Integration over the VolumeDiagram 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:

2

12 1

zz

z z z

V

zV V dz

And hence the total change in z-direction velocity is:

zz

VV z

z

Region

zz2

z1

Vz2

Vz1

Area element daR

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

2

1

zz z

Rz R z Vol

V VQ dz da d

z z

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

2

1

xx x

Rx R x Vol

V VQ dxda d

x x

2

1

y y yRy R y Vol

V VQ dy da d

y y

and

Now since the total net outflow Qnet(total) will be given by Q(x) + Q(y) + Q(z) it follows that:

( )yx z

net total Vol

VV VQ d

x y z

Flow through an element of the surface(Volume Integration)

Recognising the expression:

as the definition of the divergence of a field we have

yx zVV V

x y z

( )net total VolumeQ d divV

So we can again link the three results in the form

L U

R RR RPda Pda

Volumed divV Ssurface

a V n

3

1( )

m

mrfar near m

da

V m