Approximate controllabillity of a 2D linear system related to the motion of two fluids with surface tension

TL;DR

This paper studies the approximate controllability of a 2D fluid interface system influenced by surface tension, showing controllability under certain geometric conditions using linearized equations.

Contribution

It introduces a controllability result for a linearized 2D fluid interface model with surface tension, highlighting conditions related to Bessel function zeros.

Findings

System is approximately controllable to steady states for any positive time.

Controllability depends on the radius not being a scaled zero of a Bessel function.

Results apply to small displacements around steady states.

Abstract

We consider a coupled system of partial differential equations describing the interactions between a closed free interface and two viscous incompressible fluids. The fluids are assumed to satisfy the incompressible Navier-Stokes equations in time-dependent domains that are determined by the free interface. The mean curvature of the interface induces a surface tension force that creates a jump of the Cauchy stress tensor on both sides. It influences the behavior of the surrounding fluids, and therefore the deformation of this interface via the equality of velocities. In dimension 2, the steady states correspond to immobile interfaces that are circles with all the same volume. Considering small displacements of steady states, we are lead to consider a linearized version of this system. We prove that the latter is approximately controllable to a given steady state for any time $T>0$ by the means of additional surface tension type forces, provided that the radius of the circle of reference does not coincide with a scaled zero of the Bessel function of first kind.