Page 157 - Vector Analysis

§5.2 Eulerian and Lagrangian Coordinates 153

Let f (ψ(y, t), t) = F (y, t), A = (∇ψ)´1, and J = det(∇ψ). By (1.3) and (5.1), we find that

d ż f (x, t) dx = ż [ t), t) + ψt(y, t) ¨ (∇xf )(ψ(y, t), ] t)dy
dt ft(ψ(y, t) J(y,
Ω(t) Ω

nż

ÿ F (y, t)(JAijψti,j)(y, t) dy
+

i,j=1 Ω

ż nż[ ]
ÿ ψtiAijF,j J + F JAji ψti,j (y, t) dy
= ft(ψ(y, t), t)dy +

Ω i,j=1 Ω

ż nż (JAij ψtiF ) dy,
= (ft ˝ ψ)Jdy + ÿ ,j

Ω i,j=1 Ω

where the Piola identity (2.6) is used to conclude the last equality. The divergence theorem
then implies that

d ż f (x, t) dx = ż ˝ ψ)J dy + n ż JAji NjψtiF dSy .
dt
Ω(t) (ft ÿ B Ω

Ω i,j=1

As a consequence, changing back to the variable x on the right-hand side, by (5.3) and (5.4)

we conclude that

dż ż nż
ÿ

dt f (x, t) dx = ft(x, t) dx + (σf )(x, t) dSx . ˝

Ω(t) Ω(t) i,j=1 B Ω(t)

5.2 Eulerian and Lagrangian Coordinates

We have seen that the diffeomorphism ψ : Ω Ñ Ω(t) plays an important role in the Reynolds

transport theorem Theorem 5.1. In fluid dynamics, if the fluid domain is carried by the fluid

velocity; that is, the boundary of the fluid domain moves along with the fluid velocity, then

there is a natural map with domain Ω and range Ω(t), and we focus a little bit on this map

in this sub-section.

Let Ω(t) Ď Rn be a (time dependent) domain, and u(¨, t) : Ω(t) Ñ Rn be a smooth vector

field. We say that B Ω(t) moves along with u if any smooth curve ␣x(t) P Rn ˇ t P [0, T ](
ˇ

satisfying x(t) P B Ω(t) also satisfies that

( ) ( )( ) (5.6)
x1(t) ¨ n x(t), t = u x(t), t ¨ n x(t), t ,

where n again denotes the outward-pointing unit normal of Ω(t). We remark that using the
notation Ω(t), we include the possibility that the fluid domain may vary in time, while in a