<P> In a free liquid that is not affected by outside forces such as a gravitational field, internal attractive forces only play a role (e.g. Van der Waals forces, hydrogen bonds). Its free surface will assume the shape with the least surface area for its volume: a perfect sphere . Such behaviour can be expressed in terms of surface tension . It can be demonstrated experimentally by observing a large globule of oil placed below the surface of a mixture of water and alcohol having the same density so the oil has neutral buoyancy . </P> <P> If the free surface of a liquid is disturbed, waves are produced on the surface . These waves are not elastic waves due to any elastic force; they are gravity waves caused by the force of gravity tending to bring the surface of the disturbed liquid back to its horizontal level . Momentum causes the wave to overshoot, thus oscillating and spreading the disturbance to the neighboring portions of the surface . The velocity of the surface waves varies as the square root of the wavelength if the liquid is deep; therefore long waves on the sea go faster than short ones . Very minute waves or ripples are not due to gravity but to capillary action, and have properties different from those of the longer ocean surface waves, because the surface is increased in area by the ripples and the capillary forces are in this case large compared with the gravitational forces . Capillary ripples are damped both by sub-surface viscosity and by surface rheology . </P> <P> If a liquid is contained in a cylindrical vessel and is rotating around a vertical axis coinciding with the axis of the cylinder, the free surface will assume a parabolic surface of revolution known as a paraboloid . The free surface at each point is at a right angle to the force acting at it, which is the resultant of the force of gravity and the centrifugal force from the motion of each point in a circle . Since the main mirror in a telescope must be parabolic, this principle is used to create liquid mirror telescopes . </P> <P> Consider a cylindrical container filled with liquid rotating in the z direction in cylindrical coordinates, the equations of motion are: </P>

The free surface of a rotating fluid is