Elementary flow

The techniques involved to create more complex solutions can be for example by superposition, by techniques such as topology or considering them as local solutions on a certain neighborhood, subdomain or boundary layer and to be patched together.

Elementary flows can be considered the basic building blocks (fundamental solutions, local solutions and solitons) of the different types of equations derived from the Navier-Stokes equations.

To put it in perspective boundary layers can be interpreted as topological defects on generic manifolds, and considering fluid dynamics analogies and limit cases in electromagnetism, quantum mechanics and general relativity one can see how all these solutions are at the core of recent developments in theoretical physics such as the ads/cft duality, the SYK model, the physics of nematic liquids, strongly correlated systems and even to quark gluon plasmas.

) and two-dimensional, its velocity can be expressed in terms of a stream function,

In cylindrical coordinates 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 dimensions on the orthogonal plane.

Line sources and line sinks (below) are important elementary flows because they play the role of monopole 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 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 The case of a vertical line absorbing at a fixed rate a constant quantity of fluid Q per unit length is a line sink.

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.

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 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.

Therefore, This is derived from a stream function or from a potential function Which is dual to the previous case of a line source Given an incompressible two-dimensional flow which is also irrotational we have: Which is in cylindrical coordinates [2] 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