**Project Heads**

Etienne Emmrich, Dietmar Hömberg, Robert Lasarzik

**Project Members**

Luisa Plato

**Project Duration**

01.05.2021 − 30.04.2024

**Located at**

WIAS

An effective control of microfluids is the essential part of microfluidic devices such as so-called “Lab-on-a-chip” devices, which aim to incorporate laboratory functions on small chips. This should allow to parallelize and provide the laboratory functions cheaper and makes them easier assessable. But due to the small scales the Reynolds number is inherently small such that usual flow control techniques by pressure driven flow are only rarely applicable.

An easy way out is to control the fluid via flows induced by electric fields, so called electrokinetic phenomena. But in isotropic fluids only linear electrokinetic phenomena occur and only charged particles can be transported.These restrictions can be lifted using anisotropic fluids. In such fluids nonlinear electrokinetics occur, which may transport any particles and allows to use alternating electric fields to induce a flow.

The directed movement is rather induced by the interaction of the electric field with the singularities in the molecular orientation than the charged particles.

These phenomena are highly nonlinear and may be modelled by non-standard nonlinear systems of PDEs. Such systems of PDEs request non-standard solvability concepts. We will apply the innovative dissipative solution concept to this system and use this notion to prove several standard questions in the context of PDEs. This includes existence, uniqueness, and long-time behavior of solutions, convergence of Finite Element approximations and optimal control schemes. In the end, we plan to understand the nonlinear electrokinetic phenomena in liquid crystals better and be able to provide efficient numerical simulations and optimal control strategies, in order to help building microfluidic devices based on the nonlinear electrokinetics in anisotropic microfluids.

**External Website**

**Related Publications
**

- L. Banas, R. Lasarzik, and A. Prohl.

Numerical analysis for nematic electrolytes.

*IMA Journal of Numerical Analysis*, 2020. - E. Emmrich and R. Lasarzik.

Weak-strong uniqueness for the general Ericksen-Leslie system in three dimensions.

*Discrete & Continuous Dynamical Systems*– A, 38(1078-0947):4617, 2018. - D. Hömberg, K. Krumbiegel, and J. Rehberg.

Optimal control of a parabolic equation with dynamic boundary condition.*Applied Mathematics & Optimization*, 67(1):3–31, 2013. - D. Hömberg, R. Lasarzik, and L. Plato. Existence of weak solutions to an anisotropic electrokinetic flow model.
*WIAS preprint no. 3104, 2024.* - R. Lasarzik. Dissipative solution to the Ericksen-Leslie system equipped with the Oseen-Frank energy.
*Zeitschrift für angewandte Mathematik und Physik*, 70(1):8, 2018. - R. Lasarzik, M. Reiter. Analysis and Numerical Approximation of Energy-Variational Solutions to the Ericksen–Leslie Equations.
*Acta Applicandae Mathematicae***184**, 11 (2023).

**Related Pictures
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