$8\mathrm{.}14.$ Washburn's law

Liquid uptake by porous media is typically driven by surface tension. Suppose that a long, horizontal pore of radius $a$ is initially filled with air, and that at $t=0$ one end is immersed in a liquid. If the contact angle $$ is there will be a liquid column of growing length $L(t),$ as shown in Fig. P$8\mathrm{.}14.$ Assume that the far end of the pore is vented, keeping the air pressure constant at $P_0,$ and that the liquid outside the pore is also at $P_0.$

Lic

$1$

$($a$)$ Relate $d\frac{L}{d}t$ to the volume flow rate $Q(t)$ of the liquid.

$($b$)$ Assuming that the flow in the pore is pseudosteady and fully developed, show that

$L(t)=[\frac{acos}{2}t{]}^{\frac{1}{2}}$

which is called Washburn's law.

$($c$)$ Estimate how large $t$ must be for the pseudosteady approximation to become valid. Washburn's law will be accurate if the observation time greatly exceeds that.