The case of a vertical line emitting at a fixed rate a constant quantity of fluid Q per unit length is a line source. The problem has a cylindrical symmetry and can be treated in two dimension on the orthogonal plane.
Line sources and line sinks (below) are important elementary flows because they play the role of monopole(s) for incompressible fluids (which can also be considered examples of solenoidal fields i.e. divergence free fields). Generic flow patterns can be also de-composed in terms of multipole expansions, in the same manner as for electric and magnetic fields where the monopole is essentially the first non-trivial (e.g. constant) term of the expansion.
This flow pattern is also both irrotational and incompressible.
This is characterized by a cylindrical symmetry:
Where the total outgoing flux is constant
Therefore,
This is derived from a stream function
or from a potential function