Rate of fluid flow

In fluid dynamics, the rate of fluid flow is the volume of fluid which passes through a given area per unit time. It is also called flux.

Given an area A, and a fluid flowing perpendicularly through it with uniform speed v, then the flux is

.

If the velocity of the fluid incides on the area with an angle θ (away from the perpendicular), then the flux is

.

If the velocity of the fluid through the area is non-uniform (or if the area is non-planar) then the rate of fluid flow can be calculated by means of a surface integral:

where dS is a differential surface described by

where n is a unit vector normal to the surface and dA is the differential magnitude of the area.

If the rate of fluid flow is to be defined for a differential area which encloses a volume (see Gaussian surface), then it can be shown that the rate of fluid flow out of the volume is the divergence of the velocity vector field at a point inside that volume:

.