Thermocapillary Thin Films: Periodic Steady States and Film Rupture

TL;DR

This paper investigates stationary periodic solutions in a thermocapillary thin-film model, analyzing their bifurcation, stability, and the onset of film rupture as parameters vary, using Hamiltonian and bifurcation theory.

Contribution

It introduces a global bifurcation analysis of stationary solutions, revealing their convergence to rupture states and their instability near bifurcation points.

Findings

Periodic solutions bifurcate from constant states at critical Marangoni number

Solutions on the bifurcation branch tend to film rupture

Bifurcating solutions are unstable near the bifurcation point

Abstract

We study stationary, periodic solutions to the thermocapillary thin-film model \begin{equation*} \partial_t h + \partial_x \Bigl(h^3(\partial_x^3 h - g\partial_x h) + M\frac{h^2}{(1+h)^2}\partial_xh\Bigr) = 0,\quad t>0,\ x\in \mathbb{R}, \end{equation*} which can be derived from the B\'enard-Marangoni problem via a lubrication approximation. When the Marangoni number $M$ increases beyond a critical value $M^*$, the constant solution becomes spectrally unstable via a (conserved) long-wave instability and periodic stationary solutions bifurcate. For a fixed period, we find that these solutions lie on a global bifurcation curve of stationary, periodic solutions with a fixed wave number and mass. Furthermore, we show that the stationary periodic solutions on the global bifurcation branch converge to a weak stationary periodic solution which exhibits film rupture. The proofs rely on a Hamiltonian formulation of the stationary problem and the use of analytic global bifurcation theory. Finally, we show the instability of the bifurcating solutions close to the bifurcation point and give a formal derivation of the amplitude equation governing the dynamics close to the onset of instability.