# 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

[itex] \phi = A \cdot v \cdot \cos \theta. [itex]

In the special case where the flow is perpendicular to the area A (where θ = 0 and [itex] \cos \theta = 1 [itex]) then the flux is

[itex] \phi = A \cdot v. [itex]

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:

[itex] \phi = \iint_{S} \mathbf{v} \cdot d \mathbf{S} [itex]

where dS is a differential surface described by

[itex] d\mathbf{S} = \mathbf{n} \, dA, [itex]

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:

[itex]\iint_S\mathbf{v}\cdot d\mathbf{S}=\iiint_V\left(\nabla\cdot\mathbf{v}\right)dV.[itex]es:caudal

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