In studying Euler's equation, we became familiar with the gradient of a scalar field because the gradient of the pressure is related to force on a unit volume of the fluid. Here we introduce derivative of a vector field: The divergence.
The divergence of a velocity field (or a momentum density field) is related to conservation of mass and mass flow. Calculation of the divergence of a momentum density field leads us to the divergence theorem and the equation of continuity.
THE DIVERGENCE THEOREM
|
|
| Cube of length a, with flow in and out indicated by the velocity |
| field vectors. |
To understand divergence, we consider the flow of material in and out of a volume as shown in the figure. This volume is different from the volumes used in discussion of Euler's equation. This volume does not move. Think of it as a wire cage suspended in the flowing fluid for purposes of making measurements. It is not necessarily a small volume.
We will calculate the net change in mass of fluid in the volume as a result of the fluid flow. Positive change will indicate increase in mass in the volume. Since the volume is constant, an increase in mass must mean an increase in the density of the fluid in the volume.
The increase is equal to the mass that flows into the volume minus the mass that flows out. We will calculate these changes in terms of the fluid velocity field carrying material across each of the faces of the cube. We will do the calculation from two different points of view; one using vector formalism directly over the whole surface, and the other using a Taylor series expansion of the vector components.
Vector Calculation of Net Mass Flow
The change in mass within the cube can be calculated as the sum of the mass
transfers across each face, during a time interval,
:
For the moment, we restrict attention to the left and right faces, abbreviating left and right with L and R. The fields (density and velocity) can be different on different faces, but for the moment, we treat them as constant on any particular face. (For a nonuniform velocity field, we will use the average value of the vector field, averaged over the left face.)
In a time interval, 
, the shaded volume flows in across the left face. (We assume
positive velocity components.) The length 
is given by
so that the volume which flows into the left face is:
where 
is the area of the left face of the cube.
The mass change due to flow across the left face is then
where 
is the density of the fluid on the left (temporarily taken as
constant).
|
| A cube of edge length, a, showing a nonuniform velocity |
| field and an area element, dS. |
To allow formally for the possibility that the velocity and the density
change from point to point on the left face, we divide the surface into small
areas, 
. (These areas are small enough that the fields are constant over
.) Then we sum the mass change for each small element.
where we have dropped the subscript L on the right, since
and 
will be evaluated anew at each patch of area . In the limit of small this sum becomes an integral:
where we have factored out the time interval, since it is the same for each bit of area.
|
| A cube showing surface area vector elements on top, left, |
| and right sides. |
To account for all 6 faces, it is useful to have a vector representation in
which the surface area element has direction pointed normal to the surface, and outward from the
volume of the cage as in the figure.
Then the mass flow into the area on the left face (during time ) is
.
The minus sign appears because the velocity is opposite in direction to the
area element, making 
negative for flow into the left face. Note that the y and z
components of the velocity carry no material across the left face.
The net flow of mass across the left face in time interval
is:
.
On the right face, positive (x) velocity is in the same direction as the
area element so that the integrand is positive for positive velocity component.
However positive velocity x component causes a decrease in mass inside the
volume so 
is negative when is positive:
.
When all faces are taken into account, we have
.
Notice that the integrand on the right is the momentum per unit volume; the momentum density. When the momentum density points mostly in the direction out of the volume (parallel to the surface element) then the mass inside the volume decreases. For example, if we have burning fuel oil inside the volume, the mass of oil decreases as the burnt gases flow away from the flame.
The boxed equation represents conservation of mass because (we assert) the only for mass inside the volume to decrease is for material to flow out across the faces: Mass cannot simply disappear.
A final note: We can take the limit as 
and find the time derivative of the mass in the sample volume.
Since the location of the volume is fixed in this development, the coordinates
of the mass inside are fixed. Thus the time derivative is taken while holding
the spatial coordinates constant. We represent this derivative as a partial
derivative with respect to time:
.
Return to index.
Return to survey.
Go forward to a Taylor series calculation of net mass flow.