Answer to the Question 10/97
The question was:A small vertical tube (radius smaller than capillary length) is put in contact with the surface of a (perfectly) wetting liquid of very low viscosity. Describe the motion of the liquid in the tube. What is the ratio between the maximal height and the final height reached by the liquid surface in the tube?
3/98 We received several answers. The answers included correct qualitative discussions, but none of them actually solved the problem quantitatively. It doesn't seem, that we are going to receive a complete and correct answer. Therefore we publish the solution.
The problem was considered theoretically and experimentally by D. Quere. Detailed explanation as well as nice experimental results can be found in Europhys.Lett. 39, 533 (1997). When the tube is inserted into liquid, the liquid will start raising due to surface tension (capillary) forces. Eventually (after a long time) the height of the liquid will be such that the capillary forces will be balanced by the weight of the liquid column. However, before that final height is reached the liquid will oscillate (due to inertia) around its final height. The maximal height reached by the liquid column (due to the oscillations) is 1.5 times the final height of the column.
The following are the main points of the solution:
The final height H of the column of liquid is determined by the balance between capillary force F=2{pi}r{gamma} and the weight Mg={pi}r2 H {rho}g, where {gamma} is the surface tension, {rho} is the density, r is the radius of the tube. Thus the final height is H=2{gamma}/({rho}gr).
If viscosity can be neglected then the motion of the liquid column can be described by the equation
d(Mv)/dt = F - Mg
where v=dh/dt, and h is the height of the liquid at time t, while M is the mass of the column at time t. The solution of this equation is a parabola, with its maximum h=1.5 H. (This solution means that after reaching the maximal point the liquid interface swings back to h=0. This will not happen when viscosity is taken into account. The height will simply perform damped oscillations around H.)
Back to "front page"