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 of a given quantity that flows through a unit area per unit time¹. Flux in this definition is a vector.

In the field of Electromagnetism , flux is usually the integral of a Vector quantity over a finite surface. The result of this integration is a scalar quantity². The Magnetic Flux is thus the integral of the magnetic vector field 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, above³ p.357. It has units of watts/(meter)².

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 mathematically defined definitions, there are many more loose usages found in fields such as biology.

TRANSPORT PHENOMENA

Flux definition and theorems

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. Five of the most common forms of flux from the transport literature are defined as:

#''Momentum flux'', the rate of change of Momentum moving across a unit area (N·s·m^-2·s^-1). (Newtonian fluid, Viscous Flow )
#''Heat flux'', the rate of Heat flow across a unit area (J·m^-2·s^-1). ( Fourier's Law )
#''Chemical flux'', the rate of movement of Mole s across a unit area (mol·m^-2·s^-1). ( Fick's Law Of Diffusion )
#''Mass flux'', the rate of Mass flow across a unit area (kg·m^-2·s^-1). (An alternate form of Fick's law that includes the grams per mole term to convert moles to mass)
#''Volumetric flux'', the rate of Volume flow across a unit area (m³·m^-2·s^-1). ( Darcy's Law )

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.

The fundamental laws that govern this process include:

Newton's law of viscosity

Fourier's law of convection

Fick's law of diffusion.

Darcy's law of groundwater flow

Thermal systems

In thermal systems, the flux is the Rate Of Heat Flow per area per time (J·m^-2·s^-1). ⁵ This definition of heat flux fits Maxwell's original definition⁶.

Chemical diffusion

Flux, or diffusion, for gaseous molecules can be related to the Function :

:

\Phi = 4\pi\sigma_{ab}^2\sqrt{rac{8kT}{\pi N}}

where ''N'' is the total number of gaseous particles, ''k'' is Boltzmann's constant, ''T'' is the relative temperature in kelvins, and

\sigma_{ab}

is the mean free path between the molecules ''a'' and ''b''.

Chemical molar flux of a component A in an Isothermal , Isobaric System is also defined in Ficks's First Law as:

:

\overrightarrow{J_A} = -D_{AB} 
abla c_A

where

D_{AB}

is the molecular diffusion coefficient (m²/s) of component A diffusing through component B, and

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 flux⁸.

Note:

abla

(" Nabla ") denotes the Del operator.

Quantum mechanics

See Also: Probability current

In Quantum Mechanics , Particles of mass m in the state

\psi(r,t)

have a probability density defined as

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Flux