As described in the previous paragraph Material and Spatial Frames, lowercase letters are used to denote the spatial frame coordinates while uppercase letters denote the material frame coordinates. In the followings, a physical quantity
A will be referred to as
A(x, y, z) in the spatial frame and to as
A(X, Y, Z) in the material frame.
In these equalities, Ω0 and
Ω denote the same domain but represented in the material frame or the spatial frame, respectively. As expected, the same mass is found by integrating
ρ(X, Y, Z) over the domain in the material frame or by integrating
ρ(x, y, z) over the domain in the spatial frame. The same invariance principle applies to quantities per unit area, in particular for heat flux and heat transfer coefficients:
Here, ∂Ω0 and
∂Ω are the boundaries of the same domain in material and spatial frames, respectively.
Thermal conductivity, k, is a tensor density. The relationship between the value on the spatial frame and the material frame is:
and that the unit normal vector n verify that
n(X, Y, Z) and
FTn(x, y, z) are collinear, the total conductive heat flux through a boundary, computed in both frames according to the integrals below, gives the same result:
Here, ∂Ω0 and
∂Ω are the boundaries of the same domain in material and spatial frames, respectively.
where xTX corresponds to the tangential derivative
x with respect to
X, and so on.
where xTX corresponds to the tangential derivative
x with respect to
X, and so on. The
(nx, ny, nz) vector corresponds to the normal vector in the spatial frame, and the
(nX, nY, nZ) vector corresponds to the normal vector in the material frame.
The right-hand side of this relation shows a new term u ⋅ ∇ corresponding to convection in the case of fluids or convected quantity by translational motion in the case of a solid.