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What is the relation between pressure and velocity of a fluid in a closed pipe flow? I used the Bernoulli equation in case of a pipe having non viscous fluid. The pipe is say tilted at an angle and has uniform cross-sectional area. Then how to find pressure difference between two points having height difference between them? By continuity equation velocity at all points is the same. Then shouldn't the pressure be same at both points? I am really confused.

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  • $\begingroup$ I used the Bernoulli equation in case of a pipe having non-viscous fluid Does you pipe actually contain a non-viscous fluid or is that an assumption you are making so you can use Bernoulli? In case viscosity is not negligible you should look into Hagen-Poiseuille flow $\endgroup$ – nluigi Nov 11 '15 at 9:41
  • $\begingroup$ The fluid is actually non-viscous $\endgroup$ – hybrid1999 Nov 11 '15 at 12:54
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What is the relation between pressure and velocity of a fluid in a closed pipe flow?

Bernoulli's equation:

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By continuity equation velocity at all points is the same. Then shouldn't the pressure be same at both points?

No the pressure won't be the same at all points in the pipe. Considering Pascal's Law, a change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid. But it holds for static fluids essentially. And equation of continuity stems from the principle of conservation of mass but in this case the mass is not conserved per unit time. Consider 2 different cross-sections of the pipe and fluid flow through it per unit time is not the same since some component of $g$ along with some resultant force is acting on the water column at the 2 different cross-sections and the 2 different cross-sections have a distinct height difference.So equation of continuity is not VALID in this case and hence conclusions cannot be made from this.

Then how to find pressure difference between two points having height difference between them?

Use Bernoulli's principle and you get the relation $$P_2-P_1=(h_2-h_1)\rho g$$

The pipe is say tilted at an angle.

Use the components of pressure in that direction and use the above equation.

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