Mechanics of Fluids

362 The flow of an inviscid fluid
Fig. 9.1
which the fluid flows. The average velocities across each face of the element
are as shown. For an incompressible fluid, volume flow rate into the element
equals volume flow rate out; thus for unit thickness perpendicular to the
diagram
uδy + vδx =
�
u + ∂u
∂x δx
� �
δy + v + ∂v
∂y δy
�
δx
whence, as the size of the element becomes vanishingly small, we have in the
limit the partial differential equation
∂u ∂v
+ = 0 (9.1)
∂x ∂y
9.2 THE STREAM FUNCTION
Figure 9.2 illustrates two-dimensional plane flow. It is useful to imagine a
transparent plane, parallel to the paper and unit distance away from it; the
lines in the diagram should then be regarded as surfaces seen edge on, and
the points as lines seen end on. A is a fixed point but P is any point in the
plane. The points A and P are joined by the arbitrary lines AQP, ARP. For
an incompressible fluid flowing across the region shown, the volume rate of
flow across AQP into the space AQPRA must equal that out across ARP.
Whatever the shape of the curve ARP, the rate of flow across it is the same
as that across AQP; in other words, the rate of flow across the curve ARP
depends only on the end points A and P. Since A is fixed, the rate of flow
across ARP is a function only of the position of P. This function is known as
the stream function, ψ. The value of ψ at P therefore represents the volume
flow rate across any line joining P to A at which point ψ is arbitrarily zero.
(A may be at the origin of coordinates, but this is not necessary.)
If a curve joining A to P ′ is considered (Fig. 9.3), PP ′ being along a streamline,
then the rate of flow across AP ′ must be the same as across AP since,
by the definition of a streamline, there is no flow across PP ′ . The value of
ψ is thus the same at P ′ as at P and, since P ′ was taken as any point on the
streamline through P, it follows that ψ is constant along a streamline. Thus
the flow may be represented by a series of streamlines at equal increments
of ψ, like the contour lines on a map.