Energy Constrained Transport Maximization across a Fluid Interface

Accurate control of hyperbolic trajectories in any dimension

“…We remark that we have achieved global transport by applying localized controls at the two hyperbolic trajectories, highlighting the role hyperbolic trajectories have on the phase space. By targeting energy towards regions where it has the most impact in this fashion, we offer a new approach towards energy-constrained transport maximization [39,44,49,50]. A refinement of this idea by additionally being able to control the directions in which the stable and unstable manifolds emanate from the hyperbolic trajectory is underway.…”

“…Thus, if hyperbolic trajectories with a heteroclinic manifold connecting them are made to move in a judiciously chosen manner, it will be possible to break apart the heteroclinic manifold into intersecting stable and unstable manifolds, thereby causing complicated (i.e., chaotic) mixing. Thus, controlling hyperbolic trajectories is a yet-unexplored avenue in the topic of controlling and optimizing mixing which is eliciting much recent interest [23,24,[38][39][40][41][42][43][44][45][46][47][48][49][50].…”

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

“…If fully time-periodic and centred around zero, one can use the concept of an average flux which relates to the size of a lobe of fluid transported across per unit time. 46,[51][52][53][54][55]64 But in our case the flux function is not periodic, and indeed decays to zero as t → ±∞ since the bending part of the channel has finite length. Such a measure is therefore not possible.…”

“…The focus is on characterising how the transport occurs between the upper and lower cells of the two-cell microdroplet, as this is the instigating process for mixing between the upper and lower fluids. This shall be quantified as a time-varying function, utilising recently developed nonautonomous dynamical systems theory 46,47 (standard time-periodic analyses [48][49][50][51][52][53][54][55] are not applicable for generally varying channel boundaries).…”

“…The function M is a Melnikov function which unlike in the classical interpretation 49,50,63 is valid for general aperiodic v. The theory related to the Melnikov function has been used previously in situations in which flux is to be maximised in time-periodic flows. [52][53][54][55]64 Unlike in previous applications, 47,[52][53][54][55]64 the Melnikov function need not be multiplied by a small parameter since this smallness is encoded in v. Thus, M(t) can be thought of as the flux function, directly quantifying the area of fluid per unit time being transported from the lower to the upper cell instantaneously at a general time t, with error O(|v| 2 ).…”