Chapter 18, Problem 140P

Poiseuille’s law [Eq. (9-41)] gives the volume flow rate of a viscous fluid through a pipe, (a) Show that Poiseuille’s law can be written in the form ΔP = IR, where I = ΔV/Δt represents the volume flow rate and R is a constant of proportionality called the fluid flow resistance. (b) Find R in terms of the viscosity of the fluid and the length and radius of the pipe. (c) If two or more pipes are connected in series so that the volume flow rate through them is the same, do the resistances of the pipes add as for electrical resistors $\left(R_{\text{eq}}=R_1+R_2+\cdots \right)$? Explain. (d) If two or more pipes are connected in parallel, so the pressure drop across them is the same, do the reciprocals of the resistances add as for electrical resistors $\left(1/R_{\text{eq}}=1/R_{\text{1}}+1/R_{\text{2}}+\cdots \right)$? Explain.

To determine

Show that Poiseuille’s law can be written in the form $\Delta P=IR$.

Answer to Problem 140P

It is shown that the Poiseuille’s law can be written in the form $\Delta P=IR$.

Explanation of Solution

Write the expression for the volume flow rate.

    $I=\frac{\Delta V}{\Delta t}$        (I)

Here, $I$ is the volume flow rate, $\Delta V$ is the volume, $\Delta t$ is the time.

Write the expression for the Poiseuille’s law.

    $\frac{\Delta V}{\Delta t}=\frac{\pi r^4\Delta P}{8\eta L}$        (II)

Here, $\Delta P$ is the driving pressure, $\eta$ is the viscosity of the liquid, $L$ is the length of the tube, $r$ is the radius of the pipe.

Equate equation (I) and (II) to solve for $\Delta P$.

    $\Delta P=I\frac{8\eta L}{\pi r^4}$        (III)

Conclusion:

Therefore, it is showed that the Poiseuille’s law can be written in the form $\Delta P=IR$.

To determine

The constant of proportionality $R$ in terms of the viscosity of the liquid, the length and the radius of the pipe.

Answer to Problem 140P

The constant of proportionality $R$ in terms of the viscosity of the liquid, the length and the radius of the pipe is $\underset{\_}{\frac{8\eta L}{\pi r^4}}$.

Explanation of Solution

Write the expression for the driving pressure.

    $\Delta P=IR$        (IV)

Equate equation (III) and (IV) to solve for $R$.

    $R=\frac{8\eta L}{\pi r^4}$        (V)

Conclusion:

Therefore, the constant of proportionality $R$ in terms of the viscosity of the liquid, the length and the radius of the pipe is $\underset{\_}{\frac{8\eta L}{\pi r^4}}$.

To determine

Whether the resistance of the pipes add as for electrical resistors for pipes connected in series.

Answer to Problem 140P

Yes, the resistance of the pipes add as for electrical resistors.

Explanation of Solution

If two or more pipes are connected in series, the volume flow rate remains same.

    $I=\frac{\Delta V}{\Delta t}$        (VI)

Here, $\frac{\Delta V}{\Delta t}$ remains same.

Write the expression for the total driving pressure.

    $\Delta P_{\text{Tot}}=\Delta P_1+\Delta P_2+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$        (VII)

Here, $\Delta P_{\text{Tot}}$ is the total driving pressure.

Use equation (VI) and (IV) in (VII) to solve for the $\Delta P_{\text{Tot}}$.

    $\begin{array}{c}\Delta P_{\text{Tot}}=I{R}_1+I{R}_2+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \\ =I\left(R_1+R_2+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \right)\end{array}$        (VIII)

Write the expression electrical resistance for the series combination of resistors.

    $R_{\text{eq}}=R_1+R_2+\cdot \cdot \cdot \cdot \cdot \cdot$        (IX)

Use equation (IX) in (VIII) to solve for $\Delta P_{\text{Tot}}$.

    $\Delta P_{\text{Tot}}=I{R}_{\text{eq}}$        (X)

Conclusion:

Therefore, the resistance of the pipes add as for electrical resistors.

To determine

Whether the resistance of the pipes add as for electrical resistors for pipes connected in parallel.

Answer to Problem 140P

Yes, the resistance of the pipes add as for electrical resistors for pipes connected in parallel.

Explanation of Solution

If two or more pipes are connected as parallel, the driving pressure drops across all the pipes are the same.

Use equation (IV) to solve for $I_{\text{Tot}}$.

    $\begin{array}{c}{I}_{\text{Tot}}=\frac{\Delta P}{R_1}+\frac{\Delta P}{R_2}+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \\ =\Delta P\left(\frac{1}{R_1}+\frac{1}{R_2}+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \right)\end{array}$        (XI)

Here, $I_{\text{Tot}}$ is the total volume flow rate.

Write the expression for the electrical resistance for parallel combination.

    $\frac{1}{R_{\text{eq}}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$        (XII)

Use equation (XII) in (XI) to solve for $I_{\text{Tot}}$.

    $I_{\text{Tot}}=\frac{\Delta P}{R_{\text{eq}}}$        (XIII)

Conclusion:

Therefore, the resistance of the pipes add as for electrical resistors for pipes connected in parallel.