We now return to the scalar representation of the flow and calculate the change of mass in a second way, using a Taylor series expansion of the velocity and density fields. The net flow across the left and right faces causes a mass change
Note that for positive velocity components, flow on the left increases the mass and flow on the right decreases the mass. We make this more compact by renaming:
where 
can
refer to either the left or the right face.
The function p is clearly a function of x, but any y and z dependence of
has been "integrated away". We can calculate the
difference in p from the left to the right side using a Taylor series expansion.
We choose to expand about a special point along the x axis: The value of x for
which 
is equal to its average value. (We know that this is inside the
volume by the mean value theorem of calculus.) Call this point 
. Then on the left face,
and on the right face,
where a is the length of one edge of the cube.
In this notation, using the Taylor series expansions we can write
where 
represents the average value of the quantity within.
Using the calculus definition of the average value of a function over an interval, we write
.
Using this expression in the Taylor series for the mass change during
, we have
where 
is a volume element for integration. (Note that 
is independent of x so the partial derivative can "move
inside" the area integral.)
Adding in the contributions from flows across the other 4 surfaces,
The higher order terms in the Taylor series expansion, indicated by ... , may be ignored if we make the length of the edge of the cube small enough. Thus for a small enough volume, we can write
or
In the limit of short times, we have
Compare this to the previous result:
Return to index.
Return to survey.
Go forward to the divergence theorem.