Question

The formula for the resistance of a fluid is:
A) $R = \dfrac{{\pi {r^4}}}{{8\eta l}}$
B) $R = \dfrac{{8\eta l}}{{\pi {r^2}}}$
C) $R = \dfrac{{8\eta l}}{{\pi {r^3}}}$
D) $R = \dfrac{{8\eta l}}{{\pi {r^4}}}$

Answer 1

Hint: The resistance of fluid resists the flow of the fluid. This ideally originates due to the friction between the fluid layers. Due to this resistance we have a pressure drop in the fluid flow. You need to address the change of fluid flow with respect to the pressure drop to calculate the resistance of the fluid.

Formula Used: The Poiseuille's law for the volume flow rate is
$Q = \dfrac{{{P_1} - {P_2}}}{l}\dfrac{{\pi {r^4}}}{{8\eta }}$
where, $Q$ is the volume flow rate, ${P_1}$ and ${P_2}$ are the initial and final pressure respectively, $r$ is the width of the tube for flow, $l$ is the length of the tube for flow and $\eta $ is the coefficient of viscosity.

Complete step by step answer:
Step 1:
The flow of fluid causes due to the pressure difference. So, we can take an analogy with the electronic resistance as well.
For electric current flow, the potential difference is the cause and resistance try to prevent the flow of electrons.
We have the
$I = \dfrac{V}{R}$

where, $I$ is the current, $V$ is the potential difference and $R$ is the resistance.
The eq (1) can be represented in this form with taking the analogy with the potential difference $V$ to the pressure difference $\left( {{P_1} - {P_2}} \right)$ and with the current $I$ to the volume flow rate of the fluid $Q$.
$Q = \dfrac{{({P_1} - {P_2})}}{{\dfrac{{8\eta l}}{{\pi {r^4}}}}}$
Step 2:
The resistance $R$ can also be analogously compared to the resistance of fluid.
$Q = \dfrac{{\left( {{P_1} - {P_2}} \right)}}{{\dfrac{{8\eta l}}{{\pi {r^4}}}}} = \dfrac{{\left( {{P_1} - {P_2}} \right)}}{{{R_{fluid}}}}$
$\therefore {R_{fluid}} = \dfrac{{8\eta l}}{{\pi {r^4}}}$
Final Answer:
The formula for the resistance of the fluid is (D) $R = \dfrac{{8\eta l}}{{\pi {r^4}}}$.

Note: The analogy should be taken carefully by comparing the cause of the flow. The pressure difference causes the fluid flow similar to the fact that the potential difference causes the electric flow. That is the main key point. You can guess, that the more path the fluid travels the more will be its resistance and the viscosity is the main cause of the friction between the layers; else, the rest of the dependence should follow from the Poiseuille’s law itself. You should be very careful in writing the eq (1), otherwise the dependence with the index of $r$ shall be not correct.