Meniscus in U-shaped capillary?

Question

What does the meniscus look like for a U-shaped capillary?

From similar questions, I learned that the total height (labeled as h) reached by the water in the un-bent capillary is less than the total distance traveled by water in the bent capillary, because the force of gravity is "pulling" water further downward.

Assuming that this is true (correct me if it's not), will the water continue traveling through the bend until it reaches the opening of the tube? Or could it stop somewhere before?

Now, assuming water reaches the capillary opening, would the meniscus shape be like A, B, or C? I read that if a capillary tube height is insufficient, the water will stop at the top, and the meniscus will slowly become more and more convex, where the limit is 0 degrees (so, a flat meniscus). In a U shaped capillary, since gravity will attract the water downward, I thought the meniscus will be shaped like C.

I've read many similar questions as this, but can't find the answer, so I thought I'd make a StackExchange account to ask haha. Thanks in advance!

Answer 1

See Wikipedia's "Surface curvature and pressure" and the explanation and calculations of "Jurin's law with glass and water" and "hysteresis". Exact results require laboratory conditions with clean atomically smooth glass and pure water, surface roughness and impurities spoil the calculations.

Height of a meniscus (your first (unlabeled) tube):

The water will only rise to $h$ in a straight capillary tube.

"The height $h$ of a liquid column is given by Jurin's Law

$$h={{2 \gamma \cos{\theta}}\over{\rho g r}},$$

where $\gamma$ is the liquid-air surface tension (force/unit length), $\theta$ is the contact angle, $\rho$ is the density of liquid (mass/volume), $g$ is the local acceleration due to gravity (length/square of time[28]), and $r$ is the radius of tube. Thus the thinner the space in which the water can travel, the further up it goes.

For a water-filled glass tube in air at standard laboratory conditions, $\gamma$ = 0.0728 N/m at 20 °C, $\rho$ = 1000 kg/m3, and $g$ = 9.81 m/s2. For these values, the height of the water column is

$$ h \approx {{1.48 \times 10^{-5}}\over r} \ \mbox{m}.$$

Thus for a 2 m (6.6 ft) radius glass tube in lab conditions given above, the water would rise an unnoticeable 0.007 mm (0.00028 in). However, for a 2 cm (0.79 in) radius tube, the water would rise 0.7 mm (0.028 in), and for a 0.2 mm (0.0079 in) radius tube, the water would rise 70 mm (2.8 in).

Water height in a capillary plotted against capillary diameter

Thus a 0.2mm internal diameter tube, over 3 inches long, only has 2.8 inches if water in it.

"hysteresis"

"... liquid advances over previously dry surface but recedes from previously wet surface ...".

The first tube is drawn correctly, the shape is C for the labeled tubes (advancing from a previously wet surface), and the distance is $h$ in all cases. If the tube were a lot longer, filled with water, and the end was lower than the reservoir then you get a siphon and water extending past $h$, otherwise approximately $h$ is where the water stops (depending on the aforementioned requirements).