Confused with flow rate of water

[Koveras00]

By Bernoulli's law, if potential energy is constant, the work done, given by pV, where p is the pressure, V is the change in volume, is equal to the change in kinetic energy, 1/2mv^2.

Does that mean that the flow rate of the liquid is proportionate to the root of the pressure applied??

If it is so, why is it that by Poiseuille's law of flow, E = (pi)r4pt/8Vl,

E is the viscosity of the flowing liquid, t is the time for which the liquid flows, V is the volume of liquid which flows in that time, r is the radius of the pipe, and l is the length of the pipe,

that the rate of flow is proportionate to the pressure applied??

 

[HallsofIvy]

Viscosity is friction. Potential Energy is not conserved.

 

[Koveras00]

hmmm, i knew that, but, now i am designing a experiment and i do not know how to relate pressure to the flow rate of a liquid. What i am asking is that which formulae shld i use?? Since, both of them gave mi different proportionality, i can't possibly refer to both of them...

 

[TALewis]

The reason your Bernoulli result and your Poiseuille result aren't agreeing is that Bernoulli's equation assumes non-viscous flow.

Poiseuille's flow equation comes from some application of the Navier-Stokes equations to the laminar velocity profile in a circular pipe. Poiseuille's law is given by:

$$Q=\frac{\pi R^4 \Delta p}{8 \mu L}$$

Q is the volumetric flow rate, R is the pipe radius, $\Delta p$ is the pressure drop, $\mu$ is the dynamic viscosity, and L is the length of the pipe.

This result is only valid for laminar flow in a circular pipe (it also ignores the roughness of the pipe). Therefore, once you calculate a flow rate, you should also calculate a Reynolds number and see that it is less than 2100. The Reynolds number is given as:

$$Re=\frac{\rho V D}{\mu}$$

Re is the Reynolds number (a dimensionless parameter), $\rho$ is the fluid density, V is the flow velocity, and D is the pipe diameter. V=Q/A, where A is the cross-sectional area of the pipe.

If you find that the Reynolds number is between 2100 and 4000, then the flow is called transitional. If it is greater than 4000, the flow is considered turbulent. You might get away with applying Poiseuille in the transitional case, but in the turbulent region, there are more complicated results from Fluid Mechanics that you should apply. Any introductory text should be more than enough to help you out.

Edited for LaTeX errors, and fixing R^2 to R^4 in Poiseuille's equation.