Influence of inertia on the transient axisymmetric free-surface flow inside thin cavities of arbitrary shape

Abstract

The influence of inertia is examined for transient axisymmetric free surface flow inside a thin cavity of arbitrary shape. The flow field is obtained by solving the lubrication equations, which are averaged over the cavity gap by expanding the velocity in terms of Chandrasekhar functions and using the Galerkin projection method. The formulation accounts for the transverse flow, as well as nonlinearities stemming from inertia and front location. Both flows under an imposed flow rate, and an imposed pressure at the cavity entrance are examined. The influence of inertia, aspect ratio, gravity, and cavity geometry on the evolution of the front, flow rate, and pressure is assessed particularly in the early stage of flow. Comparison with existing results shows full qualitative agreement for cavities of various geometries and flow conditions. Inertia is found to have a significant influence on early transient behavior, leading to the development of a flow of the “boundary-layer” type upon inception. The effect of inertia is further explored by developing a multiple-scale analysis to obtain an approximate solution at small Reynolds number, Re. Comparison with the exact (numerical) solution indicates a wide range of validity for the multiple-scale approach, even in the moderately small Re range.