On hydrostatic limit of Beris-Edwards system in a thin strip

TL;DR

This paper analyzes the hydrostatic limit of the Beris-Edwards system for nematic liquid crystals in a thin strip, deriving a decoupled limit system involving the Prandtl equations and a novel Q-tensor system.

Contribution

It establishes the hydrostatic limit of the 3D Beris-Edwards system, deriving a partly decoupled system with a new Q-tensor model and proving convergence and well-posedness.

Findings

Derivation of the hydrostatic limit system for nematic liquid crystals.

Convergence proof of the rescaled Beris-Edwards system to the limit.

Well-posedness of the limit system in Sobolev spaces.

Abstract

In this paper we consider the 3D co-rotational Beris-Edwards system modeling the hydrodynamic motion of nematic liquid crystals in a thin strip. The system contains the incompressible Navier-Stokes, coupled with a parabolic system for matrix-valued functions, the $Q$-tensors. We show that under a suitable scaling, corresponding, in the Navier-Stokes part, to the hydrostatic scaling, one obtains in the limit a partly decoupled system. For the fluid part we obtain the Prandtl system while for the $Q$-tensors we obtain a non-standard system, involving fluids components and a non-standard combination of partly dissipative equations and algebraic constraints. We prove the convergence of the rescaled system and the well-posedness of the limit in Sobolev spaces.