Viscoelastic Film Wave Instability

I am investigating the evolution and possible instabilities that arise when a thin viscoelastic film flows down an inclined, wavy substrate. My analytical work already covers the long-wave model, its linear and weakly nonlinear limits, and the appearance of two distinct wave families that depend on the substrate’s steepness. What I still need is a rigorous, publication-ready consolidation of the theory together with reproducible numerical evidence. Your job is to extend and verify the existing derivations, implement a robust time-dependent solver, and map out finite-amplitude travelling solutions across a representative parameter space. If you prefer MATLAB, Python (NumPy/SciPy), or a spectral‐element COMSOL workflow, that is fine, provided the final code runs out of the box and reproduces the key figures. Deliverables – the project is complete when you hand over • a clean derivation that connects the long-wave model to the full viscoelastic Navier–Stokes equations, • stability curves and nonlinear bifurcation diagrams that clearly identify the two wave families and their dependence on bottom steepness, • time-dependent simulations that demonstrate growth, saturation and fully developed travelling waves, and • well-commented source code, inputs, and a brief usage note so the results can be regenerated. I will check the mathematics for consistency and run the code myself; if the plots and trends match the analytical predictions and converge under grid refinement, the milestone is approved.

Project ID: 40189506