components in all three coordinate directions.
The rate of increase of the momentum is:
(7.85)
This vector accounts for the left-hand side of
(7.84).
As material moves through the fixed element
(Fig. 7.19), momentum is carried in and out parallel to the three coordinate directions in proportion
to the respective velocity components. This transport of momentum is measured by the momentum
flux per unit volume, which is the product of the
velocity and the momentum, v(v). For example,
on the left-hand side of the element we have v x (v x ),
where and v x are evaluated at x – ␦x/2. This quantity times the area, ␦y␦z, is the momentum flux
through the left-hand side, v x (v x )␦y␦z. Similarly,
the momentum flux through the right-hand side is
v x (v x )␦y␦z, where and v x are evaluated at x ϩ␦x/2.
We account for the difference between the rate of
momentum in through the left side and the rate
out through the right-hand side as:
(7.86)
Because the velocity components v y and v z are parallel to these sides, they cannot contribute to this
part of the rate of momentum change. On the
other hand, v y can carry x momentum, v x
through the front and back sides of the element,
and v z can carry x momentum, v x through the
bottom and top of the element. The pairs of
arrows normal to the sides of the element in Fig.
7.19 are meant to represent these three fluxes of x
momentum across the element. The order of the
two velocity vector components is in keeping with
an “on–in” subscript convention: e.g. v y (v z )␦x␦z is
the momentum flux “on” a side with normal parallel to the y-coordinate of the momentum “in”
the z-direction.
To convert the finite difference in (7.86) to a
partial derivative a set of n elements (Fig. 7.19) is
considered with successively smaller volumes
that contain x and approach zero in the limit as n
→ ϱ so the elements converge on the central point
at x. In this limit the partial derivative of the
momentum flux, v x (v x ), with respect to x is:
v x ( v x ) | xϪ␦xր2 ␦y ␦z Ϫ v x ( v x ) | xϩ␦xր2 ␦y ␦z
Ѩ
Ѩt
(v)
(7.87)
Multiplying both sides of (7.87) by the volume and
comparing this to (7.86), we see how the negative
of the left-hand side of (7.87) accounts for one component of the rate of x-momentum change. The
components of the rate of x-momentum change
through the front and back of the element, and
through the bottom and top of the element, are
similarly defined so the net rate of change of x
momentum is:
(7.88)
Here it is understood that the derivatives are evaluated at the point x. The net rate of change of y
and z momentum each have three components
that are found using a similar procedure.
The one-dimensional relationship (7.88) for the
rate of change of momentum is generalized to
three dimensions using the differential operator
ٌ on the momentum flux, v(v). Because the
momentum flux is a product of two vectors,
referred to as a dyadic product, this operation is
somewhat different from that defined in (7.77).
The dyadic product is a special form of secondrank tensor with nine components. For example,
the dyadic product of the two arbitrary vectors, u
and w is (Bird et al., 1960):
(7.89)
Using the del operator (7.76) on this dyadic
product we have:
(7.90)
The rate of change of momentum per unit
volume is found by dividing (7.88) and its counϩ
Ѩu x w z
Ѩx
ϩ
Ѩu y w z
Ѩy
ϩ
Ѩu z w z
Ѩz e z
ϩ
Ѩu x wy
Ѩx
ϩ
Ѩuywy
Ѩy
ϩ
Ѩu z wy
Ѩz e y
ٌ · uw ϭ
Ѩu x w x
Ѩx
ϩ
Ѩu y w x
Ѩy
ϩ
Ѩu z w x
Ѩz
e x
uw ϭ
u x w x u x w y u x w z
u y w x u y w y u y w z
u z w x u z w y u z w z
Ϫ
Ѩ
Ѩz
v z ( v x )␦x ␦y ␦z
Ϫ
Ѩ
Ѩx
v x ( v x )␦y ␦z ␦x Ϫ
Ѩ
Ѩy
v y ( v x )␦x ␦z ␦y
Ѩ
Ѩx
v x ( v x ) ϭ
lim
n → ϱ
v x ( v x ) | xϩ␦xր2 Ϫ v x ( v x ) | xϪ␦xր2
␦x
270
CONSERVATION OF MASS AND MOMENTUM
The rate of increase of the momentum is:
(7.85)
This vector accounts for the left-hand side of
(7.84).
As material moves through the fixed element
(Fig. 7.19), momentum is carried in and out parallel to the three coordinate directions in proportion
to the respective velocity components. This transport of momentum is measured by the momentum
flux per unit volume, which is the product of the
velocity and the momentum, v(v). For example,
on the left-hand side of the element we have v x (v x ),
where and v x are evaluated at x – ␦x/2. This quantity times the area, ␦y␦z, is the momentum flux
through the left-hand side, v x (v x )␦y␦z. Similarly,
the momentum flux through the right-hand side is
v x (v x )␦y␦z, where and v x are evaluated at x ϩ␦x/2.
We account for the difference between the rate of
momentum in through the left side and the rate
out through the right-hand side as:
(7.86)
Because the velocity components v y and v z are parallel to these sides, they cannot contribute to this
part of the rate of momentum change. On the
other hand, v y can carry x momentum, v x
through the front and back sides of the element,
and v z can carry x momentum, v x through the
bottom and top of the element. The pairs of
arrows normal to the sides of the element in Fig.
7.19 are meant to represent these three fluxes of x
momentum across the element. The order of the
two velocity vector components is in keeping with
an “on–in” subscript convention: e.g. v y (v z )␦x␦z is
the momentum flux “on” a side with normal parallel to the y-coordinate of the momentum “in”
the z-direction.
To convert the finite difference in (7.86) to a
partial derivative a set of n elements (Fig. 7.19) is
considered with successively smaller volumes
that contain x and approach zero in the limit as n
→ ϱ so the elements converge on the central point
at x. In this limit the partial derivative of the
momentum flux, v x (v x ), with respect to x is:
v x ( v x ) | xϪ␦xր2 ␦y ␦z Ϫ v x ( v x ) | xϩ␦xր2 ␦y ␦z
Ѩ
Ѩt
(v)
(7.87)
Multiplying both sides of (7.87) by the volume and
comparing this to (7.86), we see how the negative
of the left-hand side of (7.87) accounts for one component of the rate of x-momentum change. The
components of the rate of x-momentum change
through the front and back of the element, and
through the bottom and top of the element, are
similarly defined so the net rate of change of x
momentum is:
(7.88)
Here it is understood that the derivatives are evaluated at the point x. The net rate of change of y
and z momentum each have three components
that are found using a similar procedure.
The one-dimensional relationship (7.88) for the
rate of change of momentum is generalized to
three dimensions using the differential operator
ٌ on the momentum flux, v(v). Because the
momentum flux is a product of two vectors,
referred to as a dyadic product, this operation is
somewhat different from that defined in (7.77).
The dyadic product is a special form of secondrank tensor with nine components. For example,
the dyadic product of the two arbitrary vectors, u
and w is (Bird et al., 1960):
(7.89)
Using the del operator (7.76) on this dyadic
product we have:
(7.90)
The rate of change of momentum per unit
volume is found by dividing (7.88) and its counϩ
Ѩu x w z
Ѩx
ϩ
Ѩu y w z
Ѩy
ϩ
Ѩu z w z
Ѩz e z
ϩ
Ѩu x wy
Ѩx
ϩ
Ѩuywy
Ѩy
ϩ
Ѩu z wy
Ѩz e y
ٌ · uw ϭ
Ѩu x w x
Ѩx
ϩ
Ѩu y w x
Ѩy
ϩ
Ѩu z w x
Ѩz
e x
uw ϭ
u x w x u x w y u x w z
u y w x u y w y u y w z
u z w x u z w y u z w z
Ϫ
Ѩ
Ѩz
v z ( v x )␦x ␦y ␦z
Ϫ
Ѩ
Ѩx
v x ( v x )␦y ␦z ␦x Ϫ
Ѩ
Ѩy
v y ( v x )␦x ␦z ␦y
Ѩ
Ѩx
v x ( v x ) ϭ
lim
n → ϱ
v x ( v x ) | xϩ␦xր2 Ϫ v x ( v x ) | xϪ␦xր2
␦x
270
CONSERVATION OF MASS AND MOMENTUM