A Simple Pressure Velocity Paradox

I have this conceptual problem which I asked many people regarding pressure velocity problem and many got confused, I later turned this problem into pressure velocity paradox.Please help me with this problem:

Suppose you take a pipe of uniform cross-section with which has got lots of bends( the setup in on a horizontal table). Now, water flows in and due to lots of bends there would be head losses and so, the pressure at the inlet would not be same as at the exit.
Now the problem is, if we apply continuity equation A1V1=A2V2 and the inlet and exit then the velocity must be same( since the area is same throughout).
But according to Bernoulli, change in pressure energy is equal to change in kinetic energy, and since the pressure has changed so should the velocity at different points.

These two points are contradicting. Could you please help?

Bernoulli's Equation is valid for fluid flow which is :

• Incompressible
• Inviscid or Non-Viscous
• Steady
• Irrotational
• It is assumed that No energy losses are observed in the fuid flow.

However ,in case of bends , velocity of fluid suddenly changes at bends , leading to separation of flow and thus, formation of eddies takes place.
This effect causes dissipation of energy associated with the flow of fluid.
In practical situations , flow is not frictionless , nor inviscid , nor steady and irrotational.
Even if we maintain a steady flow somehow ,and that too being kept irrotational..even then friction can't be eliminated.
Another point... Continuity equation is basically conservation of mass in terms of flow rates... and for its validity , it is assumed that no accumulation of mass occurs inside the control volume (or a certain volume of pipe under consideration).All the fluid which enters a given part of the pipe..essentially leaves that part after some time...Only under this assumption is the equation of continuity in the very simplified form valid. While in case of formation of eddies... fluid may get accumulated in the volume... and eddies lead to dissipation of energy.
Hence , the doubt you have is only due to intermixing of concepts and ignoring the very important assumptions that must be taken before considering these equations.
The differential form of the continuity equation is:


where

• ρ is fluid density,
• t is time,
• u is the flow velocity vector field.

Thank You.
Any suggestions and improvements are welcome.

Prashant Kumar Sharma

I have this conceptual problem which I asked many people regarding pressure velocity problem and many got confused, I later turned this problem into pressure velocity paradox.Please help me with this problem:

Suppose you take a pipe of uniform cross-section with which has got lots of bends( the setup in on a horizontal table). Now, water flows in and due to lots of bends there would be head losses and so, the pressure at the inlet would not be same as at the exit.
Now the problem is, if we apply continuity equation A1V1=A2V2 and the inlet and exit then the velocity must be same( since the area is same throughout).
But according to Bernoulli, change in pressure energy is equal to change in kinetic energy, and since the pressure has changed so should the velocity at different points.

These two points are contradicting. Could you please help?

You have not taken the friction into consideration. There is energy loss at the walls of the pipe. The inlet and the outlet velocity would not be the same, unless you are considering frictionless, incompressible flow. Whenever you encounter a negation to the conservation of energy, just re-think for a second - everything clears on the second thought.

Prashant Kumar Sharma

I have this conceptual problem which I asked many people regarding pressure velocity problem and many got confused, I later turned this problem into pressure velocity paradox.Please help me with this problem:

Suppose you take a pipe of uniform cross-section with which has got lots of bends( the setup in on a horizontal table). Now, water flows in and due to lots of bends there would be head losses and so, the pressure at the inlet would not be same as at the exit.
Now the problem is, if we apply continuity equation A1V1=A2V2 and the inlet and exit then the velocity must be same( since the area is same throughout).
But according to Bernoulli, change in pressure energy is equal to change in kinetic energy, and since the pressure has changed so should the velocity at different points.

These two points are contradicting. Could you please help?

If the fluid flow is steady and the liquid is incompressible, then ideally,
A1V1=A2V2, it will hold true.
Now as far as Bernoulli's equation, change of pressure energy will not be there if bend is smooth ( and cross sectional area doesnot changes).
In case of steep bends, pressure will change but cross sectional area at that point will also change, thus A2 will also change, compensating for change in velocity.

Think of it as in this way, if pipe is full of water, and u start pushing 200 ml of water each second, 200 ml of water will come out each second (as water cannot disappear in between).
I am no expert in this field, but I go with common experience that I have of physical phenomenon😀

Prashant Kumar Sharma

But according to Bernoulli, change in pressure energy is equal to change in kinetic energy, and since the pressure has changed so should the velocity at different points.

In logic, there is a statement 'A implies B'. A is true. Then B has to be true. B is true. A need not be true.
I use umbrella if it rains. It rains. So I take an umbrella. On the other hand I take an umbrella. It is not raining. I have taken the umbrella because the sun is fierce.
Likewise, decrease in pressure energy should lead to change in kinetic energy. However, it assumes that no other energy change is involved. The issue is what Shashank raised. Friction loss, which will appear as heat (Conservation law). This leads to pressure loss without affecting velocity.