I'm currently working on a problem which is really giving me some issues.
The problem concerns the force required to expel water from a syringe. We have a 20ml syringe (which is $2\times10^{-5}$ meters cubed) with a diameter of 1cm, full of water. The needle of the syringe is 40mm in length and has a diameter of 0.2mm. All of the water must be expelled from the syringe in 20s. How much force must be applied to the syringe head to achieve this?
Ordinarily this is fine, but we have to include the pressure loss as a result of the friction in the needle. I'm using the Darcy–Weisbach equation to determine this. I calculated the speed the fluid needs to flow at by dividing the flow rate by the cross-sectional area of the needle. I've used a Moody chart to get $f_D$ as 0.046, and I'm using $\rho = 998.21$. I'm guessing the pressure loss in the needle is therefore $$0.046\times\frac{0.04}{0.002}\times\frac{998.21\times31.8^2}{2} = 4.64MPa$$ Is that correct? In which case, how do I now get to the force from here?
