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Flux: General Concepts
Synthesis from Sources
Attributions to Come
Introduction [1]
In the various subfields of physics, there exist two common usages of the term flux, each with rigorous mathematical frameworks. A simple and ubiquitous concept throughout physics and applied mathematics is the flow of a physical property in space, frequently also with time variation. It is the basis of the field concept in physics and mathematics, with two principal applications: in transport phenomena and surface integrals. The terms "flux", "current", "flux density", "current density", can sometimes be used interchangeably and ambiguously, though the terms used below match those of the contexts in the literature.
Origin of the Term
The word flux comes from Latin: fluxus means "flow", and fluere is "to flow". As fluxion, this term was introduced into differential calculus by Isaac Newton.
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 dimensions [quantity]·[time]−1·[area]−1. For example, the magnitude of a river's current, i.e. the amount of water that flows through a cross-section of the river each second, or the amount of sunlight that lands on a patch of ground each second is also a kind of flux.
Flux F through a surface, dS is the differential vector area element, n is the unit normal to the surface. Left: No flux passes in the surface, the maximum amount flows normal to the surface. Right: The reduction in flux passing through a surface can be visualized by reduction in F or dSequivalently (resolved into components, θ is angle to normal n). F·dS is the component of flux passing though the surface, multiplied by the area of the surface (see dot product). For this reason flux represents physically a flow per unit area.
General Mathematical Definition (Transport)
In this definition, flux is generally a vector due to the widespread and useful definition of vector area, although there are some cases where only the magnitude is important (like in number fluxes, see below). The frequent symbol is j (or J), and a definition for scalar flux of physical quantity q is the limit. The area required to calculate the flux is real or imaginary, flat or curved, either as a cross-sectional area or a surface. The vector area is a combination of the magnitude of the area through which the mass passes through, A, and a unit vector normal to the area.
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.
—James Clerk Maxwell
Transportation Fluxes
Eight of the most common forms of flux from the transport phenomena literature are defined as follows
