If two velocity fields and can be represented by means of the velocity potentials and , respectively, then their superposition can be represented by
In other words, the algebraic sum defines the superimposed fields of and .
As an illustration, suppose that it is desired to determine the irrotational and divergenceless flow of fluid past a circular cylinder which is held fixed. Since the flow is presumed to be Laplacian, the pattern can be established merely by superimposing a velocity -U on the pattern of flow found in Art. 1.13. This will effectively stop the cylinder, and at the same time alter the pattern of the streamlines and velocity potential. The velocity -U can be represented by the potential
Thus if we let
from Eq. (240a) then the potential applicable to flow past a stationary cylinder is
We could also take
and the superposition of these streamfunctions gives
The addition of the two component stream function isolines is demonstrated graphically in Fig. 1.15-1.