Page 92 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 75

Let the flow past the line at any point Q on it be at velocity q over a small length 6s
of line where direction of q makes angle /3 to the tangent of the curve at Q. The
component of the velocity q perpendicular to the element 6s is q sin /3 and therefore,
assuming the depth of stream flow to be unity, the amount of fluid crossing the
element of line 6s is q sin /3 x 6s x 1 per second. Adding up all such quantities crossing
similar elements along the line from 0 to P, the total amount of flow past the line
(sometimes called flux) is

which is the line integral of the normal velocity component from 0 to P.
If this quantity of fluid flowing between 0 and P remains the same irrespective of
the path of integration, i.e. independent of the curve of the rope then sop q sin /3 ds is
called the stream function of P with respect to 0 and

Note: it is implicit that $0 = 0.

Sign convention for stream functions
It is necessary here to consider a sign convention since quantities of fluid are being
considered. When integrating the cross-wise component of flow along a curve, the
component can go either from left to right, or vice versa, across the path of integra-
tion (Fig. 2.16). Integrating the normal flow components from 0 to P, the flow
components are, looking in the direction of integration, either (a) from left to right or
(b) from right to left. The former is considered positive flow whilst the latter is
negative flow. The convention is therefore:

Flow across the path of integration is positive if, when looking in the direction of
integration, it crosses the path from left to right.

2.5.2 The streamline
From the statement above, $p is the flow across the line OP. Suppose there is a point
PI close to P which has the same value of stream function as point P (Fig. 2.17). Then
the flow across any line OP1 equals that across OP, and the amount of fluid flowing
into area OPPIO across OP equals the amount flowing out across OP1. Therefore, no
fluid crosses line PP1 and the velocity of flow must be along, or tangential to, PPI.

Fig. 2.16