The problem of capillary flow in interior corners that are rounded is re-visited analytically in this work. By the appropriate geometric scaling, and through the introduction of a new parameter that features the roundedness of the corner, the Navier-Stokes equation is reduced to a convenient form for both numerical and analytical solution. The scaling and analysis of the problem is expected to significantly reduce the reliance on numerical data for such problems, and the design process can be both shortened and improved as a result. For capillary flows of perfect wetting fluids in the rounded corner with an advancing tip, a finite interfacial curvature related to the corner roundedness results at the tip. Accordingly, an outer and inner region of the flow is suggested based on the impact of the corner roundedness on the flow. In this study, asymptotic solutions of the geometrical ‘cross-flow’ problem for the outer region are sought under several constraints and are expected to narrowly bracket parallel numerical solutions. A complete understanding of the flow will be obtained only after the cross-flow problem for the inner region is solved. However, for the flow in the outer region a similarity solution is obtained and presented that reveals how roundedness retards the flow.

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