Boundary layers for the upper-convected Beris–Edwards model of nematic liquid crystals

Abstract In this paper, we derive and analyze a set of Prandtl-type equations for the boundary layers in nematic liquid crystals. We focus on a two-dimensional model where the hydrodynamics are governed by the Beris–Edwards equations with a shape parameter ξ = 1, specifically emphasizing the upper convected derivative in the order-tensor equation. We introduce a novel decomposition of the order tensor, which, combined with an Ansatz inspired by Prandtl’s theory, leads to a set of limiting equations as the Reynolds, Ericksen, and Deborah numbers approach infinity. We explore two distinct regimes of the dimensionless parameters in the Beris–Edwards equations. The first regime results in a partial decoupling in the limiting equations, where the velocity field is unaffected by the order tensor, though the order tensor is influenced by the flow. In the second regime, we derive a fully coupled system. Our analytical investigation of the derived models reveals that, in the decoupled case, the limiting equations admit analytic-type solutions, while in the coupled setting, the equations allow for shear-flow type solutions.