Boundary stabilizing actuators for multi-phase fluids in a channel

TL;DR

This paper develops a boundary feedback control method to stabilize a two-fluid mixture in a channel, ensuring stability regardless of viscosity levels, with a simple and computationally feasible controller design.

Contribution

It introduces an explicit, boundary-based feedback controller for stabilizing multi-phase fluid flows in a channel, independent of viscosity and other system coefficients.

Findings

Achieves boundary stabilization for multi-phase fluids in a 3D channel.

Controller is simple, explicit, and computationally easy to implement.

Stability holds for any viscosity level, independent of other parameters.

Abstract

This work is devoted to the problem of boundary stabilization of a mixture of two viscous and incompressible fluids in a three dimensional channel-like domain $(x,y,z)\in \mathbb{R}\times (0,1)\times\mathbb{R}$. The model consists of the Navier-Stokes equations, governing the fluid velocity, coupled with a convective Cahn-Hilliard equation for the relative density of the atoms of one of the fluids. The feedback controller, we design here, is with actuation only on the tangential component of the velocity field on both walls $y=0,1$. Moreover, it is given in a simple and explicit form, easy to manipulate from the computational point of view. The stability is guaranteed no matter how large the kinematic viscosity of the mixture is. Also it is independent of the other coefficients of the system.