Rate of fluid flow - Definition and Overview

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

<math> \phi = A v <math>.

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

<math> \phi = A (\cos \theta) 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>