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Def. Divergence of a vector. Let A(x, y, z) = A1 (x, y, z) i + A2 (x, y, z) j + A3 (x, y, z) k be a vector point function that is defined and differentiable at each point (x, y, z) in a certain region of space. The divergence of A is a scalar point function defined as
The divergence can also be written, using the del operator, as
Def. Flux strength. A measure of the flow of a fluid. It is equal to the amount of fluid passing through one square unit of cross-sectional area perpendicular to the direction of flow per unit of time.
Example. Water is flowing in a river at a rate of 3 feet per second. The amount passing through a one square foot cross-section perpendicular to the direction of flow is 3 cubic feet per second. The measure of water flow, viewed as flux, would then be given as 3 cu ft/sec per sq. ft.
Divergence. Physical interpretation. Let A be a vector point function defined throughout some region R of space. Then the divergence at any point P is given by
where A represents flux strength at point P, ΔV is the volume enclosed by the surface ΔS centered at P, n is the outward drawn unit normal at dS, and the limit is obtained by shrinking ΔV to point P. See Fig. 1 where ΔV is the volume of the depicted cube centered at P, ΔS is the surface of the cube, and ds is a differential element on ΔS. Physically
represents the net flux outflow per unit volume of the flux A from the surface ΔS. The quantity A∙n ds measures the rate of flow of the flux through the surface element dS.
If the integral
vanishes, there is no net outward flow. In this case any outward flux of A over part of the surface is balanced by an equal inward flux flow over the rest of the surface. If the integral is positive, the flux flow out of ΔV exceeds that into ΔV, and one says that there are sources within the volume. A negative value of the integral indicates the existence within ΔV of sinks, or points at which flux is being destroyed. If there are both sources and sinks within ΔV, the integral gives a measure of their algebraically combined strength. Thus div A can be taken as a measure of the source or sink strength at point P. If in a region there are no sources or sinks, then div A = 0 and we call A a solenoidal vector field.
Equation 2) can be taken as a definition of the divergence of A , and all the properties may be derived from it.
Continuity equation of an incompressible fluid. The equation
div A = 0
is called the continuity equation of an incompressible fluid. It states the condition that fluid is neither created nor destroyed.
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