A.2. General Balance Equation

Consider the integral

F( t )= V( t ) d 3 x f( x,t )                        (A.13)

which gives the total amount of a quantity F for a fixed set of fluid particles occupying a volume V(t).  Here, f is called the density of F.

The convective derivative of F is

DF Dt = D Dt [ V( t ) d 3 x   f( x,t ) ]                       (A.14)

and gives the rate of change of F in the reference frame moving with V(t).  Using the results from A.1.2, we have

DF Dt = t [ V( 0 ) d 3 z   J  f[ x( z,t ),t ] ]                  [ (A.5) used ]

= t [ V( 0 ) d 3 z   J  f( z,t ) ]            where      f( z,t )=f[ x( z,t ),t ]

= V( 0 ) d 3 z   t [ J  f( z,t ) ]

= V( 0 ) d 3 z   [ f( z,t ) J t   +J f( z,t ) t ]

= V( 0 ) d 3 z   J[ f( z,t ) x v  + f( z,t ) t ]               [ (A.12) used ]

= V( 0 ) d 3 z   J[ f( x,t ) x v  + Df( x,t ) Dt ]

= V( t ) d 3 x   [ f( x,t ) x v  + Df( x,t ) Dt ]                                (A.16)

Now, the change in F can be caused by a source with density generating rate σ F , or by a flux J F  so that

DF Dt = V( t ) d 3 x   σ F S( t ) dS    J F                           (A.17)

= V( t ) d 3 x   ( σ F x J F )

where S(t) is the surface bounding V(t) and we've used the Green's theorem.

Since (A.16,17) are valid for arbitrary V(t), we must have

f x v  + Df Dt = σ F x J F

Þ            Df Dt =f x v  + σ F x J F                  (A.18)

which is known as the balance equation.  Using

Df Dt = f t +v x f

we have

f t +v x f+f x v  = σ F x J F

or

f t + x ( vf )  = σ F x J F

f t   + x ( J C + J F )= σ F                          (A.19)

f t   + x J= σ F

where J C =vf  is called the convective current of F and the total current is J= J C + J F .  Integrating (A.19) over a volume V f  fixed in space, we have

V f d 3 x    f t = V f d 3 x x ( vf+ J F )+ V f d 3 x σ F                     (A.20)

Þ            d dt V f d 3 x   f= S f dS   ( vf+ J F )+ V f d 3 x σ F               (A.21)