3.2 Motion of a Continuum
61
Consider first the scalar quantity φ(R, t) = φ(r, t) as expressed in either material
or spatial coordinates. The partial derivative ∂φ(R, t)/∂t represents the rate of
change of φ following a particle, i.e., R is constant but r is time dependent. In
contrast, ∂φ(r, t)/∂t gives the rate of change in φ at a given point in space, i.e., r is
constant but R is time dependent.
These considerations lead immediately to the equations used to compute time
derivatives following a particle (fixed R). For the material description, we have
d
dt
φ(R, t) =
∂
∂t
φ(R, t).
(3.22)
For the spatial description, since r depends on time, the chain rule yields
d
dt
φ(r, t) =
∂
∂t
φ(r, t) +
∂r
∂t
·
∂
∂r
φ(r, t),
(3.23)
which follows from Eq. (2.56).
As in Eq. (2.66) 1 , the derivative ∂/∂r can be replaced by a gradient operator. We
define the two forms
∇ =
∂
∂R
,
∇ =
∂
∂r
,
(3.24)
which represent gradient operators in material and spatial coordinates, respectively.
Only the second form is used here; the first will be used later in this chapter. With
the velocity v = ∂r/∂t, Eq. (3.23) can be written as
d
dt
φ(r, t) =
∂
∂t
φ(r, t) + v · ∇φ(r, t).
(3.25)
This equation defines the material time derivative of φ, with the operator given
by 2
d
dt
=
∂
∂t
+ v · ∇.
(3.26)
Note that if φ is expressed in the material form φ(R, t), then ∇φ = 0 and dφ/dt
reduces to Eq. (3.22). Thus, the operator d/dt is valid for both the spatial and
material descriptions. In addition, please note that an overdot denotes d/dt and not
∂/∂t, i.e., ˙
u = du/dt.
Since it is written in direct notation, Eq. (3.26) is valid for any coordinate system.
To obtain the appropriate form in Cartesian coordinates, we set v = v i e i and, with
ds i = dx i , Eq. (2.66) gives ∇ = e i ∂/∂x i . Substituting these relations into (3.25)
yields
2 Some authors use the alternative notation D/Dt for the material time derivative to emphasize that
the derivative follows the motion of a particle in a continuum.
61
Consider first the scalar quantity φ(R, t) = φ(r, t) as expressed in either material
or spatial coordinates. The partial derivative ∂φ(R, t)/∂t represents the rate of
change of φ following a particle, i.e., R is constant but r is time dependent. In
contrast, ∂φ(r, t)/∂t gives the rate of change in φ at a given point in space, i.e., r is
constant but R is time dependent.
These considerations lead immediately to the equations used to compute time
derivatives following a particle (fixed R). For the material description, we have
d
dt
φ(R, t) =
∂
∂t
φ(R, t).
(3.22)
For the spatial description, since r depends on time, the chain rule yields
d
dt
φ(r, t) =
∂
∂t
φ(r, t) +
∂r
∂t
·
∂
∂r
φ(r, t),
(3.23)
which follows from Eq. (2.56).
As in Eq. (2.66) 1 , the derivative ∂/∂r can be replaced by a gradient operator. We
define the two forms
∇ =
∂
∂R
,
∇ =
∂
∂r
,
(3.24)
which represent gradient operators in material and spatial coordinates, respectively.
Only the second form is used here; the first will be used later in this chapter. With
the velocity v = ∂r/∂t, Eq. (3.23) can be written as
d
dt
φ(r, t) =
∂
∂t
φ(r, t) + v · ∇φ(r, t).
(3.25)
This equation defines the material time derivative of φ, with the operator given
by 2
d
dt
=
∂
∂t
+ v · ∇.
(3.26)
Note that if φ is expressed in the material form φ(R, t), then ∇φ = 0 and dφ/dt
reduces to Eq. (3.22). Thus, the operator d/dt is valid for both the spatial and
material descriptions. In addition, please note that an overdot denotes d/dt and not
∂/∂t, i.e., ˙
u = du/dt.
Since it is written in direct notation, Eq. (3.26) is valid for any coordinate system.
To obtain the appropriate form in Cartesian coordinates, we set v = v i e i and, with
ds i = dx i , Eq. (2.66) gives ∇ = e i ∂/∂x i . Substituting these relations into (3.25)
yields
2 Some authors use the alternative notation D/Dt for the material time derivative to emphasize that
the derivative follows the motion of a particle in a continuum.