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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).