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# Optimal control of particle separation in inertial microfluids

Christopher Prohm:

At intermediate Reynolds numbers, particles in a micro fluidic
channel assemble at fixed distances from the channel axis and
bounding walls, an effect first discovered by Segré and Silberberg
[1]. This behavior is very different from particle movement at low
Reynolds numbers, where due to kinematic reversibility rigid particles
do not cross streamlines. The Segré-Silberberg effect can be
described in terms of an effective lift force acting on the
particles.

Devices utilizing inertial lift forces for the
separation of bacteria and red blood cells have recently been
demonstrated [2]. The separation is most efficient for large size
differences since the inertial lift force scales with the third power
of the particle radius. Here, we show that one can use optimal control
theory to determine external control forces generated, for example, by
optical tweezers to steer particles and to separate them when they are
similar in size.

We study the system by mesoscopic simulations of
the fluid using multi-particle collision dynamics (MPCD) [3]. We
determine lift forces and single-particle probability distributions in
steady state and analyze their dependence on particle radius and
Reynolds number. We show that the Boltzmann distribution for the
potential connected to the lift force reproduces the observed
distribution functions.

We use the lift force determined by MPCD to
set up a Smoluchowski equation which describes the particle motion in
lateral channel direction. We then employ the formalism of optimal
control [4] to determine proles of the external force, which help to
steer particles and thereby maximize particle separation. We verify
that the calculated force profiles are indeed able to separate
particles of similar size by performing independent simulations of the
corresponding Langevin equations.

[1] G. Segré and A. Silberberg, Nature, 189, 209 (1961).

[2] A. J. Mach and D. Di Carlo, Biotechnol. Bioeng., 107, 302 (2010).

[3] C. Prohm, M. Gierlak, and H. Stark EPJE, 35, 80 (2012).

[4] F. Tröltzsch, Optimal Control of Partial Dierential Equations, American Mathematical Society, first edition (2010).