Jonas Jansen

selected publications

1. Nonlinearity

   Thermocapillary thin films: periodic steady states and film rupture

   Gabriele Bruell, Bastian Hilder, and Jonas Jansen

   Nonlinearity, Open access

   Abstract Link Supplementary Material Blog Slides

   Abstract

   We study stationary, periodic solutions to the thermocapillary thin-film model which can be derived from the Bénard–Marangoni problem via a lubrication approximation. When the Marangoni number M increases beyond a critical value , 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.

2. J Nonlinear Sci

   Fast-Moving Pattern Interfaces Close to a Turing Instability in an Asymptotic Model for the Three-Dimensional Bénard–Marangoni Problem

   Bastian Hilder and Jonas Jansen

   Journal of Nonlinear Science, Aug 2025, Open access

   Abstract Link Supplementary Material

   Abstract

   We study the bifurcation of planar patterns and fast-moving pattern interfaces in an asymptotic long-wave model for the three-dimensional Bénard–Marangoni problem, which is close to a Turing instability with an additional neutral mode at k = 0. We derive the model from the full free-boundary Bénard–Marangoni problem for a thin liquid film on a heated substrate of low thermal conductivity via a lubrication approximation. This yields a quasilinear, fully coupled, mixed-order degenerate-parabolic system for the film height and temperature. As the Marangoni number M increases beyond a critical value M^*, the pure conduction state destabilizes via a Turing(–Hopf) instability. Close to this critical value, we formally derive a system of amplitude equations which govern the slow modulation dynamics of square or hexagonal patterns. Using center manifold theory, we then study the bifurcation of square and hexagonal planar patterns. Finally, we construct planar fast-moving modulating traveling front solutions that model the transition between two planar patterns. The proof uses a spatial dynamics formulation and a center manifold reduction to a finite-dimensional invariant manifold, where modulating fronts appear as heteroclinic orbits. These modulating fronts facilitate a possible mechanism for pattern formation, as previously observed in experiments.

3. PhysD

   Steady pattern formation and film rupture in a two-dimensional thermocapillary thin-film model of the Bénard–Marangoni problem

   Stefano Böhmer, Bastian Hilder, and Jonas Jansen

   Physica D: Nonlinear Phenomena, Dec 2025, Open access

   Abstract Link Supplementary Material

   Abstract

   We study two-dimensional, stationary square and hexagonal patterns in the thermocapillary deformational thin-film model for the fluid height $h \partial_t h+ ∇⋅\left( h^3 \sprlr∇∆h-g∇h + M\frach^2(1+h)^2∇h \right) =0 ,\quad t>0 ,\quad x∈\mathbbR^2, that can be formally derived from the Bénard–Marangoni problem via a long-wave approximation. Using a linear stability analysis, we show that the flat surface profile corresponding to the pure conduction state destabilises at a critical Marangoni number M^* via a conserved long-wave instability. For any fixed absolute wave number k_0, we find that square and hexagonal patterns bifurcate from the flat surface profile at M=M^* + 4k_0^2$. Using analytic global bifurcation theory, we show that the local bifurcation curves can be extended to global curves of square and hexagonal patterns with constant absolute wave number and mass. We exclude that the global bifurcation curves are closed loops through a global bifurcation in cones argument, which also establishes nodal properties for the solutions. Furthermore, assuming that the Marangoni number is uniformly bounded on the bifurcation branch, we prove that solutions exhibit film rupture, that is, their minimal height tends to zero. This assumption is substantiated by numerical experiments.