Page 94 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 77

\’ -sx --
I /

I qf

I >
lo X

Fig. 2.18a
aQ/ax and aQ/ay being the partial derivatives with respect to x and y respectively.
Then, equating terms:
u = a+/ay (2.56a)
and
v = -a+/ax (2.56b)
these being the velocity components at a point x, y in a flow given by stream function Q.
(b) In polar coordinates Let the point P(r, 8) be on the streamline AB (Fig. 2.18b) of
constant Q, and point Q(r + Sr, 8 + SO) be on the streamline CD of constant Q + SQ.
The velocity components are qn and qt, radially and tangentially respectively. Here
the most convenient path of integration is PRQ where OP is produced to R so that
PR = Sr, i.e. R is given by ordinates (r + Sr, 8). Then
SQ = -q& + qn(r + Sr)M
= -q& + qnrM + qJrS8
To the first order of small quantities:
SQ = -q& + qnrS8 (2.57)

Detail
’ at P,Q
Fig. 2.18b