Mechanics of Fluids

9.6.2 Flow from a line source
A source is a point from which fluid issues uniformly in all directions. If, for
two-dimensional flow, the flow pattern consists of streamlines uniformly
spaced and directed radially outwards from one point in the reference plane
(as in Fig. 9.16), the flow is said to emerge from a line source. (We remember
that for two-dimensional flow what appears on the diagram as a point is to
be regarded as a line seen end on.) The strength m of a source is the total
volume rate of flow from it, the line source of two-dimensional flow being
considered of unit length. The velocity q at radius r is given by
Volume rate of flow ÷ Area perpendicular to velocity = m/2πr
for unit depth, since the velocity is entirely in the radial direction. As r → 0,
q →∞and so no exact counterpart of a source is found in practice. However,
except at the singular point r = 0, a similar flow pattern would be
achieved by the uniform expansion of a circular cylinder forcing fluid away
from it, or, more approximately, by the uniform emission of fluid through
the walls of a porous cylinder. Nevertheless, the concept of a source has its
chief value in providing a basic mathematical pattern of flow which, as we
shall see, can be combined with other simple patterns so as to describe flows
closely resembling those found in practice.
In cases such as this, where circular symmetry is involved, polar coordinates
are more suitable. The velocity qr radially outwards = ∂ψ/∂n =
−∂ψ/r∂θ. The minus sign arises from the ‘right-to-left’ convention and the
fact that the angle θ is considered positive anti-clockwise; the length ∫ r dθ of
a circumferential line therefore increases in the anti-clockwise direction, and
to an observer on this line and facing that direction, the radially outward
flow would appear to move from left to right, that is, in the negative direction.
A tangential velocity qt is given by ∂ψ/∂r; if one looked outwards along
the radius one would see a positive (i.e. anticlockwise) tangential velocity
coming from one’s right and so the sign here is positive.
Basic patterns of flow 375
Fig. 9.16