Page 83 - Physical Principles of Sedimentary Basin Analysis

3.16 Euler and Lagrange coordinates 65

Lagrange Euler
a z

t t
Figure 3.16. (Left) A Lagrange coordinate system follows the movement of the “particles” of the
continuum. The particles are therefore at rest in the Lagrange coordinate system. (Right) An Euler
coordinate system is a fixed reference frame in which the “particles” move.

and it is normally expressed with reference to an arbitrary position x as

v = v(x, t) = V a(x, t), t . (3.139)
As already suggested, we will use an upper-case letter for a property calculated by
Lagrangian coordinates and the corresponding lower-case letter for the same property
calculated by Eulerian coordinates. As an example, we have

f (x, t) = F a(x, t), t and F(a, t) = f x(a, t), t (3.140)
where f and F represent the same property of the continuum, like for instance velocity,
temperature or density. It is important to distinguish between the velocity (or any other
property) calculated by the Lagrangian coordinates and the Eulerian coordinates, because
the functions like V and v are different. Both functions are expressions for the same veloc-
ity, but they take different arguments. The time derivative of a property is different in the
two coordinate systems. We see from equation (3.140) that
∂F(a, t) ∂ f (x, t) ∂ f (x, t) ∂x i (a, t)
= + (3.141)
∂t ∂t ∂x i ∂t
because the position x is not a constant of time. The right-hand side of equation (3.141)
accounts for the fact that we have to move along with the particle in order to measure the
time rate of change of a property of a specific particle. This derivative is called the material
derivative or convective derivative and it is often denoted
Df (x, t) ∂ f (x, t)
= + v(x, t) ·∇ f (x, t). (3.142)
dt ∂t
Notice from equation (3.141) that the material derivative in Eulerian coordinates is the
partial derivative in Lagrangian coordinates. In the next section we will need the material
derivative of the Jacobian connecting the Eulerian and Lagrangian coordinate systems. The
Jacobian is the following determinant:

∂x 1
··· ∂x 1
∂a 1 ∂a 3
··· (3.143)

J(a, t) =
∂x 3 ∂x 3

···

∂a 1 ∂a 3