# Volumetric flow rate

**Editor-In-Chief:** C. Michael Gibson, M.S., M.D. [1]

## Overview

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 [m^{3} 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 m^{3}/(m^{2} 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:

- <math> Q = A \cdot v \cdot \cos \theta. </math>

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

- <math> Q = 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> Q = \iint_{S} \mathbf{v} \cdot d \mathbf{S} </math>

where *d***S** is a differential surface described by:

- <math> d\mathbf{S} = \mathbf{n} \, dA </math>

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:

- <math>\iint_S\mathbf{v}\cdot d\mathbf{S}=\iiint_V\left(\nabla\cdot\mathbf{v}\right)dV.</math>

## See also

- Air to cloth ratio
- Discharge (hydrology)
- Flowmeter
- Flux (transport definition)
- Mass flow rate
- Poiseuille's law
- Darcy's law
- Orifice plate

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