Integration over the Volume

1. Integration over the Volume Diagram 10 is similar to diagram 5 only now we are concerned with the change in velocity between the top and bottom surfaces of the column. Since Velocity is a vector we must deal separately with its components integrating the change across each oblong element. We consider only the velocity component aligned with the long axis of the column as only this component takes fluid into or out of the column ends. Thus for the z-direction velocity change we get: Vz2 z z2 Vz1 z1 And hence the total change in z-direction velocity is: Region Area element daR

2. By combining this inner integration w.r.t z with an outer integration over the region we can integrate over the entire volume of the body. This leads to the result Where Q(z) is the net rate of flow out of the C.V. due to the z-direction components of the local velocity vectors. Similarly for the x and y directions and Now since the total net outflow Qnet(total) will be given by Q(x) + Q(y) + Q(z) it follows that:

3. Flow through an element of the surface(Volume Integration) Recognising the expression: as the definition of the divergence of a field we have So we can again link the three results in the form