FLUX, VECTOR FLUX, SURFACE INTEGRAL REPRESENTING A SUMMATION OF FLUX PASSING THROUGH A CLOSED SURFACE

Def. Flux. A term which denotes the volume or mass of fluid or particles transferred across a given area perpendicular to the direction of flow in a given time.

There are many specific examples in science of the term flux. For electromagnetic radiation, it signifies the energy per unit time, or the power passing through a surface. For photons or particles, flux is the number per unit time passing through a surface.

                                                                                    The International Dictionary of Applied Mathematics

Def. Vector flux. If V describes a vector field, for example the velocity of an incompressible fluid, then the total flux through a surface S in the field is given by

The vector V may refer to electric, magnetic, or gravitational force; heat or a fluid, etc. The surface integral may be converted to a volume integral by Gauss’s Theorem.

                                                                 The International Dictionary of Applied Mathematics

Surface integral representing a sum of the flux passing through a closed surface. The surface integral

can be viewed as representing the sum of the flux passing through the walls of a closed surface. Let A(x, y, z) be a vector point function representing a flux field defined over some region Q of space. Let S be some closed surface within region Q. The flux could be that of a moving fluid such as water. The surface S could be a sphere constructed of wire mesh submerged in a river. The water passing through the walls of the mesh sphere would correspond to flux passing through a closed surface.

The flux strength at a point can be specified as a flow rate.

Def. Flow rate. The amount of fluid passing through a square unit of cross-sectional area perpendicular to the direction of flow per unit of time. Example: Three cubic feet of water pass through a one square foot cross-sectional area perpendicular to the direction of flow per second. The flow rate is three cu. ft. / sec / sq. ft.

The product of fluid velocity times a cross-sectional area perpendicular to the direction of flow gives the amount of fluid passing through the cross-sectional area per second.

Consider the flux A at some small surface element ΔS on the surface of S. A represents the amount of fluid that passes through one square unit of cross-section perpendicular to A in one second. See Fig. 1. The vector A can be written as A = |A|a where a is a unit vector in the direction of A and |A| is the magnitude of A. The amount of fluid passing through surface element ΔS is given by |A| ΔT where ΔT is the area of the projection of surface element ΔS on a plane perpendicular to A. See figures 2 and 3. ΔT is the area of the region ABCD shown in Fig. 3. ABCD represents the projection of the surface element ΔS on a plane through point A perpendicular to vector A. We will denote the area of element ΔS by the same name, ΔS. ΔT is related to the area ΔS by

            ΔT = n∙a ΔS .

In differential form this can be written

            dT = n∙a dS .                           

The sum of all the flux passing through the wall of the closed surface S is then given by