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 through it with uniform velocity v with an angle θ (away from the perpendicular), then the flux is
: In the special case where the flow is perpendicular to the area A (where θ = 0 and ) 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
:
with n the unit vector normal to the surface and dA the differential magnitude of the area. If we have a surface S which encloses a volume V, the divergence theorem states that the rate of fluid flow through the surface is the integral of the divergence of the velocity vector field v on that volume:
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