Problem 1.

A cylindrical vessel of radius 0.1 meter is filled with water to a height of 0.5 meter. It has a capillary tube 0.15 meter long and 0.0002 meter radius fixed horizontally at its bottom. Find the time in which the water level will fall to a height of 0.2 meter.

Solution:

In the present problem we need to use the Bernoulli's law and the continuity equation (liquid is incompressible).

We introduce points B and C as shown in the figure below. The velocities at these two points are

and

From the continuity equation we know that the product of the velocity and the cross section area is constant. We apply this equation for points C and B:

The radius of vessel at point C is , then the cross section area at point C is

The radius of tube at point B is , then the cross section area at point B is

Then

and

...........................................................(1)

Therefore the velocity of liquid at point C is much smaller than the velocity of the liquid at point B.

Now we can write down the Bernoulli's law for points B and C:

Since then we can disregard the second term in the left hand side:

Then

Using the relation (1) we obtain

At the same time the velocity at point C can be expressed as the change of the height of the liquid in the vessel:

Then

Now we need to take the integral (from initial to the final moment) of both sides of the equation