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. This article or section should be merged with Fluid mechanics Fluid dynamics is the study of fluids (liquids and gases) in motion, and the effect of the fluid motion on fluid boundaries, such as solid containers or other fluids. ... This article is in need of attention. ...

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 This article explains the meaning of area as a physical quantity. ... This article is about velocity in physics. ... A perpendicular line. ...

In the special case where the flow is perpendicular to the area A (where θ = 0 and cosθ = 1) then the flux is

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: In mathematics, a surface integral is a definite integral taken over some surface that may be a curved set in space; it can be thought of as the double integral analog of the path integral. ...

where dS is a differential surface described by

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: In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Ostrogradsky-Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ...