# Fluid mechanics - Question Poiseuille exercise

A little help/direction would be very helpful:

The needle of a syringe has a diameter $$d=0.6mm$$ and its length is $$l=2cm$$. The water flow forced in the needle is $$Q=10^{-7} m^3 s^{-1}$$

Assuming laminar flow, calculate:

1. The average speed of water
2. What is the pressure drop necessary to have such a flow I think I'm OK with the 1: $$Q=S×V$$ so:

$$V=Q/S=10^{-7}/( 6×10^{-3} × 6 ×10^{-3}) = 8.84×10^{-4} m s^{-1}$$

2/ For this one I assume I have to use Poiseuille equation:

$$\Delta P = \frac{8\mu LQ}{\pi R^4}$$

but I don't know how to do as I don't have the dynamic viscosity of the water ($$\mu$$). I don't know if I suppose to know this value (as it depend on temperature I presume?) or if I have to / can express the pressure drop without knowing this value.

Can someone help me/push me a little in the right direction?

• I understand you're not a native English speaker but understand that the word I is ALWAYS capitalised, regardless of its position in a sentence.
– Gert
Dec 4, 2020 at 16:24
• Don't forget the $\pi$ for the cross-sectional area $S$, since capillaries are usually assumed to be circular. Also, you need to halve the diameter to get the radius. Dec 4, 2020 at 21:18

Less than 5 seconds of googling and I found the dynamic viscosity of water:

$$\mu=8.90\times 10^{-4}\text{ }\mathrm{Pa.s}$$

at $$25^{\circ}\mathrm{C}$$ of temperature.

A table of dynamic viscosity dependence on temperature is also provided in that link.

or if I have to / can express the pressure drop without knowing this value.

No, of course you can't calculate $$\Delta P$$ without knowing $$\mu$$.

• People doing fluid dynamics courses should know the viscosity of water at room temperature from the top of their head. At 20 degrees it is 0.001 Pa s. Dec 4, 2020 at 16:31
• @Bernhard That's silly. I've done a lot of FD in my time and still don't know that value 'of the top of my head'.
– Gert
Dec 4, 2020 at 16:38
• I've known it in my head since 1962 when I had my first course in fluid dynamics: 1 centipoise = 0.01 Poise Dec 4, 2020 at 17:45
• Sure Chet but 'Daphoque' isn't a habitual user of FD, methinks.
– Gert
Dec 4, 2020 at 17:47
• thanks, finally get an answer from our professor indicating we have to use 1x10^-3 Dec 6, 2020 at 13:07

I don't confirm your velocity calculation. The cross sectional area of the capillary is $$\frac{\pi D^2}{4}=2.83\times 10^{-7}\ m^2$$So the velocity is 0.354 m/s.

• haha yeah thanks stupid mistake ! Dec 6, 2020 at 13:05