Thermocapillary thin films: periodic steady states and film rupture

• DOI: 10.1088/1361-6544/ad2a8a
• Publication Date: 2024-03-13T11:19:21Z
• Authors: Gabriele Bruell; Bastian Hilder; Jonas Jansen
• Keywords: Mathematics - Analysis of PDEs; Fluid Dynamics (physics.flu-dyn); FOS: Mathematics; 35B36, 70K50, 35B32, 37G15, 35Q35, 35K59, 35K65, 35D30, 35Q79, 76A20, 35B10, 35B35; FOS: Physical sciences; Physics - Fluid Dynamics; Pattern Formation and Solitons (nlin.PS); 0101 mathematics; Nonlinear Sciences - Pattern Formation and Solitons; 01 natural sciences; Analysis of PDEs (math.AP)

Abstract We study stationary, periodic solutions to the thermocapillary thin-film model

∂ t

h +

∂ x

h 3

∂ x 3

h − g

∂ x

h

+ M

h 2

1 + h

2

∂ x

h

= 0 , t > 0 ,

x ∈

R

,

which can be derived from the Bénard–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.