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The lubrication approximation is a fundamental framework in fluid mechanics for modeling the flow of thin liquid layers dominated by viscous and surface tension effects. We use Lie symmetry analysis to investigate the third-order ordinary differential equation y ″ ′ = y − n representing the steady-state form of the generalized lubrication equation. We examine how the parameter n affects the possibility of finding explicit solutions using invariant differentiation and by treating the equation as a first-order system. Invariant differentiation yields explicit solutions for n = 5 / 4 and n = 1 / 2 . The first-order system yields solutions via integration by quadrature for n = 5 / 2 and n = 4 / 3 . The results highlight the strong dependence of solution existence on the value of n and provide insight into the nonlinear behavior of the film profile.
Navrátil et al. (Tue,) studied this question.