In this paper, we investigate the dynamics of a liquid film flowing over a periodic wavy wall. This study is based on a long-wave model that is valid at near-critical Reynolds number. For the periodic wall surface, we prove the existence of a periodic steady-state solution to the model by the method of abstract contraction mapping in a particular functional space. Using the Floquet–Bloch theory and asymptotic method, we establish several analytic results on the stability of the periodic steady-state solution in a weighted functional space.
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