Thin Film Instability Derivations -- 2

I am examining the hydrodynamics and instabilities of a thin viscous film flowing over a periodically corrugated, uniformly heated incline. Long-wave theory has already shown that Marangoni stresses, together with the local steepness of the substrate, can destabilise the flow and generate finite-amplitude waves. What I need now is a clean, step-by-step mathematical treatment of the problem that can stand on its own in a journal appendix. The scope includes the full hierarchy: leading-order long-wave formulation, linear stability analysis, and a weakly nonlinear expansion up to the first amplitude equation that captures secondary instabilities. Please derive every governing equation, nondimensionalise where appropriate, and state all assumptions explicitly so that another researcher can reproduce the work without referring back to me. Illustrative sketches or dispersion-relation plots are welcomed, but the priority is a logically ordered set of equations and analytical results rather than polished graphics. Deliverables • A LaTeX document (or compatible source) containing all derivations, clearly numbered equations, and concise explanatory text. • A brief note highlighting any limiting cases or parameter regimes where the model simplifies. Acceptance criteria • All steps from the Navier-Stokes equations to the amplitude evolution equation are transparent and gap-free. • Marangoni and substrate-slope effects are retained to leading order and their competing roles clearly identified. • Variables and parameters are defined once and used consistently throughout the manuscript. If you are comfortable juggling lubrication theory, perturbation methods, and Marangoni boundary conditions, I would be happy to hear how quickly you could assemble the full derivation package. Deadline: 2-3 days

Project ID: 40208569