import numpy as np
The volume flow rate is given by
$$ \frac{dV}{dt} = \frac{\pi\rho ghr^4}{8\eta L}, $$where $\eta$ and $\rho$ are the viscosity and density of the fluid, $h$ is the head of pressure across the tube, and $g$ the gravitational field strength. In an experiment, the graph of the flow rate versus height has a slope measured to 7%, the length is known to 0.5%, and the radius to 8%. What is the fractional precision to which the viscosity is known? If more experimental time is available, should this be devoted to (i) collecting more flow-rate data, (ii) measuring the length better, or (iii) measuring the radius of the tube better?
First, we need to rewrite this relation in terms that will enable us to calculate the viscosity $\eta$ from the length $L$, the radius $r$ and the slope of a plot of flow rate $F = dV/dt$ versus $h$. Let’s call that slope $s$ to simplify the notation.
The slople of the flow rate vs. $h$ graph is
$$ s = \frac{\pi\rho gr^4}{8\eta L}, $$and solving for the viscosity gives
$$ \eta = \frac{\pi\rho gr^4}{8sl}. $$The uncertainty in the viscosity is given by
\begin{eqnarray*} \alpha_\eta^2 &=& \left(\frac{\partial \eta}{\partial s}\right)^2\alpha_s^2 + \left(\frac{\partial \eta}{\partial L}\right)^2\alpha_L^2 + \left(\frac{\partial \eta}{\partial r}\right)^2\alpha_r^2 \\ &=& \left(\frac{-\eta}{s}\right)^2\alpha_s^2 + \left(\frac{-\eta}{L}\right)^2\alpha_L^2 + \left(\frac{4\eta}{r}\right)^2\alpha_r^2, \end{eqnarray*}and the fractional uncertainty is given by
$$ \left(\frac{\alpha_\eta}{\eta}\right)^2 = \left(\frac{\alpha_s}{s}\right)^2 + \left(\frac{\alpha_L}{L}\right)^2 + \left(4\frac{\alpha_r}{r}\right)^2 $$Frac_alpha_s = 0.07 # Fractional uncertainty in s
Frac_alpha_L = 0.005 # Fractional uncertainy in L
Frac_alpha_r = 0.08 # Fractional uncertainty in r
Frac_alpha_eta = np.sqrt(Frac_alpha_s**2+Frac_alpha_L**2+(4*Frac_alpha_r)**2)
Frac_alpha_eta
So, the fractional precision to which the viscosity is known is 33%. Note that this is dominated by the error in the radius measurement. If the fractional error were zero for both the slope and the length L measurement, we’d get a fractional uncertainty of 32% with the radius measurement alone because of the factor of 4 that comes in from the fact that we take the 4th power of $r$ in the relation.
So, clearly if we want to improve the measurement, the radius of the tube is the thing that we need to work on -- that dominates the error.
version_information is from J.R. Johansson (jrjohansson at gmail.com); see Introduction to scientific computing with Python for more information and instructions for package installation.
version_information is installed on the linux network at Bucknell
%load_ext version_information
%version_information numpy