Proof That Loss of Fluid Per Unit Volume Equals Divergence of Velocity Field
The x component of the velocity at the centre of the face ABCD above is
The x component of the velocity at the centre of the face EFGH above is
The volume of fluid crossing ABCD per unit time is
The volume of fluid crossing EFGH per unit time is
The rate at which water is lost is the difference of these two and equals
Similarly the net loss of fluid in the y and z directions areandrestively.
Adding these gives the rate of loss of fluid overall as