User:Dolphin51/Sandbox2

Magnus effect in potential flow

Potential flow is a mathematical model of the steady flow of a fluid with no viscosity or vorticity present. For potential flow around a circular cylinder, it provides the following results:

Non-spinning cylinder

Streamlines for the potential flow around a circular cylinder in a uniform flow.

The flow pattern is symmetric about a horizontal axis through the centre of the cylinder. At each point above the axis and its corresponding point below the axis, the spacing of streamlines is the same so velocities are also the same at the two points. Bernoulli’s principle shows that, outside the boundary layers, pressures are also the same at corresponding points. There is no lift acting on the cylinder.^[1]

Spinning cylinder

Streamlines for the potential flow around a spinning cylinder. The concentric circular streamlines of a free vortex have been superimposed on the parallel streamlines of a uniform flow.

Streamlines are closer spaced immediately above the cylinder than below, so the air flows faster past the upper surface than past the lower surface. Bernoulli’s principle shows that the pressure adjacent to the upper surface is lower than the pressure adjacent to the lower surface. The Magnus force acts vertically upwards on the cylinder.^[2]

Streamlines immediately above the cylinder are curved with radius little more than the radius of the cylinder. This means there is low pressure close to the upper surface of the cylinder. Streamlines immediately below the cylinder are curved with a larger radius than streamlines above the cylinder. This means there is higher pressure acting on the lower surface than on the upper.^[3]

Air immediately above and below the cylinder is curving downwards, accelerated by the pressure gradient. A downwards force is acting on the air.

Newton's third law predicts that the Magnus force and the downwards force acting on the air are equal in magnitude and opposite in direction.

Citations

^ "From the symmetry of the streamlines it is clear that the velocity of the air is the same at corresponding points above and below the ball, ..."
"From Bernoulli's equation we then deduce that the pressure at such corresponding points are equal and that the air exerts no upward or downward force on the ball by virtue of its motion; the dynamic lift is zero." Resnick and Halliday (1966), PHYSICS, Section 18-5
^ "When dynamic lift on an object occurs it is always associated with an unsymmetrical set of streamlines relatively close together on one side and relatively far apart on the other ... that correspond ... to circulation of fluid around the object."
“[the streamlines] are closer together above [the body] than they are below so that Bernoulli's principle predicts the observed dynamic lift." Resnick and Halliday (1966), PHYSICS, Section 18-5
^ "...if a streamline is curved, there must be a pressure gradient across the streamline..." Babinsky, Holger (November 2003), "How do wings work?", Physics Education, 38 (6): 497, Bibcode:2003PhyEd..38..497B, doi:10.1088/0031-9120/38/6/001, S2CID 1657792

[1] "From the symmetry of the streamlines it is clear that the velocity of the air is the same at corresponding points above and below the ball, ..."
"From Bernoulli's equation we then deduce that the pressure at such corresponding points are equal and that the air exerts no upward or downward force on the ball by virtue of its motion; the dynamic lift is zero." Resnick and Halliday (1966), PHYSICS, Section 18-5

[2] "When dynamic lift on an object occurs it is always associated with an unsymmetrical set of streamlines relatively close together on one side and relatively far apart on the other ... that correspond ... to circulation of fluid around the object."
“[the streamlines] are closer together above [the body] than they are below so that Bernoulli's principle predicts the observed dynamic lift." Resnick and Halliday (1966), PHYSICS, Section 18-5

[3] "...if a streamline is curved, there must be a pressure gradient across the streamline..." Babinsky, Holger (November 2003), "How do wings work?", Physics Education, 38 (6): 497, Bibcode:2003PhyEd..38..497B, doi:10.1088/0031-9120/38/6/001, S2CID 1657792

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