# Boundary conditions and polymeric drag reduction for the Navier-Stokes equations

## TL;DR

This paper develops a PDE model to analyze how wall-grafted polymers in a viscous fluid reduce drag, establishing mathematical results on well-posedness, inviscid limits, and quantifying drag reduction effects.

## Contribution

It introduces a novel PDE boundary condition modeling polymer effects and proves global well-posedness and drag reduction scaling in two-dimensional flows.

## Key findings

- Wall-friction factor scales as 1/Re, indicating drag reduction.
- Global strong solutions exist in two dimensions.
- Inviscid limit to Euler solutions with explicit rate.

## Abstract

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 are polymers which are floating in the solvent and attach (either by adsorption or through irreversible chemical binding) at one of their chain ends to the substrate (wall). We propose a PDE model to study this setup in the simple setting where the solvent is a viscous incompressible Navier-Stokes fluid occupying the bulk of a smooth domain $\Omega\subset \mathbb{R}^d$, and the wall-grafted polymer is in the so-called mushroom regime (inter-polymer spacing on the order of the typical polymer length). The microscopic description of the polymer enters into the macroscopic description of the fluid motion through a dynamical boundary condition on the wall-tangential stress of the fluid, something akin to (but distinct from) a history-dependent slip-length. We establish global well-posedness of strong solutions in two-spatial dimensions and prove that the inviscid limit to the strong Euler solution holds with a rate. Moreover, the wall-friction factor $\langle f\rangle$ and the global energy dissipation $\langle \varepsilon\rangle$ vanish inversely proportional to the Reynolds number $Re$. This scaling corresponds to Poiseuille's law for the friction factor $\langle f\rangle \sim1/ Re$ for laminar flow and thereby quantifies drag reduction in our setting. These results are in stark contrast to those available for physical boundaries without polymer additives modeled by, e.g., no-slip conditions, where no such results are generally known even in two-dimensions.