Spreading of steadily-discharged oil on water in the viscous-gravity stage

•Spreading of steadily discharged oil is investigated in the viscous-gravity regime.•A similarity solution is used along with an integral boundary layer approach.•Analytical solutions at the leading edge and the origin are used for integration.•Matching of the two solutions provides the unidirectional spreading prefactor.•The prefactor from a simplified approach is found consistent with this work.

The second stage of oil spreading on water, in which gravity forces promote spreading against the resisting viscous effects, is investigated using a similarity solution in combination with the integral boundary layer approach in the case where oil is discharged at a constant rate. Analytical solutions near the leading edge and the origin are developed, and used to start integration in both unidirectional and axisymmetric cases of spreading to provide a solution for the oil thickness, the oil velocity, and the boundary-layer thickness profiles. The existence of a singularity during integration required “matching” of two solutions originating from the leading edge and the origin. This led to the spreading law prefactor for the unidirectional spreading case. The prefactor reached with a simplified approach in a published theoretical work is found consistent with the present investigation value obtained based on a detailed analysis of the boundary layer in the water phase. The unidirectional spreading results are compared with the constant oil volume case. “Matching” was not possible in the axisymmetric spreading case, which is attributed to the infinite velocity at the origin, as given by the asymptotic solution. The analysis suggests the need for additional theoretical investigations in the axisymmetric spreading case. The order of magnitude analysis shows the need for further experimental work for both spreading geometries to ensure surface tension forces are negligible, which can be achieved by using sufficiently large oil discharge rates.

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