Mechanics of Fluids

source is unable to move to the left beyond S, and so it diverges from the
axis θ = π and is carried to the right.
In two-dimensional flow a streamline represents a surface (viewed edge
on) along which the velocity must everywhere be tangential; hence there can
be no flow perpendicular to the surface. So, for steady two-dimensional flow
of a frictionless fluid, any streamline may be replaced by a thin, solid barrier.
In particular, the resultant streamline diverging from S may be considered
the barrier dividing the previously uniform stream from the source flow.
Since, however, the flows on the two sides of this barrier do not interact, the
pattern of streamlines outside the barrier is identical with that which would
be obtained if the barrier were the contour of a solid body with no flow
inside it. Thus the source is simply a hypothetical device for obtaining the
form of the contour of the body deflecting the originally uniform flow.
Adding the stream functions for the uniform flow and the source we
obtain, for the resultant flow,
ψ =−Uy +
�
− mθ
�
=−Ur sin θ −
2π
mθ
2π
At the stagnation point, θ = π; thus the value of ψ there is −m/2 and this
value must be that everywhere along the streamline corresponding to the
contour of the body. The contour is therefore defined by the equation
−Uy − mθ
2π =−m
2
It extends to infinity towards the right, the asymptotic value of y being given
by m/2U (when θ → 0) or −m/2U (when θ → 2π).
The velocity components at any point in the flow are given by
qt = ∂ψ/∂r =−U sin θ
Combinations of basic flow patterns 385
Fig. 9.21