Papers › Thermocapillary Thin Films: Periodic Steady States and Film Rupture
Thermocapillary Thin Films: Periodic Steady States and Film Rupture
Gabriele Brüll, Bastian Hilder, Jonas Jansen
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We study stationary, periodic solutions to the thermocapillary thin-film model ∂ₜ h + ∂ₓ (h³(∂ₓ³ h - g∂ₓ h) + Mh²/((1+h)²)∂ₓh) = 0, t>0, x∈ℝ, 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.
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