Knowlpedia

Definition

Transport equation

An equation prescribing the material derivative of a scalar field along a given velocity.

A transport equation for a scalar a(t,x)a(t,x), with prescribed velocity b(t,x)b(t,x) and source g(t,x)g(t,x), is

∂ta+b⋅∇a=g.\partial_ta+b\cdot\nabla a=g.

For sufficiently regular fields and a satisfying X′=b(t,X)X'=b(t,X), it becomes

ddta(t,X(t))=g(t,X(t)).\frac{d}{dt}a(t,X(t))=g(t,X(t)).

Thus source-free transport preserves the scalar value along each trajectory.

Density transport

The conservative density equation is ∂ta+∇⋅(ab)=g\partial_ta+\nabla\cdot(ab)=g. It equals the displayed transport equation when ∇⋅b=0\nabla\cdot b=0; otherwise it has the additional term a∇⋅ba\nabla\cdot b.

Adding νΔa\nu\Delta a on the right gives an advection-diffusion equation. Transport and diffusion then act simultaneously.

References