Geography Reference
In-Depth Information
2.5.1
An Eulerian Derivation
We consider a volume element δx δy δz that is fixed in a Cartesian coordinate
frame as shown in Fig. 2.5. For such a fixed control volume the net rate of mass
inflow through the sides must equal the rate of accumulation of mass within the
volume. The rate of inflow of mass through the left-hand face per unit area is
ρu
∂x (ρu) δx
2
whereas the rate of outflow per unit area through the right-hand face is
ρu
∂x (ρu) δx
+
2
Because the area of each of these faces is δyδz, the net rate of flow into the
volume due to the x velocity component is
ρu
δyδz
ρu
δyδz
∂x (ρu) δx
∂x (ρu) δx
+
2
2
∂x (ρu)δxδyδz
=−
Similar expressions obviously hold for the y and z directions. Thus, the net rate of
mass inflow is
∂x (ρu)
∂z (ρw) δxδyδz
∂y (ρv)
+
+
Fig. 2.5
Mass inflow into a fixed (Eulerian) control volume due to motion parallel to the x axis.
 
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