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. 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) s of a
vector field through surfaces with
unit normal , the angle from to is . Flux is a measure of how much of the field passes through a given surface. is decomposed into components perpendicular (⊥) and parallel to . 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.
Top: Three field lines through a plane surface, one normal to the surface, one parallel, and one intermediate.
Bottom: Field line through a
curved surface, showing the setup of the unit normal and surface element to calculate flux. of with the unit normal vector
(blue arrows) at the point multiplied by the area . The sum of 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: j = \frac{I}{A}, where I = \lim_{\Delta t \to 0}\frac{\Delta q}{\Delta t} = \frac{\mathrm{d}q}{\mathrm{d}t}. 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: j(\mathbf{p}) = \frac{\partial I}{\partial A}(\mathbf{p}), I(A,\mathbf{p}) = \frac{\mathrm{d}q}{\mathrm{d}t}(A, \mathbf{p}). 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: \mathbf{j}(\mathbf{p}) = \frac{\partial \mathbf{I}}{\partial A}(\mathbf{p}), \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}}). 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 \mathbf{\hat{n}}), 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 "argmax" cannot directly compare vectors; we take the vector with the biggest norm instead.)
Properties These direct definitions can be difficult to apply directly in practice. For example, the arg max construction does not correspond straightforwardly to 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 \mathbf{\hat{n}}, then the
dot product \mathbf{j} \cdot \mathbf{\hat{n}} = j\cos\theta. That is, the component of flux passing through the surface (i.e. normal to it) is
jcos
θ, while the component of flux passing tangential to the area is
jsin
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 S, gives the proper flowing per unit of time through the surface: \frac{\mathrm{d}q}{\mathrm{d}t} = \iint_S \mathbf{j} \cdot \mathbf{\hat{n}}\, dA = \iint_S \mathbf{j} \cdot d\mathbf{A}, where
A (and its infinitesimal) is the
vector area combination \mathbf{A} = A \mathbf{\hat{n}} of the magnitude of the area
A through which the property passes and a
unit vector \mathbf{\hat{n}} 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
t1 to
t2, getting the total amount of the property flowing through the surface in that time (
t2 −
t1): q = \int_{t_1}^{t_2}\iint_S \mathbf{j}\cdot d\mathbf A\, dt.
Transport fluxes Eight of the most common forms of flux from the transport phenomena literature are defined as follows: •
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) (This definition of heat flux fits Maxwell's original definition.) This flux has units of mol·m−2·s−1, and fits Maxwell's original definition of flux. 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{2}{3 n\sigma}\sqrt{\frac{kT}{\pi m}} where the second factor is the
mean free path and the square root (with the
Boltzmann 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
quantum state ψ(
r,
t) have a
probability density defined as \rho = \psi^* \psi = |\psi|^2. So the probability of finding a particle in a differential
volume element d3
r is dP = |\psi|^2 \, d^3\mathbf{r}. Then the number of particles passing perpendicularly through unit area of a
cross-section per unit time is the probability flux; \mathbf{J} = \frac{i \hbar}{2m} \left(\psi \nabla \psi^* - \psi^* \nabla \psi \right). This is sometimes referred to as the probability current or current density, or probability flux density. == Flux as a surface integral ==