Viscous free-surface flows past cylinders

Free-surface flows of viscous liquid down an inclined plane and past cylinders of various cross-sections are investigated theoretically and experimentally. The cylinders are oriented with their axis perpendicular to the plane and are sufficiently tall that they are not overtopped. A lubrication model is applied to derive the steady governing equation for the flow depth, which is integrated numerically and analyzed asymptotically to calculate how the depth of a steady uniform flow is perturbed as it flows past the cylinder. Flows past cylinders that are narrow relative to the depth of the oncoming flow are only slightly perturbed, but for relatively wide cylinders, there forms a pond of nearly stationary fluid upstream of the cylinder and a dry region in which there is no fluid downstream of the cylinder. The structure of the flow in the regime of a relatively wide cylinder depends in detail upon the curvature of the cylinder at the upstream stagnation point. For flows past cylinders with circular and square cross-sections, the maximum flow depth occurs at the upstream stagnation point. Its numerical value may be predicted analytically on the basis of the asymptotic expressions and exhibits different dependencies upon the variables that characterize the motion. In addition, wedge-shaped obstructions are analyzed for which the flow depth increases along the wedge wall and the maximum flow depth occurs away from the upstream stagnation point. The results from new laboratory experiments of flows past circular cylinders are reported and these corroborate the theory, confirming the occurrence of both pond and dry regions. The investigation has direct relevance to the deflection of lava flows by barriers and buildings and the theory is employed to deduce simplified asymptotic expressions of the force exerted on the cylinders.