page_title: Free surface
section_title: Rotation
hierarchical_section_title: Free surface / Rotation
caption_attribution_description: In Fig 409 the particle of water the free surface of the rotating vessel at distance x from the center is moving with a velocity xw...
context_page_description: In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress,
  such as the interface between two homogeneous fluids,
  for example liquid water and the air in the Earth's atmosphere. Unlike liquids, gases cannot form a free surface on their own.Fluidized/liquified solids, including slurries, granular materials, and powders may form a free surface.
  A liquid in a gravitational field will form a free surface if unconfined from above.
  Under mechanical equilibrium this free surface must be perpendicular to the forces acting on the liquid; if not there would be a force along the surface, and the liquid would flow in that direction. Thus, on the surface of the Earth, all free surfaces of liquids are horizontal unless disturbed.
  In a free liquid that is not affected by outside forces such as a gravitational field, internal attractive forces only play a role. 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.
context_section_description: 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.
  Consider a cylindrical container filled with liquid rotating in the z direction in cylindrical coordinates, the equations of motion are:
  where  is the density of the fluid,  is the radius of the cylinder,  is the angular frequency, and  is the gravitational acceleration. Taking a surface of constant pressure  the total differential becomes
  Integrating, the equation for the free surface becomes
  where  is the distance of the free surface from the bottom of the container along the axis of rotation. If one integrates the volume of the paraboloid formed by the free surface and then solves for the original height, one can find the height of the fluid along the centerline of the cylindrical container:
  The equation of the free surface at any distance  from the center becomes
  If a free liquid is rotating about an axis, the free surface will take the shape of an oblate spheroid: the approximate shape of the Earth due to its equatorial bulge.
caption: Free surface of a liquid in a rotating vessel is a paraboloid
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