Flux

' In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.

In the study of transport phenomena (heat transfer, mass transfer and fluid dynamics), flux is defined as the amount that flows through a unit area per unit time^[1] Flux in this definition is a vector.
In the field of electromagnetism and mathematics, flux is usually the integral of a vector quantity over a finite surface. It is an integral operator and acts on a vector field as do the gradient, divergence and curl found in vector analysis. The result of this integration is a scalar quantity.^[2] The magnetic flux is thus the integral of the magnetic vector field B over a surface, and the electric flux is defined similarly. Using this definition, the flux of the Poynting vector over a specified surface is the rate at which electromagnetic energy flows through that surface. Confusingly, the Poynting vector is sometimes called the power flux, which is an example of the first usage of flux, above.^[3] It has units of watts per square metre (W/m²)

Flux has a primary mathematical definition in terms of a surface integral which uses the vectors that represent the force which is causing the flux being studied:
$Flux=\iint _{S}{\vec {F}}\cdot {\hat {n}}dS=\iint _{R}{\vec {F}}\cdot <-{\frac {\partial z}{\partial x}},-{\frac {\partial z}{\partial y}},1>dxdy$
Where ${\vec {F}}$ is the vector field, ${\hat {n}}$ is the normal unit vector which is perpendicular to the surface S, dS is the differential surface element and $\partial$ is to denote the partial derivative.

One could argue, based on the work of James Clerk Maxwell,^[4] that the transport definition precedes the more recent way the term is used in electromagnetism. The specific quote from Maxwell is "In the case of fluxes, we have to take the integral, over a surface, of the flux through every element of the surface. The result of this operation is called the surface integral of the flux. It represents the quantity which passes through the surface".

In addition to these common mathematical definitions, there are many more loose usages found in fields such as biology.

Transport phenomena

Origin of the term

The word flux comes from Latin: fluxus means "flow", and fluere is "to flow".^[5] As fluxion, this term was introduced into differential calculus by Newton.

Flux definition and theorems

Flux is surface bombardment rate. There are many fluxes used in the study of transport phenomena. Each type of flux has its own distinct unit of measurement along with distinct physical constants. Six of the most common forms of flux from the transport literature are defined as:

Momentum flux, the rate of transfer of momentum across a unit area (N·s·m⁻²·s⁻¹). (Newton's law of viscosity,)
Heat flux, the rate of heat flow across a unit area (J·m⁻²·s⁻¹). (Fourier's law of conduction)^[6] (This definition of heat flux fits Maxwell's original definition.^[4])
Chemical flux, the rate of movement of molecules across a unit area (mol·m⁻²·s⁻¹). (Fick's law of diffusion)
Volumetric flux, the rate of volume flow across a unit area (m³·m⁻²·s⁻¹). (Darcy's law of groundwater flow)
Mass flux, the rate of mass flow across a unit area (kg·m⁻²·s⁻¹). (Either an alternate form of Fick's law that includes the molecular mass, or an alternate form of Darcy's law that includes the density)
Radiative flux, the amount of energy moving in the form of photons at a certain distance from the source per steradian per second (J·m⁻²·s⁻¹). Used in astronomy to determine the magnitude and spectral class of a star. Also acts as a generalization of heat flux, which is equal to the radiative flux when restricted to the infrared spectrum.
Energy flux, the rate of transfer of energy through a unit area (J·m⁻²·s⁻¹). The radiative flux and heat flux are specific cases of energy flux.

These fluxes are vectors at each point in space, and have a definite magnitude and direction. Also, one can take the divergence of any of these fluxes to determine the accumulation rate of the quantity in a control volume around a given point in space. For incompressible flow, the divergence of the volume flux is zero.

Chemical diffusion

Chemical molar flux of a component A in an isothermal, isobaric system is defined in above-mentioned Fick's first law as:

{\overrightarrow {J_{A}}}=-D_{AB}\nabla c_{A}

where:

$D_{AB}$ is the diffusion coefficient (m²/s) of component A diffusing through component B,
$c_{A}$ is the concentration (mol/m³) of species A.^[7]

This flux has units of mol·m⁻²·s⁻¹, and fits Maxwell's original definition of flux.^[4]

Note: $\nabla$ ("nabla") denotes the del operator.

For dilute gases, kinetic molecular theory relates the diffusion coefficient D to the particle density n = N/V, the molecular mass m, the collision cross section $\sigma$ , and the absolute temperature T by

D={\frac {1}{3}}{\frac {1}{{\sqrt {2}}n\sigma }}{\sqrt {\frac {8kT}{\pi m}}}

where the second factor is the mean free path and the square root (with Boltzmann's constant k) is the mean velocity of the particles.

In turbulent flows, the transport by eddy motion can be expressed as a grossly increased diffusion coefficient.

Quantum mechanics

In quantum mechanics, particles of mass m in the state $\psi (r,t)$ have a probability density defined as

\rho =\psi ^{*}\psi =|\psi |^{2}.\,

So the probability of finding a particle in a unit of volume, say $d^{3}x$ , is

|\psi |^{2}d^{3}x.\,

Then the number of particles passing through a perpendicular unit of area per unit time is

\mathbf {J} =-i{\frac {\hbar }{2m}}\left(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*}\right).\,

This is sometimes referred to as the "flux density".^[8]

Electromagnetism

Flux definition and theorems

An example of the second definition of flux is the magnitude of a river's current, that is, the amount of water that flows through a cross-section of the river each second. The amount of sunlight that lands on a patch of ground each second is also a kind of flux.

To better understand the concept of flux in Electromagnetism, imagine a butterfly net. The amount of air moving through the net at any given instant in time is the flux. If the wind speed is high, then the flux through the net is large. If the net is made bigger, then the flux would be larger even though the wind speed is the same. For the most air to move through the net, the opening of the net must be facing the direction the wind is blowing. If the net opening is parallel to the wind, then no wind will be moving through the net. (These examples are not very good because they rely on a transport process and as stated in the introduction, transport flux is defined differently than E+M flux.) Perhaps the best way to think of flux abstractly is "How much stuff goes through your thing", where the stuff is a field and the thing is the imaginary surface.

The rings show the surface boundaries. The red arrows stand for the flow of charges, fluid particles, subatomic particles, photons, etc. The number of arrows that pass through each ring is the flux.

As a mathematical concept, flux is represented by the surface integral of a vector field,

\Phi _{f}=\int _{S}\mathbf {E} \cdot \mathbf {dA}

where:

E is a vector field of Electric Force,
dA is the vector area of the surface S, directed as the surface normal,
$\Phi _{f}$ is the resulting flux.

The surface has to be orientable, i.e. two sides can be distinguished: the surface does not fold back onto itself. Also, the surface has to be actually oriented, i.e. we use a convention as to flowing which way is counted positive; flowing backward is then counted negative.

The surface normal is directed accordingly, usually by the right-hand rule.

Conversely, one can consider the flux the more fundamental quantity and call the vector field the flux density.

Often a vector field is drawn by curves (field lines) following the "flow"; the magnitude of the vector field is then the line density, and the flux through a surface is the number of lines. Lines originate from areas of positive divergence (sources) and end at areas of negative divergence (sinks).

See also the image at right: the number of red arrows passing through a unit area is the flux density, the curve encircling the red arrows denotes the boundary of the surface, and the orientation of the arrows with respect to the surface denotes the sign of the inner product of the vector field with the surface normals.

If the surface encloses a 3D region, usually the surface is oriented such that the influx is counted positive; the opposite is the outflux.

The divergence theorem states that the net outflux through a closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the divergence).

If the surface is not closed, it has an oriented curve as boundary. Stokes' theorem states that the flux of the curl of a vector field is the line integral of the vector field over this boundary. This path integral is also called circulation, especially in fluid dynamics. Thus the curl is the circulation density.

We can apply the flux and these theorems to many disciplines in which we see currents, forces, etc., applied through areas.

Maxwell's equations

The flux of electric and magnetic field lines is frequently discussed in electrostatics. This is because Maxwell's equations in integral form involve integrals like above for electric and magnetic fields.

For instance, Gauss's law states that the flux of the electric field out of a closed surface is proportional to the electric charge enclosed in the surface (regardless of how that charge is distributed). The constant of proportionality is the reciprocal of the permittivity of free space.

Its integral form is:

\oint _{A}\epsilon _{0}\mathbf {E} \cdot d\mathbf {A} =Q_{A}

where:

$\mathbf {E}$ is the electric field,
$d\mathbf {A}$ is the area of a differential square on the surface A with an outward facing surface normal defining its direction,
$Q_{A}\$ is the charge enclosed by the surface,
$\epsilon _{0}\$ is the permittivity of free space
$\oint _{A}$ is the integral over the surface A.

Either $\oint _{A}\epsilon _{0}\mathbf {E} \cdot d\mathbf {A}$ or $\oint _{A}\mathbf {E} \cdot d\mathbf {A}$ is called the electric flux.

If one considers the flux of the electric field vector, E, for a tube near a point charge in the field the charge but not containing it with sides formed by lines tangent to the field, the flux for the sides is zero and there is an equal and opposite flux at both ends of the tube. This is a consequence of Gauss's Law applied to an inverse square field. The flux for any cross-sectional surface of the tube will be the same. The total flux for any surface surrounding a charge q is q/ε₀.^[9]

In free space the electric displacement vector D = ε₀ E so for any bounding surface the flux of D = q, the charge within it. Here the expression "flux of" indicates a mathematical operation and, as can be seen, the result is not necessarily a "flow".

Faraday's law of induction in integral form is:

\oint _{C}\mathbf {E} \cdot d\mathbf {l} =-\int _{\partial C}\ {d\mathbf {B}  \over dt}\cdot d\mathbf {s} =-{\frac {d\Phi _{D}}{dt}}

where:

$\mathrm {d} \mathbf {l}$ is an infinitesimal element (differential) of the closed curve C (i.e. a vector with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the curve C, with the sign determined by the integration direction).

The magnetic field is denoted by $\mathbf {B}$ . Its flux is called the magnetic flux. The time-rate of change of the magnetic flux through a loop of wire is minus the electromotive force created in that wire. The direction is such that if current is allowed to pass through the wire, the electromotive force will cause a current which "opposes" the change in magnetic field by itself producing a magnetic field opposite to the change. This is the basis for inductors and many electric generators.

Poynting vector

The flux of the Poynting vector through a surface is the electromagnetic power, or energy per unit time, passing through that surface. This is commonly used in analysis of electromagnetic radiation, but has application to other electromagnetic systems as well.

Biology

In general, 'flux' in biology relates to movement of a substance between compartments. There are several cases where the concept of 'flux' is important.

The movement of molecules across a membrane: in this case, flux is defined by the rate of diffusion or transport of a substance across a permeable membrane. Except in the case of active transport, net flux is directly proportional to the concentration difference across the membrane, the surface area of the membrane, and the membrane permeability constant.
In ecology, flux is often considered at the ecosystem level - for instance, accurate determination of carbon fluxes using techniques like eddy covariance (at a regional and global level) is essential for modeling the causes and consequences of global warming.
Metabolic flux refers to the rate of flow of metabolites along a metabolic pathway, or even through a single enzyme. A calculation may also be made of carbon (or other elements, e.g. nitrogen) flux. It is dependent on a number of factors, including: enzyme concentration; the concentration of precursor, product, and intermediate metabolites; post-translational modification of enzymes; and the presence of metabolic activators or repressors. Metabolic control analysis and flux balance analysis provide frameworks for understanding metabolic fluxes and their constraints.

Notes

^ Bird, R. Byron (1960). Transport Phenomena. Wiley. ISBN 0-471-07392-X. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
^ Lorrain, Paul (1962). Electromagnetic Fields and Waves. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
^ Wangsness, Roald K. (1986). Electromagnetic Fields (2nd ed.). Wiley. ISBN 0-471-81186-6. p.357
^ ^a ^b ^c Maxwell, James Clerk (1892). Treatise on Electricity and Magnetism.
^ Weekley, Ernest (1967), An Etymological Dictionary of Modern English, Courier Dover Publications, p. 581, ISBN 0486218732
^ Carslaw, H.S. (1959). Conduction of Heat in Solids (Second ed.). Oxford University Press. ISBN 0-19-853303-9. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
^ Welty (2001). Fundamentals of Momentum, Heat, and Mass Transfer (4th ed.). Wiley. ISBN 0-471-38149-7. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
^ Sakurai, J. J. (1967). Advanced Quantum Mechanics. Addison Wesley. ISBN 0-201-06710-2.
^ Feynman, Richard P (1964). The Feynman Lectures on Physics. Vol. II. Addison-Wesley. pp. 4–8, 9.

Transport phenomena

Origin of the term

Flux definition and theorems

Chemical diffusion

Quantum mechanics

Electromagnetism

Flux definition and theorems

Maxwell's equations

Poynting vector

Biology

See also

Notes

Further reading