In fluid dynamics, the volumetric flow rate, also volume flow rate and rate of fluid flow, is the volume of fluid which passes through a given area per unit time. It is also called flux. It is usually represented by the symbol Q.

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 flux is:

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

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

$Q\; =\; A\; \backslash cdot\; v.$

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:

$Q\; =\; \backslash iint\_\{S\}\; \backslash mathbf\{v\}\; \backslash cdot\; d\; \backslash mathbf\{S\}$

where dS is a differential surface described by:

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

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:

$\backslash iint\_S\backslash mathbf\{v\}\backslash cdot\; d\backslash mathbf\{S\}=\backslash iiint\_V\backslash left(\backslash nabla\backslash cdot\backslash mathbf\{v\}\backslash right)dV.$

See also

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• Discharge (hydrology)
• Mass flow rate
• Flowmeter
• Orifice plate

Fluid dynamics

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