Referential Domain

At any location in space x and time t, there is one material point, identified by its space coordinates x at time t=0, and one grid point identified by its coordinates

ξ

at time t=0. Figure 1 provides a pictorial representation and defines the velocities in each formulation.

The derivative of any physical quantity can be computed either following the material point or following the grid point. They can then be related to each other.

Given that

F

is a function f of space and time representing a physical property:

The spatial domain is given by $f (x, t)$
The material domain is given by $f^{*} (X, t)$
The mixed domain is given by $f^{* *} (ξ, t)$

Therefore,(1)

Also:(2)

\frac{\partial f^{*}}{\partial t} |_{x} = \frac{\partial f^{* *}}{\partial t} |_{ξ} + (V_{j} - W_{j}) \frac{\partial f (x, t)}{\partial x_{j}}

This relates to acceleration by:(3)

\vec{γ} = \frac{d \vec{v}}{\partial t} |_{x} = \frac{\partial}{\partial t} \vec{v} |_{ξ} + (v_{j} - w_{j}) \frac{\partial}{\partial x} \vec{v} |_{t}

Where,

$ν$: Material velocity
$w$: Grid velocity