If we apply Bernoulli's equation to a horizontal pipe of uniform diameter having the same velocity throughout the length, the equation concludes that preassure is the same along the direction of flow throughout the pipe. If the pressure difference is zero then what makes the fluid flow?

 If we apply Bernoulli's equation to a horizontal pipe of uniform diameter having the same velocity throughout the length, the equation concludes that preassure is the same along the direction of flow throughout the pipe. If the pressure difference is zero then what makes the fluid flow?

Solution

Let us have a situation like that mentioned, point 1 and point 2 along a pipe of same diameter and frictionless conditions. According to Bernoulli's, the case will result in same velocity and static pressure between the two points. So where's the pressure difference that cause the velocity of the flow?

To answer this question, we have to broaden our view and look at some point downstream of point 2, call it point 3. The condition at 3 should be different from 1 and 2 in order to drive the flow. Usually this could happen due to different elevations (i.e. Point 3 is lower than 1 and 2) or at different static pressure (i.e. 1 and 2 are pressurized while 3 is atmospheric).
Similar case is the electric circuit, when drawing a closed circuit of flowing current, point 1 and 2 will be at the same line while 3 will be after them and at least an electric resistance is in between.
This is the only case to have a flow, otherwise there will be no velocity. The proposed case in the question is just taking a small section for comparison.
I hope it helps. I'll try to add some figures to illustrate better.