Free surface

Last updated August 15, 2022

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]

Waves

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.^[7] Capillary ripples are damped both by sub-surface viscosity and by surface rheology.

Rotation

Free surface of a liquid in a rotating vessel is a paraboloid Rotating Vessel.png — Free surface of a liquid in a rotating vessel is a paraboloid

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.^[8]

Related terms

In hydrodynamics, the free surface is defined mathematically by the free-surface condition,^[9] 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,^[10] for example when a bathtub is drained.^[11]
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.^[12]
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.^[13]
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.^[14]

Related Research Articles

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Gravity wave Wave in or at the interface between fluids where gravity is the main equilibrium force

In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere and the ocean, which gives rise to wind waves.

The vorticity equation of fluid dynamics describes the evolution of the vorticity $ω$ of a particle of a fluid as it moves with its flow; that is, the local rotation of the fluid. The governing equation is:

In fluid dynamics, the baroclinity of a stratified fluid is a measure of how misaligned the gradient of pressure is from the gradient of density in a fluid. In meteorology a baroclinic flow is one in which the density depends on both temperature and pressure. A simpler case, barotropic flow, allows for density dependence only on pressure, so that the curl of the pressure-gradient force vanishes.

In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

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A capillary wave is a wave traveling along the phase boundary of a fluid, whose dynamics and phase velocity are dominated by the effects of surface tension.

Internal waves are gravity waves that oscillate within a fluid medium, rather than on its surface. To exist, the fluid must be stratified: the density must change with depth/height due to changes, for example, in temperature and/or salinity. If the density changes over a small vertical distance, the waves propagate horizontally like surface waves, but do so at slower speeds as determined by the density difference of the fluid below and above the interface. If the density changes continuously, the waves can propagate vertically as well as horizontally through the fluid.

In fluid mechanics, the Taylor–Proudman theorem states that when a solid body is moved slowly within a fluid that is steadily rotated with a high angular velocity $, the fluid velocity will be uniform along any line parallel to the axis of rotation. must be large compared to the movement of the solid body in order to make the Coriolis force large compared to the acceleration terms.$

In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium.

In fluid mechanics and hydraulics, open-channel flow is a type of liquid flow within a conduit with a free surface, known as a channel. The other type of flow within a conduit is pipe flow. These two types of flow are similar in many ways but differ in one important respect: open-channel flow has a free surface, whereas pipe flow does not.

Inertial waves, also known as inertial oscillations, are a type of mechanical wave possible in rotating fluids. Unlike surface gravity waves commonly seen at the beach or in the bathtub, inertial waves flow through the interior of the fluid, not at the surface. Like any other kind of wave, an inertial wave is caused by a restoring force and characterized by its wavelength and frequency. Because the restoring force for inertial waves is the Coriolis force, their wavelengths and frequencies are related in a peculiar way. Inertial waves are transverse. Most commonly they are observed in atmospheres, oceans, lakes, and laboratory experiments. Rossby waves, geostrophic currents, and geostrophic winds are examples of inertial waves. Inertial waves are also likely to exist in the molten core of the rotating Earth.

The intent of this article is to highlight the important points of the derivation of the Navier–Stokes equations as well as its application and formulation for different families of fluids.

In fluid dynamics, Airy wave theory gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.

A Taylor column is a fluid dynamics phenomenon that occurs as a result of the Coriolis effect. It was named after Geoffrey Ingram Taylor. Rotating fluids that are perturbed by a solid body tend to form columns parallel to the axis of rotation called Taylor columns.

In fluid dynamics, the mild-slope equation describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in coastal engineering to compute the wave-field changes near harbours and coasts.

In fluid dynamics, the radiation stress is the depth-integrated – and thereafter phase-averaged – excess momentum flux caused by the presence of the surface gravity waves, which is exerted on the mean flow. The radiation stresses behave as a second-order tensor.

In fluid mechanics, dynamic similarity is the phenomenon that when there are two geometrically similar vessels with the same boundary conditions and the same Reynolds and Womersley numbers, then the fluid flows will be identical. This can be seen from inspection of the underlying Navier-Stokes equation, with geometrically similar bodies, equal Reynolds and Womersley Numbers the functions of velocity (u’,v’,w’) and pressure (P’) for any variation of flow.

In fluid dynamics, Hicks equation or sometimes also referred as Bragg–Hawthorne equation or Squire–Long equation is a partial differential equation that describes the distribution of stream function for axisymmetric inviscid fluid, named after William Mitchinson Hicks, who derived it first in 1898. The equation was also re-derived by Stephen Bragg and William Hawthorne in 1950 and by Robert R. Long in 1953 and by Herbert Squire in 1956. The Hicks equation without swirl was first introduced by George Gabriel Stokes in 1842. The Grad–Shafranov equation appearing in plasma physics also takes the same form as the Hicks equation.

In physical oceanography and fluid mechanics, the Miles-Phillips mechanism describes the generation of wind waves from a flat sea surface by two distinct mechanisms. Wind blowing over the surface generates tiny wavelets. These wavelets develop over time and become ocean surface waves by absorbing the energy transferred from the wind. The Miles-Phillips mechanism is a physical interpretation of these wind-generated surface waves.
Both mechanisms are applied to gravity-capillary waves and have in common that waves are generated by a resonance phenomenon. The Miles mechanism is based on the hypothesis that waves arise as an instability of the sea-atmosphere system. The Phillips mechanism assumes that turbulent eddies in the atmospheric boundary layer induce pressure fluctuations at the sea surface. The Phillips mechanism is generally assumed to be important in the first stages of wave growth, whereas the Miles mechanism is important in later stages where the wave growth becomes exponential in time.

References

↑ "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…
↑ Free surface. McGraw-Hill Dictionary of Scientific and Technical Terms. McGraw-Hill Companies, Inc., 2003. Answers.com. Retrieved on 2007-12-02.
1 2 White, Frank (2003). Fluid mechanics. New York: McGraw-Hill. p. 4. ISBN 0-07-240217-2.
1 2 3 4 5 6 Rowland, Henry Augustus; Joseph Sweetman Ames (1900). "Free Surface of Liquids". Elements of Physics. American Book Co. pp. 70–71.
↑ 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.
↑ 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.
↑ Gilman, Daniel Coit; Peck, Harry Thurston; Colby, Frank Moore, eds. (1903). "Hydrostatics". The New International Encyclopædia. Dodd, Mead and Company. p. 739.
↑ "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.
↑ "Free surface". Glossary of Meteorology. American Meteorological Society. Archived from the original on 2007-12-09. Retrieved 2007-11-27.
↑ 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.
↑ "Ricerca Italiana - PRIN - Global stability of three-dimensional flows" . Retrieved 2007-12-02. The free-surface vortex (whirlpool) that occurs during the draining of a basin has received different interpretations along its history;
↑ "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, 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…
↑ 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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[titleGlossary:_Free_Surface-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.

[White-3] 1 2 White, Frank (2003). Fluid mechanics. New York: McGraw-Hill. p. 4. ISBN 0-07-240217-2.

[Rowland-4] 1 2 3 4 5 6 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.

[International-7] Gilman, Daniel Coit; Peck, Harry Thurston; Colby, Frank Moore, eds. (1903). "Hydrostatics". The New International Encyclopædia. Dodd, Mead and Company. p. 739.

[8] "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-9] "Free surface". Glossary of Meteorology. American Meteorological Society. Archived from the original on 2007-12-09. Retrieved 2007-11-27.

[isbn0-07-031118-8-10] 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-11] "Ricerca Italiana - PRIN - Global stability of three-dimensional flows" . 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-12] "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-13] 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…

[White2-14] 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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