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viscosity

is quickly removed but increases for a time and becomes permanent if the stress is maintained. The deformation is generally a combination of the elastic deformation and viscous flow. The simplest viscoelastic materials are Maxwell and Kelvin materials in which both the elastic and viscous deformation obey linear constitutive laws. The general expression for the relation between deviatoric stress (σ ij ) and deviatoric strain (ε ij ) for a linearly viscoelastic rheology is

where

 

 

 

P σij = Qεij

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P =

∂m

 

∂m−1

 

∂

 

 

 

+ am−1

 

 

 

+ · · · + a1

 

 

+ a0

∂tm

∂tm−1

∂t

Q =bn

∂n

 

 

∂n−1

∂

+ b0

 

+bn−1

 

+· · · + b1

 

 

∂tn

∂tn−1

∂t

and a0, a1, . . . am−1, and b0, b1, . . . bn are constants. Nonlinear viscoelastic rheology involves nonlinear elastic or viscous deformation.

viscosity The kinematic viscosity ν [m2s−1], expressing the molecular viscosity (µ; [kgm−1

s−1],) of a fluid, is defined by ν = µ/ρ (typically 1.0 · 10−6 to 1.5 · 10−6 m2 s−1 in water).

The turbulent viscosity νt of a fluid is a property of the eddy-like motions in the fluid and usually not dependent on the kinematic viscosity ν which is solely a physical property of the fluid. Turbulence implies νt >> ν.

viscosity (η) The proportionality factor between shear rate and shear stress, defined through the equation f = ηA(dv/dx), where F is the tangential force required to move a planar surface of area A relative to a parallel surface. The result depends on dv/dx, the rate of increase of velocity with distance from the fixed plate. Sometimes called dynamic or absolute viscosity. The term kinematic viscosity (symbol ν) is defined as η divided by the mass density. η is also denoted by µ, the molecular viscosity.

viscous-like force

See quasi-viscous force.

viscous shear stress

The stress resulting

from the intermolecular forces (proportional to viscosity) between water parcels of different velocities (shear). In a Newtonian fluids, such as

water, the viscous shear stress, in general 9 tensor terms τij (= force per unit area in i-direction acting on the surface that is normal to the j- direction) is linearly dependent on the shear. In turbulent fluids, the viscous shear stress is much smaller than the turbulent Reynolds stresses and is usually neglected for large scale processes. Viscous shear stress becomes the dominant force at small scales, such as microstructure (see Kolmogorov scale) and in viscous sublayer.

viscous sublayer The layer along a contiguous boundary, within which momentum (shear stress) is transferred by intermolecular forces; i.e., τ = νρ∂u/∂z.

visible light Light that is detectable by the normal human eye.

visible wavelengths

Approximately 400 to

700 nm.

 

visual binary system

An orbiting pair of

stars far enough apart from each other and close enough to us that two points of light can be resolved in the sky (generally only with a telescope). This requires a separation of nearly 1 arcsec. For instance, a pair of stars like the sun with an orbit period of 20 years would have to be less than about 10 pc from us to be seen as a visual binary. Most orbit periods of visual binaries are long as a result, and analysis requires data to be collected over centuries or more. Once the distance to the system is known, the masses of the two stars can be found directly from the observed orbits of the stars using Kepler’s third law of motion, (M1 + M2)P 2 = a3M1/M2 = a2/a1, where P is the orbit period in years, a1 and a2 are the semi-major axes of the two orbits around their center of mass in astronomical units, a = a1 + a2, and M1 and M2 are the larger and smaller masses in solar masses. Only about 100 visual binary systems have been studied well enough to yield masses with 15% accuracy, and most of them are stars of roughly solar mass.

VLA Very Large Array, a “Y ”-shaped array of 27 dish antennas, each of 25 m aperture, mounted on railway tracks that can be moved to form an interferometer up to 36 km across.

© 2001 by CRC Press LLC

Voigt profile

Located near Socorro, NM at 34◦04 43".497N, 107◦37 03".819W. The array has a maximum resolution of 0".04 and can be used over the frequency range 300 to 50,000 MHz (90 to 0.7 cm).

Vlasov equation The kinetic equation governing the velocity distributions of electrons and ions in an environment where Coulomb collisions may be neglected. For non-relativistic particles of charge q and mass m in the presence of an electric field E(x, t) and a magnetic field B(x, t), the Vlasov equation is (cgs units)

∂t

+ v·∂x

+ m

E + c ×B

·∂v f = 0 ,

 

∂

 

∂

 

q

 

v

 

∂

where f (v, x, t) is the velocity distribution. Here v and x are, respectively, the velocity and spatial coordinates, and t is the time.

Moment equations for macroscopic quantities are obtained by multiplying the Vlasov equation by 1, v, etc. and integrating over velocity space. For example, the zero-order moment equation is

∂n + · (nV) = 0 , ∂t

where the zeroth moment of f

n = f d3v

is the particle number density, and the first moment (divided by n)

V= 1 vf d3v n

is the mean vector velocity. Another useful moment is the stress (or pressure) tensor

P = m (v − V) (v − V) f d3v ;

note that the stress tensor need not be isotropic in a collisionless plasma.

The electric charge and current densities in the plasma are

qα fαd3v

α

and

qα vfαd3v ,

α

respectively, where the summation is taken over all charge species α.

Vlasov–Maxwell equations The fundamental equations governing the behavior of a plasma in which Coulomb collisions are negligible. A solution of the Vlasov–Maxwell system is comprised of velocity distributions for all charge-species and the associated electric charge and current densities, together with electric and magnetic fields, all consistent with the Maxwell equations governing electrodynamics. See Vlasov equation.

voids Large regions of the universe in which the matter-density is distinctly lower than the average. Voids are surrounded by relatively thin layers into which nearly all galaxies are crowded. Typical diameters of voids are between 20h−1Mpc and 60h−1Mpc (h is the dimensionless Hubble parameter, i.e., H0 /100/km/sec/Mpc), and the mass-density within a void is believed to be about 10% of the cosmic average matter-density. Hence, the large-scale distribution of matter in the universe is somewhat similar to a foam, with the voids being analogs of the air-bubbles and the layers with galaxies being analogs of the soap films.

Voids were observationally discovered in 1978–1979 and hailed at that time as a surprising discovery. However, papers were already published in 1934 that predicted that the Friedmann–Lemaître cosmological models were unstable against the formation of local minima in matter-density. Had the faith in the cosmological principle not prevailed later, voids would have been, in fact, expected.

Voigt body Also called Kelvin body. A kind of material with both properties of elastic body and viscous fluid. A Voigt body is represented by the sum of two terms for which stresses are proportional to elastic strain and viscous strain rate. Most elastic bodies containing many pores filled with viscous fluid are examples of Voigt bodies.

Voigt profile An expression that describes the shape of an absorption line considering the effects of natural and thermal Doppler broadening mechanisms. The total absorption coeffi-

© 2001 by CRC Press LLC

volatile

cient, α, is then

α = α(natural) α(thermal)

where indicates convolution. Convolving the natural and Doppler absorption coefficients, we have the normalized Voigt profile,

V ( ν, νD , γ ) =

 

 

 

 

 

 

+∞

γ / 4π2

 

1

 

−∞

[( ν −

 

−

 

]

νD√π

 

 

ν)2

 

(γ /4π)2

 

 

 

e−( ν1/ νD)2 d ( ν1) ,

where νD is the Doppler width at half maximum, and γ is the radiation damping constant (see natural line broadening).

The absorption coefficient becomes

α= πe2 f V ( ν, νD) . mec

This expression can be calculated numerically or simplified slightly by expressing it as a Hjerting function. In any case, the profile of a line is then formed by integrating α over all frequencies. This simplified form is used in calculating model atmospheres for hot stars.

volatile Easily evaporated, or in geology, elements or compounds with low molecular weight and low melting points that are easily driven from formations by heating. See refractory.

volcanic front A front line of areas of volcano distribution nearest to a trench at a subduction zone. The closer to the volcanic front, the greater the density of volcano distribution and the number of eruptions. Although most volcanic fronts are almost parallel to a trench, there are some exceptions such as the northern part of the central American arc. In many cases, there exists a deep seismic plane at around a depth of 110 km just below a volcanic front. In general, the amount of alkali in volcanic rocks tends to increase as we move away from a volcanic front toward the back-arc side.

volcanic line A well-defined line of volcanos generally is associated with subduction zones. One example is the line of volcanos extending from Mount Shasta in California to Mount Baker

in Washington. These volcanos systematically lie close to 125 km above the subducted oceanic plate.

volcanism Planetary interiors are often quite hot due to a number of heat producing processes, including radioactive decay, heat from accretion, and differentiation. This heat can cause the interior rocks to melt, producing a magma, which can make its way to the surface and erupt in a volcanic event. The types of volcanic features resulting from this eruption depend on the rate of the eruption and the viscosity (i.e., the stickiness) of the lava. Low viscosity, fluid lavas will produce flat volcanic flows (i.e., Columbia River Basalts, Washington). Slightly stickier lavas will produce low-sloped shield volcanoes (i.e., Hawaii). Magma with substantial amounts of gas incorporated in it will produce cinder cones (i.e., Sunset Crater, Arizona). Explosive eruptions are associated with stratovolcanoes (composite volcanoes) which consist of alternate episodes of quiet lava flows and explosive ash eruptions (i.e., Mt. St. Helens, Washington). The most explosive eruptions are associated with ignimbrites, which produce large ash flows (i.e., Long Valley Caldera, California). Magma composed of molten rock is the most common type of volcanic material on Earth, but elsewhere in the solar system magmas composed of sulfur and various types of volatiles (such as water, ammonia, etc.) are seen.

volcanos Volcanos are mountains, generally conical in shape, created by magmas erupted from the interior of the Earth. Most of the famous volcanos are associated with subduction. Examples include Mount Fuji in Japan, Mount St. Helens in the U.S., Penetubo in the Philippines, and Etna in Italy. This class of volcanos often has explosive eruptions. Volcanos are also associated with ocean ridges, but almost all of these volcanos are in the deep oceans and cannot be observed. An exception is the group of volcanos in Iceland. A third class of volcanos occur within plate interiors and are known as hotspots. These volcanos are generally associated with mantle plumes, and their eruptions are usually quiescent; an example is Hawaii.

© 2001 by CRC Press LLC

vorton

volume scattering function (VSF) The ratio of the scattered intensity [W sr−1] to the incident irradiance [W m−2] per unit volume [m3], given in [m−1 sr−1]; the integral of the volume scattering function over all directions and all final wavelengths is the scattering coefficient; the VSF can be written as the product of the phase function [sr−1] and the scattering coefficient [m−1].

volumetric water content (or simply water content) (θ) The ratio of liquid water volume to the total soil or rock volume: θ = Vw/Vs , where Vw is the volume of water in the sample, and Vs is the total volume of soil or rock in the sample. The theoretical range of the volumetric water content is from 0 (completely dry) to saturation (porosity or φ), but the range in natural systems is generally much less than this.

von Mises criterion The condition by which a polycrystal can deform coherently to large strains by dislocation slip. von Mises criterion points out that five independent slip systems are needed in order to deform a polycrystalline material by crystallographic slip. This is because arbitrary shape change must be possible to satisfy the displacement compatibility at grain boundaries.

vortex defect of the vacuum

See cosmic

string.

 

vortex street The series of vortices shed systematically downstream from a body in a rapid fluid flow (sufficiently high Reynolds number). For a cylindrically symmetrical body, a pair of symmetrically placed opposite vortices (with a sign which can be deduced from the streamflow) form behind the body on either side of the down-

stream direction. Small disturbances then lead to a shedding of one of the two vortices, and subsequently alternate shedding of vortices of opposite sign. The trail of vortices behind the body is the vortex street.

vorticity A vector measure, ω, of local rotation in a fluid flow:

ω = × v .

vorton Equilibrium configurations of conducting cosmic string loops, where the string tension, tending to make the loop collapse, is balanced by the centrifugal repulsion due to rotation. Symmetry reasons (Lorentz invariance along the string world sheet) forbid rotating configurations for Goto–Nambu strings and, therefore, vortons can only arise in theories where cosmic strings have a microscopic internal structure, like currents flowing along the core.

The existence of vorton equilibrium configurations would prevent the loops from decaying away in the form of radiation. They would then evolve like ordinary (eventually charged) nonrelativistic matter very early on in the history of the universe, eventually disrupting some of the predictions of standard cosmology.

This possible incompatibility is known as the vorton excess problem. Many mechanisms aiming at diluting vorton overdensity are currently being studied. If successful, vortons could stop being a problem and would become one more interesting candidate for the non-baryonic dark matter that combined observations and theoretical models tend to require. See CHUMP, conducting string, cosmic string, Goto–Nambu string.

© 2001 by CRC Press LLC

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