Applications of complex analysis

Many fascinating problems in mechanics lead to Laplace's equation or Poisson's equation in two space dimensions; two famous examples are inviscid irrotational flow and flow of a viscous liquid in a Hele-Shaw cell, which is a pair of parallel plates separated by a very thin gap. In both cases, the velocity is the gradient of a harmonic potential. It is sometimes possible to use the powerful techniques of complex variables to find explicit solutions and analyse their behaviour, for example when thin inviscid fluid sheets collide, or in a "negative squeeze film" in a lubricated bearing. The picture shows some forms of solution blow-up in a Hele-Shaw cell.

Another application of complex analysis is to solutions of the biharmonic equation, especially in slow viscous flow and elasticity. For example, complex variable methods can be used to construct a vast range of explicit solution to slow flow with free surfaces, and their analysis reveals striking parallels with the Hele-Shaw case mentioned above.

People working in this area within OCIAM are