Chemistry - Rate of fluid flow

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 through it with uniform velocity v with an angle θ (away from the perpendicular), then the flux is

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

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

$\phi = A \cdot v.$

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:

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

where dS is a differential surface described by

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

with n the unit 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:

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

Categories: Fluid dynamics