Page 98 - Bird R.B. Transport phenomena

§3.5 The Equations of Change in Terms of the Substantial Derivative 83

corresponds, in Eq. 0.3-8, to equating the cross-product terms and the internal angular
momentum terms separately.
Eq. 3.4-1 will be referred to only in Chapter 7, where we indicate that the macro-
scopic angular momentum balance can be obtained from it.

§3.5 THE EQUATIONS OF CHANGE IN TERMS
OF THE SUBSTANTIAL DERIVATIVE

Before proceeding we point out that several different time derivatives may be encoun-
tered in transport phenomena. We illustrate these by a homely example—namely, the ob-
servation of the concentration of fish in the Mississippi River. Because fish swim around,
the fish concentration will in general be a function of position (x, y, z) and time (f).

The Partial Time Derivative dldt

Suppose we stand on a bridge and observe the concentration of fish just below us as a
function of time. We can then record the time rate of change of the fish concentration at a
fixed location. The result is (dc/dt)\ , the partial derivative of с with respect to t, at con-
xyz
stant x, y, and z.
The Total Time Derivative dldt

Now suppose that we jump into a motor boat and speed around on the river, sometimes
going upstream, sometimes downstream, and sometimes across the current. All the time
we are observing fish concentration. At any instant, the time rate of change of the ob-
served fish concentration is
dc = (dc\ dxfdc) + d l(dc\ dz(dc) ( 3 5 1 )
dt U/,.y, dt \dx) + dt U y L dt Ы*,у,<
2 yAl
in which dx/dt, dy/dt, and dz/dt are the components of the velocity of the boat.

The Substantial Time Derivative D/Dt

Next we climb into a canoe, and not feeling energetic, we just float along with the cur-
rent, observing the fish concentration. In this situation the velocity of the observer is the
same as the velocity v of the stream, which has components v v , and v . If at any instant
z
y
x/
we report the time rate of change of fish concentration, we are then giving
DC dC , dC , „ dC , dC Dc dC , ( « v ,~ c о ч
=
Dt H + v *~di + v >4 + v *~di o r D T ^ + ( V > V C ) ( 3 5 - 2 )
The special operator D/Dt = d/'dt + v • V is called the substantial derivative (meaning that
the time rate of change is reported as one moves with the "substance"). The terms mater-
ial derivative, hydrodynamic derivative, and derivative following the motion are also used.
Now we need to know how to convert equations expressed in terms of dldt into
equations written with D/Dt. For any scalar function f(x, y, z, t) we can do the following
manipulations:

(3.5-3)