additional appeal since it can be considered a particular realisation of a stochastic perturbation, a theme which is attracting considerable recent attention [41,53,62]. From the microfluidic perspective, this assumption has value when the idea is to disturb a steady laminar flow (in which transport is suppressed) by imposing an agitation velocity in order to promote transport [72,76,75,10,9]. [28,49,30,17] have been suggested to tackle this.)…”