10.1 The Macroscopic Description of Nonequilibrium States
443
one term in equations like Eq. (10.1-5) is nonzero. We assume that the time derivatives
of the temperature, pressure, and concentrations are dependent variables, so that it is
not necessary to specify the time derivatives to specify the state of the system.
E X A M P L E 10.1
Assume that the concentration of substance number 2 is represented by the function
c 2 c 2 (z, t) c 0 + acos(bz)e −t/τ
where c 0 , a, b, and τ are constants and where t represents the time. Write the expressions for
∇c 2 and for ∂c 2 /∂t.
Solution
∇c 2 −kabsin(bz)e −t/τ
∂c 2
∂t
atcos(bz)e −t/τ
where k is the unit vector in the z direction.
If a fluid system is flowing the flow velocity is also needed to specify the state of
the system. The flow velocity u is a vector quantity that depends on position and time:
u u(r, t) u(x, y, z, t) iu x (r, t) + ju y (r, t) + ku z (r, t)
(10.1-6)
Just as with the temperature, pressure, and concentration, we need a specification of
how strongly the flow velocity depends on position. Each term in Eq. (10.1-6) can be
differentiated with respect to x, y, and z, so the gradient of u has nine components. We
will avoid a full discussion of this gradient and focus on single terms. For example, the
derivative ∂u y /∂z gives the rate of change of the y component of the velocity in the z
direction. This quantity is called the rate of shear, or the rate at which one layer of the
fluid is sliding (shearing) past an adjacent layer.
P R O B L E M S
Section 10.1: The Macroscopic Description of
Nonequilibrium States
10.1 Write a formula for the gradient of each of the following
functions, where a, b, and c represent constants.
a. f e −x 2 +y 2 +z 2
b. f rcos(θ) (transform to Cartesian coordinates or use
Eq. (B-46) in Appendix B).
c. f sin(bx) cos(cy)
d. f e −az 2
sin(bx)
10.2 Assume that in a two-component solution the temperature
is given by
T T 0 + Bcos(a 1 z) e
−t/b 1
and the concentration of component 2 is given by
c 2 c 0 + Csin(a 2 z) e
−t/b 2
where B, C, a 1 , a 2 , b 1 , and b 2 are constants and where t
represents the time.
a. Write the expressions for the gradient of the
temperature and the gradient of the concentration of
component 2.
b. Write the expressions for the time derivatives of the
temperature and the concentration of component 2.
443
one term in equations like Eq. (10.1-5) is nonzero. We assume that the time derivatives
of the temperature, pressure, and concentrations are dependent variables, so that it is
not necessary to specify the time derivatives to specify the state of the system.
E X A M P L E 10.1
Assume that the concentration of substance number 2 is represented by the function
c 2 c 2 (z, t) c 0 + acos(bz)e −t/τ
where c 0 , a, b, and τ are constants and where t represents the time. Write the expressions for
∇c 2 and for ∂c 2 /∂t.
Solution
∇c 2 −kabsin(bz)e −t/τ
∂c 2
∂t
atcos(bz)e −t/τ
where k is the unit vector in the z direction.
If a fluid system is flowing the flow velocity is also needed to specify the state of
the system. The flow velocity u is a vector quantity that depends on position and time:
u u(r, t) u(x, y, z, t) iu x (r, t) + ju y (r, t) + ku z (r, t)
(10.1-6)
Just as with the temperature, pressure, and concentration, we need a specification of
how strongly the flow velocity depends on position. Each term in Eq. (10.1-6) can be
differentiated with respect to x, y, and z, so the gradient of u has nine components. We
will avoid a full discussion of this gradient and focus on single terms. For example, the
derivative ∂u y /∂z gives the rate of change of the y component of the velocity in the z
direction. This quantity is called the rate of shear, or the rate at which one layer of the
fluid is sliding (shearing) past an adjacent layer.
P R O B L E M S
Section 10.1: The Macroscopic Description of
Nonequilibrium States
10.1 Write a formula for the gradient of each of the following
functions, where a, b, and c represent constants.
a. f e −x 2 +y 2 +z 2
b. f rcos(θ) (transform to Cartesian coordinates or use
Eq. (B-46) in Appendix B).
c. f sin(bx) cos(cy)
d. f e −az 2
sin(bx)
10.2 Assume that in a two-component solution the temperature
is given by
T T 0 + Bcos(a 1 z) e
−t/b 1
and the concentration of component 2 is given by
c 2 c 0 + Csin(a 2 z) e
−t/b 2
where B, C, a 1 , a 2 , b 1 , and b 2 are constants and where t
represents the time.
a. Write the expressions for the gradient of the
temperature and the gradient of the concentration of
component 2.
b. Write the expressions for the time derivatives of the
temperature and the concentration of component 2.