Flow reversal of the Stokes system with localized boundary data in the half space

TL;DR

This paper demonstrates that localized boundary data in the half-space can induce flow reversal in the unsteady Stokes system, with velocity components changing signs at different regions.

Contribution

It provides a detailed analysis of flow reversal phenomena in the Stokes system with localized boundary conditions, including explicit constructions of solutions exhibiting sign changes.

Findings

Flow reversal occurs for certain boundary influxes.

Velocity components can change signs near and far from the boundary.

Normal and tangential velocity components can have opposite signs near the boundary.

Abstract

We consider the unsteady Stokes system in the half-space with zero initial data and nonzero, space-time localized boundary data. We show that there exist boundary influxes for which the induced flow exhibits flow reversal, in the sense that at least one component of the velocity field changes its sign in the half-space. This phenomenon is demonstrated by a careful analysis of the representation formula for the Stokes system in the half-space, including pointwise estimates, based on the Green tensor with nonzero boundary data. We construct solutions of the Stokes system such that the tangential components of the velocity field exhibit at least one sign change, while the normal component exhibits at least two sign changes. Moreover, the normal component of the constructed velocity field has the opposite sign to the tangential components near the boundary, whereas it has the same sign as the tangential components sufficiently far from the boundary.