Abstract
A new theoretical framework is applied to the steady fluid flow past a solid smooth sphere. Bernoulli’s law along a streamline is combined with the cross-stream force balance: centrifugal force on the curved flow equals a pressure gradient. When compared with the standard potential theory for flow past a sphere in a text book, the prospect of a major discrepancy is found. Whereas the decay rate of the velocity perturbation away from the sphere goes as the inverse cube of the distance in the text book, the decay rate computed here is in all likelihood very different, and it depends on an unknown constant function, the radius of curvature of the streamlines versus distance from the sphere. When that function is supplied either from another theory or from detailed observations (probably streak photographs), then the new approach can be solved completely. In any case, accurate measurements of flow rates at different positions with respect to the solid are badly needed.
Highlights
Leonardo da Vinci wrote down in a notebook his observation of the flow past a rock in a stream: the flow is fastest at the sides of a covered rock rather than above it [1]
Potential flow is the name of the theory, which has been applied to flow past a circular cylinder in many text books as well as flow past a sphere in one text that I have [2]
There is another theoretical framework available, which has some advantages over the classical method, and it has already been applied to the steady flow past a circular cylinder [3]
Summary
Leonardo da Vinci wrote down in a notebook his observation of the flow past a rock in a stream: the flow is fastest at the sides of a covered rock rather than above it [1]. He did not write about the rate at which the fastest flow decayed outward from the rock toward the normal mean stream velocity. Potential flow is the name of the theory, which has been applied to flow past a circular cylinder in many text books as well as flow past a sphere in one text that I have [2].
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