Free surface

In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress,^[1] such as the interface between two homogeneous fluids.^[2] An example of two such homogeneous fluids would be a body of water (liquid) and the air in the Earth's atmosphere (gas mixture). Unlike liquids, gases cannot form a free surface on their own.^[3] 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.^[3] 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.^[4] Thus, on the surface of the Earth, all free surfaces of liquids are horizontal unless disturbed (except near solids dipping into them, where surface tension distorts the surface in a region called the meniscus).^[4]

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.^[5]^[6]

Flatness refers to the shape of a liquid's free surface. On Earth, the flatness of a liquid is a function of the curvature of the planet, and from trigonometry, can be found to deviate from true flatness by approximately 19.6 nanometers over an area of 1 square meter, a deviation which is dominated by the effects of surface tension. This calculation uses Earth's mean radius at sea level, however a liquid will be slightly flatter at the poles.^[7]^[8] Over large distances or planetary scale, the surface of an undisturbed liquid tends to conform to equigeopotential surfaces; for example, mean sea level follows approximately the geoid.

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.^[4] 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.^[4] 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,^[4] because the surface is increased in area by the ripples and the capillary forces are in this case large compared with the gravitational forces.^[9] Capillary ripples are damped both by sub-surface viscosity and by surface rheology.

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.^[4] 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:

{\frac {\partial P}{\partial r}}=\rho r\omega ^{2},\quad {\frac {\partial P}{\partial \theta }}=0,\quad {\frac {\partial P}{\partial z}}=-\rho g,

where $P$ is the pressure, $\rho$ is the density of the fluid, $r$ is the radius of the cylinder, $\omega$ is the angular frequency, and $g$ is the gravitational acceleration. Taking a surface of constant pressure $(dP=0)$ the total differential becomes

dP=\rho r\omega ^{2}dr-\rho gdz\to {\frac {dz_{\text{isobar}}}{dr}}={\frac {r\omega ^{2}}{g}}.

Integrating, the equation for the free surface becomes

z_{s}={\frac {\omega ^{2}}{2g}}r^{2}+h_{c},

where $h_{c}$ 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:

h_{c}=h_{0}-{\frac {\omega ^{2}R^{2}}{4g}}.

The equation of the free surface at any distance $r$ from the center becomes

z_{s}=h_{0}-{\frac {\omega ^{2}}{4g}}(R^{2}-2r^{2}).

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.^[10]

In hydrodynamics, the free surface is defined mathematically by the free-surface condition,^[11] that is, the material derivative on the pressure is zero: ${\frac {Dp}{Dt}}=0.$
In fluid dynamics, a free-surface vortex, also known as a potential vortex or whirlpool, forms in an irrotational flow,^[12] for example when a bathtub is drained.^[13]
In naval architecture and marine safety, the free surface effect occurs when liquids or granular materials under a free surface in partially filled tanks or holds shift when the vessel heels.^[14]
In hydraulic engineering a free-surface jet is one where the entrainment of the fluid outside the jet is minimal, as opposed to submerged jet where the entrainment effect is significant. A liquid jet in air approximates a free surface jet.^[15]
In fluid mechanics a free surface flow, also called open-channel flow, is the gravity driven flow of a fluid under a free surface, typically water flowing under air in the atmosphere.^[16]

[1]
"Glossary: Free Surface". Interactive Guide. Vishay Measurements Group. Retrieved 2007-12-02. Surface of a body with no normal stress perpendicular or shear stresses parallel to it…
[2]
Free surface. McGraw-Hill Dictionary of Scientific and Technical Terms. McGraw-Hill Companies, Inc., 2003. Answers.com. Retrieved on 2007-12-02.
[3]
White, Frank (2003). Fluid mechanics. New York: McGraw-Hill. p. 4. ISBN 0-07-240217-2.
[4]
Rowland, Henry Augustus; Joseph Sweetman Ames (1900). "Free Surface of Liquids". Elements of Physics. American Book Co. pp. 70–71.
[5]
Millikan, Robert Andrews; Gale, Henry Gordon (1906). "161. Shape assumed by a free liquid". A First Course in Physics. Ginn & company. p. 114. Since, then, every molecule of a liquid is pulling on every other molecule, any body of liquid which is free to take its natural shape that is which is acted on only by its own cohesive forces, must draw itself together until it has the smallest possible surface compatible with its volume; for, since every molecule in the surface is drawn toward the interior by the attraction of the molecules within, it is clear that molecules must continually move toward the center of the mass until the whole has reached the most compact form possible. Now the geometrical figure which has the smallest area for a given volume is a sphere. We conclude, therefore, that if we could relieve a body of liquid from the action of gravity and other outside forces, it would at once take the form of a perfect sphere.
[6]
Dull, Charles Elwood (1922). "92. Shape Assumed by a Free Liquid". Essentials of Modern Physics. New York: H. Holt. Since the molecules of liquids slide over one another readily, the force of gravity causes the surface of liquids to become level. If the force of gravity can be nullified, a small portion of free liquid will then assume a spherical form.
[7]
Dew, G.D. (March 1966), "The measurement of optical flatness", Journal of Scientific Instruments, 43 (7): 409–415, Bibcode:1966JScI...43..409D, doi:10.1088/0950-7671/43/7/301, PMID 5941575
[8]
Bünnagel, R.; Oehring, H.-A.; Steiner, K. (1968), "Fizeau interferometer for measuring the flatness of optical surfaces", Applied Optics, 7 (2): 331–335, Bibcode:1968ApOpt...7..331B, doi:10.1364/AO.7.000331, PMID 20062467
[9]
Gilman, Daniel Coit; Peck, Harry Thurston; Colby, Frank Moore, eds. (1903). "Hydrostatics". The New International Encyclopædia. Dodd, Mead and Company. p. 739.
[10]
"Hydrostatics". Appletons' Cyclopædia of Applied Mechanics. New York: D. Appleton and company. 1880. p. 123. If a perfectly homogeneous mass of liquid be acted upon by a force which varies directly as the distance from the centre of the mass, the free surface will be of spherical form; if the mass rotates about an axis, the form assumed will be that of an oblate spheroid, which is the shape of the earth.
[11]
"Free surface". Glossary of Meteorology. American Meteorological Society. Archived from the original on 2007-12-09. Retrieved 2007-11-27.
[12]
Brighton, John A.; Hughes, William T. (1999). Schaum's outline of theory and problems of fluid dynamics. Boston, Mass: McGraw Hill. p. 51. ISBN 0-07-031118-8. A simple example of irrotational flow is a whirlpool, which is known as a potential vortex in fluid mechanics.
[13]
"Ricerca Italiana - PRIN - Global stability of three-dimensional flows". Archived from the original on 2012-02-09. Retrieved 2007-12-02. The free-surface vortex (whirlpool) that occurs during the draining of a basin has received different interpretations along its history;
[14]
"The Free Surface Effect - Stability". Retrieved 2007-12-02. In a partly filled tank or fish hold, the contents will shift with the movement of the boat. This "free surface" effect increases the danger of capsizing.
[15]
Suryanarayana, N. V. (2000). "3.2.2 Forced Convection - External Flows". In Kreith, Frank (ed.). The CRC Handbook of Thermal Engineering (Mechanical Engineering). Berlin: Springer-Verlag and Heidelberg. pp. 3–44. ISBN 3-540-66349-5. In free-surface jets — a liquid jet in an atmosphere of air is a good approximation to a free-surface jet — the entrainment effect is usually negligible…
[16]
White, Frank M. (2000). "2.5 Open Channel Flow". In Kreith, Frank (ed.). The CRC Handbook of Thermal Engineering (Mechanical Engineering). Berlin: Springer-Verlag and Heidelberg. pp. 2–61. ISBN 3-540-66349-5. The term open channel flow denotes the gravity-driven flow of a liquid with a free surface.

[titleGlossary:_Free_Surface-1] [1]
"Glossary: Free Surface". Interactive Guide. Vishay Measurements Group. Retrieved 2007-12-02. Surface of a body with no normal stress perpendicular or shear stresses parallel to it…

[2] [2]
Free surface. McGraw-Hill Dictionary of Scientific and Technical Terms. McGraw-Hill Companies, Inc., 2003. Answers.com. Retrieved on 2007-12-02.

[White-3] [3]
White, Frank (2003). Fluid mechanics. New York: McGraw-Hill. p. 4. ISBN 0-07-240217-2.

[Rowland-4] [4]
Rowland, Henry Augustus; Joseph Sweetman Ames (1900). "Free Surface of Liquids". Elements of Physics. American Book Co. pp. 70–71.

[5] [5]
Millikan, Robert Andrews; Gale, Henry Gordon (1906). "161. Shape assumed by a free liquid". A First Course in Physics. Ginn & company. p. 114. Since, then, every molecule of a liquid is pulling on every other molecule, any body of liquid which is free to take its natural shape that is which is acted on only by its own cohesive forces, must draw itself together until it has the smallest possible surface compatible with its volume; for, since every molecule in the surface is drawn toward the interior by the attraction of the molecules within, it is clear that molecules must continually move toward the center of the mass until the whole has reached the most compact form possible. Now the geometrical figure which has the smallest area for a given volume is a sphere. We conclude, therefore, that if we could relieve a body of liquid from the action of gravity and other outside forces, it would at once take the form of a perfect sphere.

[6] [6]
Dull, Charles Elwood (1922). "92. Shape Assumed by a Free Liquid". Essentials of Modern Physics. New York: H. Holt. Since the molecules of liquids slide over one another readily, the force of gravity causes the surface of liquids to become level. If the force of gravity can be nullified, a small portion of free liquid will then assume a spherical form.

[7] [7]
Dew, G.D. (March 1966), "The measurement of optical flatness", Journal of Scientific Instruments, 43 (7): 409–415, Bibcode:1966JScI...43..409D, doi:10.1088/0950-7671/43/7/301, PMID 5941575

[8] [8]
Bünnagel, R.; Oehring, H.-A.; Steiner, K. (1968), "Fizeau interferometer for measuring the flatness of optical surfaces", Applied Optics, 7 (2): 331–335, Bibcode:1968ApOpt...7..331B, doi:10.1364/AO.7.000331, PMID 20062467

[International-9] [9]
Gilman, Daniel Coit; Peck, Harry Thurston; Colby, Frank Moore, eds. (1903). "Hydrostatics". The New International Encyclopædia. Dodd, Mead and Company. p. 739.

[10] [10]
"Hydrostatics". Appletons' Cyclopædia of Applied Mechanics. New York: D. Appleton and company. 1880. p. 123. If a perfectly homogeneous mass of liquid be acted upon by a force which varies directly as the distance from the centre of the mass, the free surface will be of spherical form; if the mass rotates about an axis, the form assumed will be that of an oblate spheroid, which is the shape of the earth.

[AMS-11] [11]
"Free surface". Glossary of Meteorology. American Meteorological Society. Archived from the original on 2007-12-09. Retrieved 2007-11-27.

[isbn0-07-031118-8-12] [12]
Brighton, John A.; Hughes, William T. (1999). Schaum's outline of theory and problems of fluid dynamics. Boston, Mass: McGraw Hill. p. 51. ISBN 0-07-031118-8. A simple example of irrotational flow is a whirlpool, which is known as a potential vortex in fluid mechanics.

[titleRicerca_Italiana_-_PRIN_-_Global_stability_of_three-dimensional_flows-13] [13]
"Ricerca Italiana - PRIN - Global stability of three-dimensional flows". Archived from the original on 2012-02-09. Retrieved 2007-12-02. The free-surface vortex (whirlpool) that occurs during the draining of a basin has received different interpretations along its history;

[titleThe_Free_Surface_Effect_-_Stability_|_TP_10038_|_Marine_Safety-14] [14]
"The Free Surface Effect - Stability". Retrieved 2007-12-02. In a partly filled tank or fish hold, the contents will shift with the movement of the boat. This "free surface" effect increases the danger of capsizing.

[Suryanarayana-15] [15]
Suryanarayana, N. V. (2000). "3.2.2 Forced Convection - External Flows". In Kreith, Frank (ed.). The CRC Handbook of Thermal Engineering (Mechanical Engineering). Berlin: Springer-Verlag and Heidelberg. pp. 3–44. ISBN 3-540-66349-5. In free-surface jets — a liquid jet in an atmosphere of air is a good approximation to a free-surface jet — the entrainment effect is usually negligible…

[White2-16] [16]
White, Frank M. (2000). "2.5 Open Channel Flow". In Kreith, Frank (ed.). The CRC Handbook of Thermal Engineering (Mechanical Engineering). Berlin: Springer-Verlag and Heidelberg. pp. 2–61. ISBN 3-540-66349-5. The term open channel flow denotes the gravity-driven flow of a liquid with a free surface.

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