# Why does (potential) fluid flow bend around a solid surface in the flow?

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.

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?

• 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. – Selene Routley Apr 16 '14 at 14:00
• 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? – Dipole Apr 16 '14 at 14:05
• 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. – Dipole Apr 16 '14 at 14:09