Standard flows in terms of у and ф

There are three basic two-dimensional flow fields, from combinations of which all other steady flow conditions may be modelled. These are the uniform parallel flow, source (sink) and point vortex.

The three flows, the source (sink), vortex and uniform stream, form standard flow states, from combinations of which a number of other useful flows may be derived.

3.3.1 Two-dimensional flow from a source (or towards a sink)

A source (sink) of strength m( — m) is a point at which fluid is appearing (or disappearing) at a uniform rate of m( — m) m2 s. Consider the analogy of a small hole in a large flat plate through which fluid is welling (the source). If there is no obstruction and the plate is perfectly fiat and level, the fluid puddle will get larger and larger all the while remaining circular in shape. The path that any particle of fluid will trace out as it emerges from the hole and travels outwards is a purely radial one, since it cannot go sideways, because its fellow particles are also moving outwards.

Also its velocity must get less as it goes outwards. Fluid issues from the hole at a rate of mm2s . The velocity of flow over a circular boundary of lm radius is m/27rm s_l. Over a circular boundary of 2 m radius it is m/(2ir x 2), i. e. half as much, and over a circle of diameter 2 r the velocity is mj2nrms~l. Therefore the velocity of flow is inversely proportional to the distance of the particle from the source.

All the above applies to a sink except that fluid is being drained away through the hole and is moving towards the sink radially, increasing in speed as the sink is approached. Hence the particles all move radially, and the streamlines must be radial lines with their origin at the source (or sink).