Editing Flux (section)

== Flux as flow rate per unit area ==
In [[transport phenomena]] ([[heat transfer]], [[mass transfer]] and [[fluid dynamics]]), flux is defined as the ''rate of flow of a property per unit area'', which has the [[dimensional analysis|dimensions]] [quantity]·[time]<sup>−1</sup>·[area]<sup>−1</sup>.<ref>{{cite book | first=R. Byron | last=Bird | author-link=Robert Byron Bird | author2=Stewart, Warren E. | author3=Lightfoot, Edwin N. | author3-link=Edwin N. Lightfoot | year=1960 | title=Transport Phenomena | publisher=Wiley | isbn=0-471-07392-X | url-access=registration | url=https://archive.org/details/transportphenome00bird }}</ref> The area is of the surface the property is flowing "through" or "across". For example, the amount of water that flows through a cross section of a river each second divided by the area of that cross section, or the amount of sunlight energy that lands on a patch of ground each second divided by the area of the patch, are kinds of flux.

=== General mathematical definition (transport) ===
[[File:General flux diagram.svg|thumb|upright=1.5|The [[field line]]s of a [[vector field]] {{math|'''F'''}} through surfaces with [[unit vector|unit]] normal {{math|'''n'''}}, the angle from {{math|'''n'''}} to {{math|'''F'''}} is {{mvar|θ}}. Flux is a measure of how much of the field passes through a given surface. {{math|'''F'''}} is decomposed into components perpendicular (⊥) and parallel {{nowrap|( ‖ )}} to {{math|'''n'''}}. Only the parallel component contributes to flux because it is the maximum extent of the field passing through the surface at a point, the perpendicular component does not contribute. <br />'''Top:''' Three field lines through a plane surface, one normal to the surface, one parallel, and one intermediate. <br />'''Bottom:''' Field line through a [[curved surface]], showing the setup of the unit normal and surface element to calculate flux.]]
[[Image:Surface integral - definition.svg|thumb|upright=1.5|To calculate the flux of a vector field {{math|'''F'''}} ''(red arrows)'' through a surface {{mvar|S}} the surface is divided into small patches {{mvar|dS}}.  The flux through each patch is equal to the normal (perpendicular) component of the field, the [[dot product]] of {{math|'''F'''('''x''')}} with the unit normal vector {{math|'''n'''('''x''')}} ''(blue arrows)'' at the point {{math|'''x'''}} multiplied by the area {{mvar|dS}}.  The sum of {{math|'''F''' · '''n''', ''dS''}} for each patch on the surface is the flux through the surface.]]
Here are 3 definitions in increasing order of complexity. Each is a special case of the following. In all cases the frequent symbol ''j'', (or ''J'') is used for flux, ''q'' for the [[physical quantity]] that flows, ''t'' for time, and ''A'' for area. These identifiers will be written in bold when and only when they are vectors.

First, flux as a (single) scalar:
<math display="block">j = \frac{I}{A},</math>
where
<math display="block">I = \lim_{\Delta t \to 0}\frac{\Delta q}{\Delta t} = \frac{\mathrm{d}q}{\mathrm{d}t}.</math>
In this case the surface in which flux is being measured is fixed and has area ''A''. The surface is assumed to be flat, and the flow is assumed to be everywhere constant with respect to position and perpendicular to the surface.

Second, flux as a [[scalar field]] defined along a surface, i.e. a function of points on the surface:
<math display="block">j(\mathbf{p}) = \frac{\partial I}{\partial A}(\mathbf{p}),</math>
<math display="block">I(A,\mathbf{p}) = \frac{\mathrm{d}q}{\mathrm{d}t}(A, \mathbf{p}).</math>
As before, the surface is assumed to be flat, and the flow is assumed to be everywhere perpendicular to it. However the flow need not be constant. ''q'' is now a function of '''p''', a point on the surface, and ''A'', an area. Rather than measure the total flow through the surface, ''q'' measures the flow through the disk with area ''A'' centered at ''p'' along the surface.

Finally, flux as a [[vector field]]:
<math display="block">\mathbf{j}(\mathbf{p}) = \frac{\partial \mathbf{I}}{\partial A}(\mathbf{p}),</math>
<math display="block">\mathbf{I}(A,\mathbf{p}) = \underset{\mathbf{\hat{n}}}{\operatorname{arg\,max}}\; \mathbf{\hat{n}}_{\mathbf p} \frac{\mathrm{d}q}{\mathrm{d}t}(A,\mathbf{p}, \mathbf{\hat{n}}).</math>
In this case, there is no fixed surface we are measuring over. ''q'' is a function of a point, an area, and a direction (given by a unit vector <math>\mathbf{\hat{n}}</math>), and measures the flow through the disk of area A perpendicular to that unit vector. ''I'' is defined picking the unit vector that maximizes the flow around the point, because the true flow is maximized across the disk that is perpendicular to it. The unit vector thus uniquely maximizes the function when it points in the "true direction" of the flow. (Strictly speaking, this is an [[abuse of notation]] because the "arg{{nnbsp}}max" cannot directly compare vectors; we take the vector with the biggest norm instead.)

==== Properties ====

These direct definitions, especially the last, are rather unwieldy {{Citation needed|date=May 2025}}. For example, the arg{{nnbsp}}max construction is artificial from the perspective of empirical measurements, when with a [[weathervane]] or similar one can easily deduce the direction of flux at a point. Rather than defining the vector flux directly, it is often more intuitive to state some properties about it. Furthermore, from these properties the flux can uniquely be determined anyway.

If the flux '''j''' passes through the area at an angle θ to the area normal <math>\mathbf{\hat{n}}</math>, then the [[dot product]]
<math display="block">\mathbf{j} \cdot \mathbf{\hat{n}} = j\cos\theta.</math>
That is, the component of flux passing through the surface (i.e. normal to it) is ''j''{{nnbsp}}cos{{nnbsp}}''θ'', while the component of flux passing tangential to the area is ''j''{{nnbsp}}sin{{nnbsp}}''θ'', but there is ''no'' flux actually passing ''through'' the area in the tangential direction. The ''only'' component of flux passing normal to the area is the cosine component.

For vector flux, the [[surface integral]] of '''j''' over a [[Surface (mathematics)|surface]] ''S'', gives the proper flowing per unit of time through the surface:
<math display="block">\frac{\mathrm{d}q}{\mathrm{d}t} = \iint_S \mathbf{j} \cdot \mathbf{\hat{n}}\, dA = \iint_S \mathbf{j} \cdot d\mathbf{A},</math>
where '''A''' (and its infinitesimal) is the [[vector area]]{{snd}} combination <math>\mathbf{A} = A \mathbf{\hat{n}}</math> of the magnitude of the area ''A'' through which the property passes and a [[unit vector]] <math>\mathbf{\hat{n}}</math> normal to the area.
Unlike in the second set of equations, the surface here need not be flat.

Finally, we can integrate again over the time duration ''t''<sub>1</sub> to ''t''<sub>2</sub>, getting the total amount of the property flowing through the surface in that time (''t''<sub>2</sub>&nbsp;−&nbsp;''t''<sub>1</sub>):
<math display="block">q = \int_{t_1}^{t_2}\iint_S \mathbf{j}\cdot d\mathbf A\, dt.</math>

=== Transport fluxes ===
Eight of the most common forms of flux from the transport phenomena literature are defined as follows: {{Citation needed|date=May 2025}}
# [[Transport phenomena#Momentum transfer|Momentum flux]], the rate of transfer of [[momentum]] across a unit area (N·s·m<sup>−2</sup>·s<sup>−1</sup>). ([[Newton's law of viscosity]])<ref name="Physics P.M">{{cite book|title=Essential Principles of Physics |author1=P.M. Whelan |author2=M.J. Hodgeson |edition=2nd|year=1978|publisher=John Murray|isbn=0-7195-3382-1}}</ref>
# [[Heat flux]], the rate of [[heat]] flow across a unit area (J·m<sup>−2</sup>·s<sup>−1</sup>). ([[Fourier's law|Fourier's law of conduction]])<ref>{{cite book | last=Carslaw | first=H.S. |author2=Jaeger, J.C. | title=Conduction of Heat in Solids  | edition=Second | year=1959 | publisher=Oxford University Press | isbn=0-19-853303-9 }}</ref> (This definition of heat flux fits Maxwell's original definition.)<ref name="Maxwell" />
# [[Diffusion flux]], the rate of movement of molecules across a unit area (mol·m<sup>−2</sup>·s<sup>−1</sup>). ([[Fick's law of diffusion]])<ref name="Physics P.M" />
# [[Volumetric flux]], the rate of [[volume]] flow across a unit area (m<sup>3</sup>·m<sup>−2</sup>·s<sup>−1</sup>). ([[Darcy's law|Darcy's law of groundwater flow]])
# [[Mass flux]], the rate of [[mass]] flow across a unit area (kg·m<sup>−2</sup>·s<sup>−1</sup>). (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 transferred in the form of [[photons]] at a certain distance from the source per unit area per second (J·m<sup>−2</sup>·s<sup>−1</sup>). Used in astronomy to determine the [[Magnitude (astronomy)|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 electromagnetic spectrum.
# [[Energy flux]], the rate of transfer of [[energy]] through a unit area (J·m<sup>−2</sup>·s<sup>−1</sup>). The radiative flux and heat flux are specific cases of energy flux.
# [[Particle flux]], the rate of transfer of particles through a unit area ([number of particles] m<sup>−2</sup>·s<sup>−1</sup>)

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 ====
As mentioned above, chemical [[mass flux#Molar fluxes|molar flux]] of a component A in an [[isothermal]], [[Isobaric process|isobaric system]] is defined in [[Fick's law of diffusion]] as:
<math display="block">\mathbf{J}_A = -D_{AB} \nabla c_A</math>
where the [[nabla symbol]] ∇ denotes the [[gradient]] operator, ''D<sub>AB</sub>'' is the diffusion coefficient (m<sup>2</sup>·s<sup>−1</sup>) of component A diffusing through component B, ''c<sub>A</sub>'' is the [[concentration]] ([[mole (unit)|mol]]/m<sup>3</sup>) of component A.<ref>{{cite book | last=Welty |author2=Wicks, Wilson and Rorrer | year=2001 | title=Fundamentals of Momentum, Heat, and Mass Transfer | edition=4th | publisher=Wiley | isbn=0-471-38149-7 }}</ref>

This flux has units of mol·m<sup>−2</sup>·s<sup>−1</sup>, and fits Maxwell's original definition of flux.<ref name="Maxwell">{{cite book | last=Maxwell | first=James Clerk| author-link=James Clerk Maxwell | year=1892 | title=Treatise on Electricity and Magnetism | publisher=Courier Corporation| isbn=0-486-60636-8}}</ref>

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 (physics)|cross section]] <math>\sigma</math>, and the [[Thermodynamic temperature|absolute temperature]] ''T'' by
<math display="block">D = \frac{2}{3 n\sigma}\sqrt{\frac{kT}{\pi m}}</math>
where the second factor is the [[mean free path]] and the square root (with the [[Boltzmann constant]] ''k'') is the [[Maxwell–Boltzmann distribution#Typical speeds|mean velocity]] of the particles.

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

=== Quantum mechanics ===
{{Main|Probability current}}

In [[quantum mechanics]], particles of mass ''m'' in the [[quantum state]] ''ψ''('''r''', ''t'') have a [[probability amplitude|probability density]] defined as
<math display="block">\rho = \psi^* \psi = |\psi|^2. </math>
So the probability of finding a particle in a differential [[volume element]] d<sup>3</sup>'''r''' is
<math display="block"> dP = |\psi|^2 \, d^3\mathbf{r}. </math>
Then the number of particles passing perpendicularly through unit area of a [[Cross section (geometry)|cross-section]] per unit time is the probability flux;
<math display="block">\mathbf{J} = \frac{i \hbar}{2m} \left(\psi \nabla \psi^* - \psi^* \nabla \psi \right). </math>
This is sometimes referred to as the probability current or current density,<ref>{{cite book |author=D. McMahon |url=https://archive.org/details/quantumfieldtheo0000mcma |title=Quantum Mechanics Demystified |publisher=Mc Graw Hill |year=2008 |isbn=978-0-07-145546-6 |edition=2nd |url-access=registration}}</ref> or probability flux density.<ref>{{cite book | author=Sakurai, J. J. | title=Advanced Quantum Mechanics | publisher=Addison Wesley | year=1967 | isbn=0-201-06710-2}}</ref>