Rate of fluid flow
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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
- <math> \phi = A \cdot v \cdot \cos \theta. <math>
In the special case where the flow is perpendicular to the area A (where θ = 0 and <math> \cos \theta = 1 <math>) then the flux is
- <math> \phi = A \cdot v. <math>
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:
- <math> \phi = \iint_{S} \mathbf{v} \cdot d \mathbf{S} <math>
where dS is a differential surface described by
- <math> d\mathbf{S} = \mathbf{n} \, dA, <math>
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:
- <math>\iint_S\mathbf{v}\cdot d\mathbf{S}=\iiint_V\left(\nabla\cdot\mathbf{v}\right)dV.<math>es:caudal