Transient displacement of Newtonian and viscoplastic liquids by air in complex tubes

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We study the transient displacement of Newtonian and viscoplastic liquids by highly pressurized air in cylindrical tubes of finite length with an expansion followed by a contraction in their cross section. Papanastasiou's formula is employed to regularize the discontinuous Bingham model. For both fluid models considered, the distribution of the remaining film on the inner tube wall is non-uniform and only partly follows the tube geometry: it is thinner in the expanding section of the tube, thicker in the contracting one, and as thin as observed in relevant experiments for straight tube segments, if these are long enough. The effect of changes in the diameter of the narrow introductory tube on the width of the remaining film depends on liquid inertia and yield stress. It is confined near the expansion corner for small Reynolds numbers, but causes extended distortions on the free surface for larger ones due to the development of ‘lip’ or ‘corner’ vortices on the expanding side of the tube. The increased viscosity of viscoplastic materials, especially where velocity gradients are smaller, reduces the extent of flow and interface fluctuations. Unyielded regions appear along the film remaining on the tube wall, the core area of the main and exit tubes and around the concave corners of the tube. These cause the thinning of the remaining material inside the entrance tube, around the centerline of the main section and the flattening of the bubble front. The size of the solid-like areas increases at higher Reynolds numbers. The tip velocity for both viscous and viscoplastic materials increases along the introductory tube until it attains a maximum, then decreases down to a plateau while it moves inside the main tube, and finally increases again as it approaches the contraction corner. Its values decrease as the Bingham number increases.