A.1.1. Convective Time Derivative

[For a more rigorous discussion, see chapter 1 of the book by R.E.Meyer, "Introduction to Mathematical Fluid Dynamics", Wiley (1971).  Part I of the book "Fluid Mechanics", Springer (1997), by J.Spurk, gives a good introduction to the foundamentals of the subject. ]

 

For our purposes, a fluid can be thought of as a collection of "fluid particles" that is dense enough to be considered as a continuum.  Operationally, this means the characteristic lengths of the phenomena under study must be significantly larger than the average nearest neighbor particle distances.  Here, a particle means something that can be treated mathematically as occupying a geometric point in space but large enough physically so that fluctuations on the atomic scale can be neglected.

 

As a direct extension of the mechanics of discrete particles, the basic dynamical variables are the particle positions x(z,t) and velocities v(z,t) where z is the label of the particles.  Similarly, any property F of the system will be decribed by F( z,t ) .  This is called the Lagrangian formulism.  Usually, one simply uses the positions of the particles at, say t = 0, to label them, i.e., z=x( z,0 )=x( 0 ) .  On the other hand, the natural way to describe a property F of a continuum is by means of a (field) function f( x,t ) .  This is called the Eulerian formulism.

 

Now, at time t, the position of the particle z can be written as

x( t )=x( z,t )=x[ x( 0 ),t ]                               (A.1)

Consider now a density

f( x,t )=f[ x( z,t ),t ]   =F( z,t )                      (A.1a)

The convective, or material, time derivative is defined as

        D Dt f( x,t ) ( F( z,t ) t ) z

which gives the change rate of f seen in a frame moving with the fluid particle z.  Written in terms of f, we have

        Df Dt = ( f[ x( z,t ),t ] t ) z   

                = ( f( x,t ) t ) x + ( f( x,t ) x ) t ( x( z,t ) t ) z                 (A.2)

Now, ( x( z,t ) t ) z  is simply the velocity   Dx Dt =v( x,t )  of the fluid particle z.  Hence

        Df Dt = ( f( x,t ) t ) x +v ( f( x,t ) x ) t

Since f is arbitrary, this implies the relation

       

        D Dt = t +v x                        (A.4)