The Equation of Continuity — Question 5

Question 5

Radial flow spreads over a growing circumference

An incompressible liquid flows steadily outward between two parallel horizontal disks separated by uniform gap $b=2.0\,\mathrm{mm}$. Beyond the central inlet the flow is axisymmetric. The volume rate is $Q=6.0\times10^{-5}\,\mathrm{m^3/s}$. Use the gap-averaged radial speed.

1. Find the mean radial speed at $r=0.020\,\mathrm{m}$ and $r=0.060\,\mathrm{m}$.

2. If the gap instead varies as $b(r)=b_0r_0/r$, what is the radial dependence of the mean speed? Explain why the singular limit near $r=0$ is excluded.

Question 5 - Solution

Radial flow spreads over a growing circumference

(a) Derivation and calculation

The radial flux crosses a cylindrical surface:$$A_r=2\pi rb,\quad Q=A_r\bar v_r\Rightarrow\boxed{\bar v_r=\frac Q{2\pi rb}}.$$$$\boxed{\bar v_r(.0200)=\frac{6.0\times10^{-5}}{2\pi(.0200)(.00200)}=.238732\,\mathrm{m/s}}.$$$$\boxed{\bar v_r(.0600)=.0795775\,\mathrm{m/s}}.$$

(b) Result and interpretation

If $b(r)=b_0r_0/r$,$$A_r=2\pi r(b_0r_0/r)=2\pi b_0r_0,\quad\boxed{\bar v_r=\frac Q{2\pi b_0r_0}=\text{constant}}.$$The finite inlet region cannot be modeled by extending radial-only motion to the axis. Fixed gap would imply $\bar v_r\to\infty$; the alternate gap law itself diverges there.

Physical check

Tripling radius at fixed gap triples flow area and divides mean speed by three.