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I was wondering if we have a very thin glass tube placed in a tub of liquid and the portion of the tube outside the liquid is lesser than the height to which the liquid can rise because of capillarity, what will happen? Will the liquid overflow?

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No, it won't overflow. That should be obvious since doing so would create a constant flow, constantly using energy, but without any energy input. Put another way, that would be a perpetual motion machine, one you could actually extract free power from.

The same force that pulls the water along the inside of the capillary tube also holds it there when it reaches the end. This force doesn't just pull upward, it pulls the water along the glass. At a certain height, the weight of the water column ballances this pull. In that case the pull is upward since there is water below but not above.

In your case, there is nothing to pull the water column higher when it reaches the end of the tube, so it just stays there. The pull only exists at the boundary between water and not-water. The bulk of the water in the tube isn't being pulled any particular way.

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It is explained very nicely by Olin Lathrop.

The surface tension is an inter-facial force. The surface tension force pulls the liquid tangential to the wall of the capillary. When a capillary is dipped in water, it starts rising up due to pulling force from the solid-vapor interface. If the capillary has insufficient length, as the water rises it accelerates till the end of the capillary. When it reaches the edge of the capillary, water would gain some kinetic energy and due to inertia it slightly overshoots the edge of the tube. Then water surface bulges out due to which the tension at the solid-vapor interface turns downward which tries to pull the water column downward.

The liquid column will make some oscillations and finally after sometime the oscillations dampen out due to viscous effects. At steady state, the water column will fill the entire capillary and the radius of curvature of the minuscus will be just about adjusted such that the vertical component of surface tension force would balance the weight of water column.

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In the formula for capillary rise only θ is a variable. So when the capillary tube of insufficient length is kept then the θ will change accordingly so that capillary length h comes out to be height of the capillary tube. Limiting case is when θ = 90 degrees when height of capillary tube is 0. This is meaningless so this is not possible. Concluding thing is the angle increases a bit to adjust accordingly.. Hope it helps

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The equation for capillary rise can be represented as

$$h=\frac{2T}{rρg}\cosθ$$

if the height of the capillary tube is less than the height calculated by this equation,the water comes up to the height of the capillary tube and stops there.then the value of $θ$ (angle of contact) keeps changing until the equation balances out, i.e., the meniscus of the liquid becomes more convex until it attains equilibrium.

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As if you will see that a capillary tube kept in a beaker filled with water,so the water level rises but if the length of the capillary tube is insufficient than the angle of cos theta will be of 90 it means that its just impossible,and the surface tension force will be just stopped.

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    $\begingroup$ I don't really think this adresses the question well at all. $\endgroup$ – Brandon Enright Mar 21 '15 at 8:19
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The water will never overflow . if the tube is of insufficient length , the radius of curvature of the liquid meniscus goes on increasing , making it more and more float till water is in equilibrium.

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As soon as fluid reaches maximum height it's angle of contact changes this leads to change in shape of meniscus it may or may not overflow depends upon fluid

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    $\begingroup$ Please, read the answer of @olin-lathrop and if you state that it can overflow, put arguments, why he is wrong with the energy conservation concept $\endgroup$ – jaromrax Feb 16 '17 at 7:52

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