Hopf instability of a Rayleigh–Taylor unstable thin film heated from the gas side
Institute of Physics, Brandenburg University of Technology, 03046, Cottbus, Germany
2 Department of Mechanical Engineering, Technion-Israel Institute of Technology, 3200003, Haifa, Israel
Accepted: 30 January 2023
Published online: 17 February 2023
A thin liquid film located on the underside of a horizontal solid substrate can be stabilized by the Marangoni effect if the liquid is heated at its free surface. Applying long-wave approximation and projecting the velocity and temperature fields onto a basis of low-order polynomials, we derive a dimension-reduced set of three coupled evolution equations where nonlinearities of both the Navier–Stokes and the heat equation are included. We find that in a certain range of fluid parameters and layer depth, the first bifurcation from the motionless state is oscillatory which sets in with a finite but small wave number. The oscillatory branch is determined using a linear stability analysis of the long-wave model, but also by solving the linearized original hydrodynamic equations. Finally, numerical solutions of the reduced nonlinear model equations in three spatial dimensions are presented.
© The Author(s) 2023
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