# Capillary action meniscus height in a tube fitted inside another tube?

I was thinking about how would capillary action change in a tube (classic example) and in a tube fitted inside another tube (considering water as the liquid involved).

Height of liquid column: where:

$\gamma$ = liquid-air surface tension

$\theta$ = contact angle

$\rho$ = density of liquid

$g$ = gravity acceleration

$r$ = radius

I tried my best to draw the examples I'm interested in order to help my explanation.

I didn't consider the capillarity inside the smaller tube in both example #2 and #3 because I'd like to assume that $a/2$ in example #1 is close to $c$ in example #2 and #3 (drawings not to scale).

Since from what I understand the column height is given, among other things (most of which can't be changed, like liquid-air surface tension, contact angle, density of liquid and gravity acceleration), by the tube radius, I'd like to know if $c$ in example #2 can be considered as $a/2$ in example #1 to calculate column height using above formula.

Also I'd like to know how having beads of slightly smaller diameter than $c$ between the two tubes (example #3) would affect the column height.

If said beads were less dense than water, could they still improve column height or would they just form a floating mat on top of 1 unit thickness?

What'd be the column height of example #2 and #3 assuming $c$ as 1mm?

I'm quite sure that given the same reached height $h$ in example #2 and #3, $c$ of #2 has to be smaller than $c$ in #3.

If $c\ll R$ the radius of your outer tube, the total curvature is approximately $\cos \theta/c$, so you will get $$h = \frac{\gamma}{\rho g} \frac{\cos \theta}{c}.$$