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# Volumetric flow rate

In fluid dynamics and hydrometry, the volumetric flow rate, also volume flow rate and rate of fluid flow, is the volume of fluid which passes through a given surface per unit time (for example cubic meters per second [m3 s-1] in SI units, or cubic feet per second [cu ft/s]). It is usually represented by the symbol Q. Volumetric flow rate should not be confused with volumetric flux, represented by the symbol q, with units of m3/(m2 s), that is, m s-1. The integration of a flux over an area gives the volumetric flow rate. Volumetric flow rate is also linked to viscosity.

Given an area A, and a fluid flowing through it with uniform velocity v with an angle θ away from the perpendicular to A, the flow rate is:

$Q = A \cdot v \cdot \cos \theta.$

In the special case where the flow is perpendicular to the area A, that is, θ = 0, the volumetric flow rate is:

$Q = A \cdot v.$

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:

$Q = \iint_{S} \mathbf{v} \cdot d \mathbf{S}$

where dS is a differential surface described by:

$d\mathbf{S} = \mathbf{n} \, dA$

with n the unit surface normal and dA the differential magnitude of the area.

If a surface S 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:

$\iint_S\mathbf{v}\cdot d\mathbf{S}=\iiint_V\left(\nabla\cdot\mathbf{v}\right)dV.$