Chapter 7 - Some Mathematics:
The Equations of Motion

Chapter 7 Contents

7.5 The Total Derivative (D/Dt)

If the number of boxes in a system increases to a very large number as the size of each box shrinks, we eventually approach limits used in differential calculus.

For example, if we subdivide the flow of water into boxes a few meters on a side, and if we use conservation of mass, momentum, or other properties within each box, we can derive the differential equations governing fluid flow.

Consider the simple example of acceleration of flow in a small box of fluid. The resulting equation is called the total derivative. It relates the acceleration of a particle Du/Dt to derivatives of the velocity field at a fixed point in the fluid. We will use the equation to derive the equations for fluid motion from Newton's Second Law which requires calculating the acceleration of a particles passing a fixed point in the fluid.

Figure 7.2 Sketch of flow used for deriving the total derivative.

We begin by considering the flow of a quantity qin into and qout out of the small box sketched in Figure 7.2. If q can change continuously in time and space, the relationship between qin and qout is:

(7.5)

The rate of change of the quantity q within the volume is:

(7.6)

But δx/δt is the velocity u; and therefore:

In three dimensions, the total derivative becomes:

(7.7a)
(7.7b)

where u is the vector velocity and ∇ is the operator del of vector field theory (See Feynman, Leighton, and Sands 1964: 2-6).

This is an amazing result. The simple transformation of coordinates from one following a particle to one fixed in space converts a simple linear derivative into a non-linear partial derivative. Now let's use the equation to calculate the change of momentum of a parcel of fluid.

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