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Potential flow obeys Laplace's equation with certain boundary conditions (i.e. no fluid penetrates the solid body in flow, and far away from the body, the flow is uniform with a given velocity and pressure).

So let's condider the potential flow around a cylinder. After the fluid moves to the "top" of the cylinder (above the "+" in the image) it then bends around the surface instead of continuing in a straight line.

Cylinder flow
(source: thermopedia.com)

Why does it bend and not continue in a straight line from the point before the bending?

Since the flow is potential (no viscosity) the only thing causing the fluid to bend around can be the pressure field, but im not sure how this works.

EDIT: Thanks to the first comment I do realize that at the "top" of the cylinder surface there is a pressure gradient normal to the flow (i.e. lower pressure nearer the surface) which would cause flow turning at that point. But how is this pressure gradient established in the first place? It seems like one is arguing that the flow bends because of the pressure gradient, but there exists a pressure gradient only because the flow bends... Am I missing something here?

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  • $\begingroup$ Have you tried plotting the pressure field? The pressure at all the streamlines will be equal (set it arbitrarily to 0 at $x\to-\infty$, when $v = v_0 \hat{x}$) and then use Bernoulli's equation $p + \rho v^2 / 2 = const$ to find the pressure. I plotted $\left|1-(x+i\,y)^{-2}\right|^2$ to give the $v^2$ field - equal to $-p$ and it seems pretty clear how the fluid is bent. $\endgroup$ – WetSavannaAnimal Apr 16 '14 at 14:00
  • $\begingroup$ Thanks for your comment. I'm not sure why it would be apparent. The pressure field has low pressure at the "top" of the cylinder and high pressure at the "back" so wouldn't this mean the flow would NOT bend around towards the back? $\endgroup$ – Dipole Apr 16 '14 at 14:05
  • $\begingroup$ Furthermore after some thought, relying on the pressure field seems like a cicular argument, in that the pressure field is set up by the curvature of the streamlines, but the streamlines are curved because of the pressure field. What am I missing out here? I guess I want to explain the bending phenomena as the cylinder is impulsively started. $\endgroup$ – Dipole Apr 16 '14 at 14:09
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As you correctly point out there is a pressure gradient behind the object that causes the flow to return to its original path.

As to why that pressure gradient exists, image what would happen if the flow continued straight after bending round the object. Behind the cylinder there would be no fluid as it is blocked by the cylinder, essentially there would be a vacuum / much lower pressure (this is essentially a pressure gradient). Obviously, such a situation is unstable and the liquid would move back in, just as in reality.

At high speeds a similar process does actually happen. This is known as cavitation and can be a big problem for things like propeller blades.

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  • $\begingroup$ Great, yes that does make sense. So basically if we view the cylinder starting from rest, the fluid will make its way to the "top", and there is instantaneously a vacuum behind it causing the fluid to fill this void, but in doing so it also slows down and the pressure increases. Does that make sense? $\endgroup$ – Dipole Apr 16 '14 at 15:57

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