Water flows from a tank and through a pipe with a nonuniform

Water flows from a tank and through a pipe with a non-uniform diameter. The cross sectional area of the pipe is given by A= A_0[1 - x/2l(1 - x/1)], where A_0 is the area at the beginning (x = 0) and end (x = l) of the pipe. The volumetric flow rate () of an inviscid fluid is related to the cross sectional area of the flow channel (A) by the relationship = vA. Plot graphs of the gage pressure within the pipe as a function of x\', where x\' = x/l is the fraction of distance along the pipe. Overlay plots on the same graph for water depths of h = 1, 4, 10, and 25 m. What height is needed to induce cavitation within the pipe if the water is at 25 degree C?

Solution

Bernoulli’s equation is an approximation based on setting the viscosity equal to zero. For any real fluid, viscosity is not zero. At the walls, there will be viscous drag (i.e. friction). A pressure difference is needed along the pipe to overcome this friction. Bernoulli’s equation does not include that contribution. It’s up to us to understand the limitations of our approximations and account for them appropriately.

But once a fluid is in motion, it actually does not need pressure to keep it moving. Just like any other mass, in the absence of external forces, it keeps going at constant speed. The friction is an external force, but if that force were zero, then the fluid would just keep moving without needing anything to keep pushing it along. Bernoulli’s equation gets that part right. The pressure difference would indeed be zero in the circumstances that you asked about.

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.