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From what I've read, the cause for the force acting as the result of the Magnus effect is the formation of a bended wake behind the moving sphere.

Take a look at that picture:

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From that picture, it appears to me that since the pressure gradient is directed into the wake region, the vertical force acting on the ball should be directed downwards, and not upwards how it is indicated in the picture.

Is this picture wrong and the wake should be reflected about the x-axis?

From my understanding, the effect responsible for detachment of the boundary layer is the development of turbulence, and turbulence should occur faster on the side where the relative velocity of the air is bigger - on top of a backspinning ball, since in that case the ball will be giving its momentum to the surronding air faster, through friction, therefore reaching the speed where turbulence occurs at the point of the x-axis located more to the right (and not as indicated on the picture, where it's more to the left (on the top side of the ball)).

Edit: Sorry, the last paragraph is wrong. I incorrectly determined where the relative velocity is bigger. My question therefore changes to the following - If this picture of the wake is correct, then why is the force directed upwards, and not downwards?

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The direction of force is determined by the Bernoulli's theorem. If I have to explain briefly Bernoulli Theorem suggest that in a fluid, if one region of the fluid have more kinetic energy than the other then the former have less pressure than the latter (if the region is on the same level i.e.no gravitation energy and atmospheric pressure is constant). So you can see that in the figure the upper region of the air will be faster due to the frictional force exerting by the ball on the air as compared to the lower region. This means that upper region has less pressure as compared to lower one. We know that fluid tend to move from higher pressure region to lower one and that's why fluid exert upward pressure on the ball.

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  • $\begingroup$ But the Bernoulli principle is not applicable, since these are viscous forces responsible for the change in velocity. $\endgroup$ – Nick The Dick Jun 10 '20 at 14:22
  • $\begingroup$ You can see this by remembering that Bernoulli's principle is basically the conservation of energy along a streamline, and when viscous forces are present, they are responsible for a change in energy. Correct me if im wrong $\endgroup$ – Nick The Dick Jun 10 '20 at 14:29
  • $\begingroup$ That's an interesting point as I remember the Bernoulli's theorem was derived by taking the assumption that the fluid must be incompressible but there was no assumption of non viscosity. But yes you have a valid point. $\endgroup$ – sslucifer Jun 10 '20 at 17:17
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The figure shows that the fluid is being directed downward. Some force on the fluid has to be doing this by giving the fluid downward momentum. By Newton's third law the fluid must be giving the thing pushing it a push in the opposite -- i.e upward -- direction. Hence the fluid pushes the ball upward.

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  • $\begingroup$ What are the points of application and direction of these forces? Are they shear friction forces at the points of flow separation? So, for the ball to fly up, should their sum's vertical component exceed the vertical downward component of the force caused by the pressure gradient? Why can't the sum of these shear forces be smaller than force from pressure gradient, apart from the argument of the Newton's third law? $\endgroup$ – Nick The Dick Jun 11 '20 at 3:33
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The answers by mike stone and sslucifer give two parts of the full explanation, respectively Newton's laws of motion and Bernoulli's principle. The third part of the explanation is the circulation theory of lift.

For a conventional airfoil travelling forwards, the pressure differences create a circulation component of flow; backwards over the top, down at the rear, forward underneath and up round the leading edge. This circulation creates a positive feedback which increases the lift several times compared to a flat plate. The faster you fly, the greater the effect.

A spinning rotor creates its own circulation through friction across the boundary layer. It only needs to travel forwards relatively slowly for the circulation to create and magnify the Bernoulli effect, deflect a significant amount of air downwards and, in Newtonian reaction to that, create useful lift. This is the Magnus effect.

However with respect to the tennis ball illustrated, it was the then Lord Rayleigh who first experimented with spinning tennis balls and established that their curved flight was due to the Magnus effect, back around 1850.

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The picture is correct because backspin tennis balls go up.

The reason is simple and is related to air drag. A ball at about 100 km/h faces a huge air resistance to slow down its velocity. And the force increases with velocity.

If it is spinning, the relative velocity is not symmetric, and the ball goes to the direction of smaller relative velocity => smaller air drag.

A top spin ball goes down and a backspin one goes up.

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