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## Homework Statement

A tank contains an incompressible liquid of density ##\rho = 1. 4\cdot10^{3}kg / m^3 ## and viscosity ##\eta=1. 4\cdot10^{-3} Pa\cdot s##. The tank is open at the summit and the level of the liquid is kept constant through a sink. At the base of the tank there is a circular pipe, negligible compared to that of the tank, of radius ##r = 0.5 cm## and length ##2L##, ##L = 98 cm##. At the midpoint of a pipe there is a free air vertical manometer that shows a vertical height of the liquid of ##h_1 = 8 cm##.

Determine:

a) the average velocity ##v## of the liquid in the tube;

b) the height ##h## of the liquid in the tub.

With the same height of the liquid in the tank, determine

c) what the speed ##v_0## of the liquid in the tube would be if ##\eta=0 Pa \cdot s## (Consider the verical manometer long enough to prevent the liquid to exit from it).

d) the power that the pump ##\mathscr{P}## in the following picture must deliver in order to balance the viscosity and to have, with the value of ##\eta## indicated initially, an average speed of ##v_0## of the liquid in the tube.

## Homework Equations

Hagen Pouiseille law $$Q=\frac{\pi r^4 \Delta p}{8\eta L}$$

## The Attempt at a Solution

**a)**Since I do not know the pressure difference between A and B (I do not know ##h##), I used H.P. law between B and C.Then I used the fact that the fluid is incompressible, which means that ##Q= \pi r^2 v_{average}= \mathrm{constant}##, to find the average speed, so

$$Q=\frac{\pi r^4 [p_{atm}-(p_{atm}+\rho g h_1)] L}{8 \eta L}\implies v_{average}=...$$

**b)**Then, using again ##Q= \pi r^2 v_{average}= \mathrm{constant}##, I used H.P. law between A and B to find ##h##

$$Q=\frac{\pi r^4 (\rho g h-\rho g h_1)}{8 \eta L}\implies h=...$$

Is the reasoning correct so far?

**c)**I think that ##v_0=\sqrt{2gh}## but in that case I would have ##h_1=0 m## which is strange, looking at the sentence "Consider the verical manometer long enough to prevent the liquid to exit from it"

**d)**Here is the most confusing one. I considered the idraulic resistance from H.P. law ##R=\frac{8 \eta L}{\pi r^4}## and I tried with ##\Delta p= Q R## and ##\mathscr{P}=Q \Delta p## ##\implies## ##\mathscr{P}=Q^2 R##

I tried to give an answer to all the points but I do not have the results and I am not convinced of what I have tried. Is there any mistake in what I have written?

Thanks so much in advice

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