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Let's say we have a pipe with water flowing through it and obstacle is inserted into the pipe and is rigidly fixed to the inner wall so that it does not get washed away.

What forces are exerted by the fluid on the object? The only one I can think of is drag (which is this). I was also thinking that the restriction in the pipe caused by the obstacle would induce a pressure drop across that obstacle which would also generate a force on top of the drag force. Is this true?

What other fluid forces would this obstacle experience?

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Both cases are true. Basically the force the object will feel is due to the net differences in pressure and viscous forces acting on its upstream and downstream faces. How the pressure field and viscous stress are distributed around the object is as function of a number of factors: the shape of the object, its size relative to the pipe, the wall roughness, flow speed, fluid viscosity etc. These processes are beautifully explained in these videos, particularly those dealing with drag.

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