On the stability of traveling wave solutions to thin-film and long-wave models for film flows inside a tube

Abstract

Traveling wave solutions are studied numerically and theoretically for models of viscous core–annular flows and falling film flows inside a tube. The models studied fall into one of two classes, referred to here as ‘thin-film’ or ‘long-wave’. One model of each type is studied for three problems: a falling viscous film lining the inside of a tube, and core–annular flow with either equal- or unequal-density fluids. In recent work, traveling wave solutions for some of these equations were found using a smoothing technique that removes a degeneracy and allows for continuation onto a periodic family of solutions from a Hopf bifurcation. This paper has three goals. First, the smoothing technique used in earlier studies is justified for these models using asymptotics. Second, this technique is used to find numerically families of traveling wave solutions not previously explored in detail, including some which have multiple turning points due to the interaction between gravity, viscous forces, surface tension, and pressure-driven flow. Third, the stability of these solutions is studied using asymptotics near the Hopf bifurcation point, and numerically far from this point. In particular, a simple theory using the constant solution at the Hopf bifurcation point produces estimates for the eigenvalues in good agreement with numerics, with the exception of the eigenvalues closest to zero; higher-order asymptotics are used to predict these eigenvalues. Far from the Hopf bifurcation point, the stabilizing role of increasing surface tension is quantified numerically for the thin-film models, while multiple changes in stability occur along families of solutions for some of the long-wave models.