Page 158 - Vector Analysis

154 CHAPTER 5. Additional Topics

lot of applications, Ω ” Ω(0) = Ω(t) for all t. Now suppose that B Ω(t) moves along with u.
Let η : Ω Ñ Rn be the unique solution to the ODE

() @ α P Ω , t P (0, T ) , (5.7a)
ηt(α, t) = u η(α, t), t @ α P Ω, (5.7b)
η(α, 0) = α

here we assume that the solution exists up to time T . The value of x = η(α, t) is the location
of the fluid particle at time t which is initially positioned at α P Ω. By (5.6), we must have
η(Ω, t) = Ω(t).

A time independent coordinate system used in the co-domain of η is called the Eulerian
coordinate. We note that since in general Ω(t) varies continuously in time, the Eulerian
coordinate is usually defined on a subset of Rn larger than Ω(t). In fact, the Cartesian
coordinate is one of the most important Eulerian coordinate system since Ω(t) Ď Rn for all
t ą 0. On the contrary, the coordinate used in the domain of η is called the Lagrangian
coordinate. Since the Lagrangian coordinate is used to identify the initial position of fluid
particles, it is often called the material coordinate as well. In short, the Eulerian coordinate
is used to describe the (larger) background space (so each x corresponds to a point in space
which might not a point in the fluid), while the Lagrangian coordinate is used to describe
the particle in the fluid (so each α corresponds to a particle in the fluid).

Let us explain what these two coordinate systems are doing. Suppose that a kind of
censor (whose volume and mass are both zero in the mathematical setting so that it does
not affect any physics) is designed to measure certain physical quantity. The censor can be
fixed at a point x in space so that the readings indicate the value of that physical quantity
at x for various time. On the other hand, we may set the censor to flow with the fluid
(the fluid will carry the censor). If the censor initially is position at a given point α, then
the readings of the censor indicate the value of the quantity at the particle which initially
locates at position α. In other words, a function with variables in Lagrangian coordinate is
a function defined on material particles inside the fluids, while a function with variables in
Eulerian coordinate is a function defined on space.

Theorem 5.2. Let u : Ω(t) ˆ (0, T ) Ñ Rn be a smooth vector field, and the flow map η(¨, t) :

" Ω Ñ Ω(t) be defined by (5.7). Then u is divergence-free if and only if det(∇η) ” 1 for
α ÞÑ η(α, t)

all t ą 0.