Boundary Conditions and Polymeric Drag Reduction for the Navier–Stokes Equations

Reducing wall drag in turbulent pipe and channel flows is an issue of great practical importance. In engineering applications, end-functionalized polymer chains are often employed as agents to reduce drag. These polymers which floating the solvent attach (either by adsorption or through irreversible chemical binding) at one their chain ends substrate (wall). We propose a PDE model study this setup simple setting where viscous incompressible Navier–Stokes fluid occupying bulk smooth domain $$\Omega \subset {\mathbb {R}}^d$$ , wall-grafted so-called mushroom regime (inter-polymer spacing on order typical length). The microscopic description enters into macroscopic motion dynamical boundary condition wall-tangential stress fluid, something akin (but distinct from) history-dependent slip-length. establish global well-posedness strong solutions two-spatial dimensions prove that inviscid limit Euler solution holds with rate. Moreover, wall-friction factor $$\langle f\rangle $$ energy dissipation \varepsilon \rangle vanish inversely proportional Reynolds number $$\mathbf{Re } . This scaling corresponds Poiseuille’s law for friction \sim 1/\mathbf{Re laminar flow thereby quantifies reduction our setting. results stark contrast those available physical boundaries without additives modeled by, example, no-slip conditions, no such generally known even two-dimensions.