Question

An incompressible viscous fluid flows through a pipe with a flow rate of 1 mL/s. The pipe has a uniform diameter D0 and a length L0. A pressure difference of P0 between the ends of the pipe is required to maintain the flow rate. What would be the flow rate if the pressure difference was increased to 2P0 and the diameter was increased to 2D0

Answer 1

Answer:

$Q_2 = 32$ mL/s

Explanation:

Given :

The flow is incompressible viscous flow.

The initial flow rate, $Q_1$ = 1 mL/s

Initial diameter, $D_1= D_0$

Initial length, $L_1=L_0$

The initial pressure difference to maintain the flow, $P_1=P_0$

We know for a viscous flow,

$\Delta P = \frac{32 \mu V L}{D^2}$

$\Delta P = \frac{32 \mu Q L}{\frac{\pi}{4}D^4}$

$Q \propto \Delta P \times D^4$

$\frac{Q_1}{Q_2}= \frac{P_1}{P_2} \times \left( \frac{D_1}{D_2} \right)^4$

$\frac{1}{Q_2}= \frac{P_0}{2P_0} \times \left( \frac{D_0}{2D_0} \right)^4$

$\frac{1}{Q_2}= \frac{1}{2} \times \left( \frac{1}{2} \right)^4$

$\frac{1}{Q_2}= \frac{1}{32}$

∴ $Q_2 = 32$ mL/s