Elementary flow
Elementary flow is a collection of basic flows from which is possible to construct more complex flows by superposition. Some of the flows reflect specific cases and constraints such as incompressible, irrotational or both as in the case of Potential flow. Two Dimensional uniform flow. Given a uniform velocity of a fluid at any position in space: This flow is incompressible because the velocity is constant, the first derivatives of the velocity components are zero, and the total divergence is zero: formula_2 Given the circulation is always zero the flow is also irrotational, we can derive this from  the Kelvin's circulation theorem and from the explicit computation of the vorticity: Being incompressible and two dimensional this flow is constructed from a stream function from which And in cylindrical coordinates: from which As usual the stream function is defined up to a constant value which here we take as zero. We can also confirm that the flow is irrotational from Being irrotational, The potential function is instead: and therefore And in cylindrical coordinates Two Dimensional line source. 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  flow. 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 Two Dimensional line sink. The case of a vertical line absorbing at a fixed rate a constant quantity of fluid "Q" per unit length is a line sink. Everything is the same as the case of a line source a part from the negative sign. This is derived from a stream function or from a potential function Given that the two results are the same a part from a minus sign we  can treat transparently both line sources and line sinks with the same stream and potential functions permitting "Q" to assume both positive and negative values and absorbing the minus sign into the definition of "Q". Two Dimensional doublet or dipole line source. If we consider a line source and a line sink at a distance d we can reuse the results above and the stream function will be The last approximation is to the first order in d. Given It remains The velocity is then And the potential instead Two Dimensional vortex line. This is the case of a vortex  filament rotating at constant speed, there is a cylindrical symmetry and the problem can be solved in the orthogonal plane. Dual to the case above of line sources, vortex lines play the role of monopoles for irrotational flows. Also in this case the flow is also both irrotational and incompressible and therefore a case of Potential flow. This is characterized by a cylindrical symmetry: Where the total circulation is constant for every closed line around the central vortex and is zero for any line not including the vortex. Therefore, This is derived from a stream function or from a potential  function Which is dual to the previous case of a line source Generic Two Dimensional potential flow. Given an incompressible two dimensional flow which is also irrotational we have: Which is in cylindrical coordinates We look for a solution with separated variables: which gives Given the left part depends only on "r" and the right parts depends only on formula_40, the two parts must be equal to a constant independent from "r" and formula_40. The constant shall be positive. Therefore, The solution to the second equation is a linear combination of formula_44 and formula_45 In order to have a single  valued velocity (and also a single valued stream function) "m" shall be a positive integer. therefore the most generic solution is given by The potential is instead given by