The third partial derivative is the local rate of
change of velocity (7.61), which may be calculated
from (7.60). The fourth partial derivative evaluates
to one.
Using the above interpretations for the partial
derivatives in (7.64) and rearranging them, we
have an expression for the particle acceleration
written in terms of the local velocity using a
spatial description of motion:
(7.65)
The first term on the right-hand side is the local
rate of change of velocity at the current position
x. What distinguishes this term from the particle
acceleration at that position is the second term,
which is a product of the local velocity and the
spatial derivative of velocity at the current time t.
The second term may be interpreted as the rate of
change of velocity due to the flow of material at
velocity v through a spatially varying velocity
field. For steady flow, such as that within the
rising viscous sphere (Fig. 5.12), the first term on
the right-hand side of (7.65) is zero, so particle
accelerations are entirely due to the spatial variations in velocity as described by the second term.
For example, the x-component of acceleration
from (7.65) is written:
(7.66)
The components a y and a z follow by change of subscripts. Here is it understood that the velocity
components are known functions of the current
position and time, v ϭ g(x, t), so the partial derivatives are taken with the appropriate independent
variables held constant as indicated in (7.65).
The operation characterized in (7.65) for calculating the particle acceleration from the local
velocity may be generalized to calculate the
material time derivative of any quantity associated
with the material, given a spatial description of
its kinematics:
(7.67)
The operator D/Dt is also referred to as the substantial derivative, but material time derivative is more
descriptive of its role. For example in the context
D
Dt
ϭ
Ѩ
Ѩt | X
ϭ
Ѩ
Ѩt | x
ϩ v
Ѩ
Ѩx | t
a x ϭ
Ѩv x
Ѩt
ϩ v x
Ѩv x
Ѩx
ϩ v y
Ѩv x
Ѩy
ϩ v z
Ѩv x
Ѩz
a ϭ
Ѩv
Ѩt | x
ϩ v
Ѩv
Ѩx | t
of fluid mechanics, as developed using the spatial
description of motion, the operator (7.67) is used to
calculate the rate of change of material properties
that play roles in the fundamental principles of
conservation of mass and momentum. Apparently
the concepts embodied in (7.67) can be traced back
to publications of Euler (Fig. 7.16) in 1770 and
Lagrange (Fig. 7.14) in 1783 (Malvern, 1969).
The material time derivative operator (7.67)
may be applied to scalar, vector (7.65), or tensor
functions that describe a property of the material
in terms of the spatial coordinates, x, and time, t
(Malvern, 1969). Suppose the mass density is
known as ϭ (x, t). The material time derivative
of this scalar quantity is:
(7.68)
The first term on the right-hand side of (7.68)
describes the local rate of change of density at the
current position x. The second term describes the
rate of change of density at the current time t due
to the flow of material at velocity v with a spatially
varying density field. In terms of the velocity components (7.68) is:
(7.69)
Here is it understood that the velocity components are known functions of the current position
and time (7.60). Furthermore, the mass density is
a known function of the current position and
time, so the partial derivatives are taken with the
appropriate independent variables held constant.
Other scalar properties of the material are operated upon and interpreted similarly.
7.3.2 Conservation of mass: the
equation of continuity
What constraints must be imposed to assure that
the relative motions of particles in a deforming
rock mass obey the fundamental law of mass conservation? To address this question we adopt the
spatial description of motion and consider a fixed
volume element within a deforming material continuum (Fig. 7.17). The center of the element is at
the arbitrary point specified by the position vector
x with components (x, y, z) which are the current
coordinates. The sides of the element are parallel
D
Dt
ϭ
Ѩ
Ѩt
ϩ v x
Ѩ
Ѩx
ϩ v y
Ѩ
Ѩy
ϩ v z
Ѩ
Ѩz
D
Dt
ϭ
Ѩ
Ѩt | X
ϩ v
Ѩ
Ѩx | t
266
CONSERVATION OF MASS AND MOMENTUM
change of velocity (7.61), which may be calculated
from (7.60). The fourth partial derivative evaluates
to one.
Using the above interpretations for the partial
derivatives in (7.64) and rearranging them, we
have an expression for the particle acceleration
written in terms of the local velocity using a
spatial description of motion:
(7.65)
The first term on the right-hand side is the local
rate of change of velocity at the current position
x. What distinguishes this term from the particle
acceleration at that position is the second term,
which is a product of the local velocity and the
spatial derivative of velocity at the current time t.
The second term may be interpreted as the rate of
change of velocity due to the flow of material at
velocity v through a spatially varying velocity
field. For steady flow, such as that within the
rising viscous sphere (Fig. 5.12), the first term on
the right-hand side of (7.65) is zero, so particle
accelerations are entirely due to the spatial variations in velocity as described by the second term.
For example, the x-component of acceleration
from (7.65) is written:
(7.66)
The components a y and a z follow by change of subscripts. Here is it understood that the velocity
components are known functions of the current
position and time, v ϭ g(x, t), so the partial derivatives are taken with the appropriate independent
variables held constant as indicated in (7.65).
The operation characterized in (7.65) for calculating the particle acceleration from the local
velocity may be generalized to calculate the
material time derivative of any quantity associated
with the material, given a spatial description of
its kinematics:
(7.67)
The operator D/Dt is also referred to as the substantial derivative, but material time derivative is more
descriptive of its role. For example in the context
D
Dt
ϭ
Ѩ
Ѩt | X
ϭ
Ѩ
Ѩt | x
ϩ v
Ѩ
Ѩx | t
a x ϭ
Ѩv x
Ѩt
ϩ v x
Ѩv x
Ѩx
ϩ v y
Ѩv x
Ѩy
ϩ v z
Ѩv x
Ѩz
a ϭ
Ѩv
Ѩt | x
ϩ v
Ѩv
Ѩx | t
of fluid mechanics, as developed using the spatial
description of motion, the operator (7.67) is used to
calculate the rate of change of material properties
that play roles in the fundamental principles of
conservation of mass and momentum. Apparently
the concepts embodied in (7.67) can be traced back
to publications of Euler (Fig. 7.16) in 1770 and
Lagrange (Fig. 7.14) in 1783 (Malvern, 1969).
The material time derivative operator (7.67)
may be applied to scalar, vector (7.65), or tensor
functions that describe a property of the material
in terms of the spatial coordinates, x, and time, t
(Malvern, 1969). Suppose the mass density is
known as ϭ (x, t). The material time derivative
of this scalar quantity is:
(7.68)
The first term on the right-hand side of (7.68)
describes the local rate of change of density at the
current position x. The second term describes the
rate of change of density at the current time t due
to the flow of material at velocity v with a spatially
varying density field. In terms of the velocity components (7.68) is:
(7.69)
Here is it understood that the velocity components are known functions of the current position
and time (7.60). Furthermore, the mass density is
a known function of the current position and
time, so the partial derivatives are taken with the
appropriate independent variables held constant.
Other scalar properties of the material are operated upon and interpreted similarly.
7.3.2 Conservation of mass: the
equation of continuity
What constraints must be imposed to assure that
the relative motions of particles in a deforming
rock mass obey the fundamental law of mass conservation? To address this question we adopt the
spatial description of motion and consider a fixed
volume element within a deforming material continuum (Fig. 7.17). The center of the element is at
the arbitrary point specified by the position vector
x with components (x, y, z) which are the current
coordinates. The sides of the element are parallel
D
Dt
ϭ
Ѩ
Ѩt
ϩ v x
Ѩ
Ѩx
ϩ v y
Ѩ
Ѩy
ϩ v z
Ѩ
Ѩz
D
Dt
ϭ
Ѩ
Ѩt | X
ϩ v
Ѩ
Ѩx | t
266
CONSERVATION OF MASS AND MOMENTUM