Page 97 - Bird R.B. Transport phenomena

82 Chapter 3 The Equations of Change for Isothermal Systems

The term (—T:VV) is always positive for Newtonian fluids, 3 because it may be written
as a sum of squared terms:

= /хФ, + KV (3.3-3)
V
which serves to define the two quantities Ф, and 4? . When the index / takes on the val-
г
v
ues 1, 2, 3, the velocity components v { become v , v , v z and the Cartesian coordinates x,
x
y
j
become x, y, z. The symbol 5 is the Kronecker delta, which is 0 if i Ф j and 1 if / = .
/y
The quantity (—T:VV) describes the degradation of mechanical energy into thermal
energy that occurs in all flow systems (sometimes called the viscous dissipation heating). 4
This heating can produce considerable temperature rises in systems with large viscosity
and large velocity gradients, as in lubrication, rapid extrusion, and high-speed flight.
(Another example of conversion of mechanical energy into heat is the rubbing of two
sticks together to start a fire, which scouts are presumably able to do.)
When we speak of "isothermal systems," we mean systems in which there are no ex-
ternally imposed temperature gradients and no appreciable temperature change result-
ing from expansion, contraction, or viscous dissipation.
The most important use of Eq. 3.3-2 is for the development of the macroscopic me-
chanical energy balance (or engineering Bernoulli equation) in Section 7.8.

§3.4 THE EQUATION OF ANGULAR MOMENTUM

Another equation can be obtained from the equation of motion by forming the cross
product of the position vector r (which has Cartesian components x, y, z) with Eq. 3.2-9.
The equation of motion as derived in §3.2 does not contain the assumption that the stress
(or momentum-flux) tensor т is symmetric. (Of course, the expressions given in §2.3 for
the Newtonian fluid are symmetric; that is, r /y = т .)
у7
When the cross product is formed, we get—after some vector-tensor manipula-
tions—the following equation of change for angular momentum:

| p [ r X v ] = -[V- pv[r X v]] - [V • {r X p6} ] -[V • (r X } ] + [rX pg] - [е:т] (3.4-1)
т
+
+ +
Here e is a third-order tensor with components E (the permutation symbol defined in
i]k
§A.2). If the stress tensor т is symmetric, as for Newtonian fluids, the last term is zero.
According to the kinetic theories of dilute gases, monatomic liquids, and polymers, the
tensor т is symmetric, in the absence of electric and magnetic torques. 1 If, on the other
hand, т is asymmetric, then the last term describes the rate of conversion of bulk angular
momentum to internal angular momentum.
The assumption of a symmetric stress tensor, then, is equivalent to an assertion that
there is no interconversion between bulk angular momentum and internal angular mo-
mentum and that the two forms of angular momentum are conserved separately. This

3 An amusing consequence of the viscous dissipation for air is the study by H. K. Moffatt [Nature,
404, 833-834 (2000)] of the way in which a spinning coin comes to rest on a table.
4 G. G. Stokes, Trans. Camb. Phil Soc, 9, 8-106 (1851), see pp. 57-59.
1 J. S. Dahler and L. E. Scriven, Nature, 192, 36-37 (1961); S. de Groot and P. Mazur, Nonequilibrium
Thermodynamics, North Holland, Amsterdam (1962), Chapter XII. A literature review can be found in
G. D. C. Kuiken, Ind. Eng. Chem. Res., 34, 3568-3572 (1995).