We discuss the self-similar solutions of the second kind associated with the propagation of turbulent bursts in a fluid at rest. Such solutions involve an eigenvalue parameter μ, which cannot be determined from dimensional analysis. Existence and uniqueness are established and the dependence of μ on a physical parameter λ in the problem is studied: estimates are obtained and the asymptotic behaviour as λ → ∞ is established.

This paper reviews the governing equations for a plane Hele–Shaw flow of a power-law fluid. We find two closely related partial differential equations, one for the pressure and one for the stream function. Some mathematical results for these equations are presented, in particular some exact solutions and a representation theorem. The results are applied to Hele–Shaw flow. It is then possible to determine the flow near an arbitrary corner for any power-law fluid. Other examples are also given.

We consider the initial value problem for the equation ut = uxx + H(u), where H is the Heaviside graph, on a bounded interval with Dirichlet boundary conditions, and discuss existence, regularity and uniqueness of solutions and interfaces.

In this paper, new non-classical symmetry reductions and exact solutions for the 2+1- dimensional, time-independent and time-dependent, dissipative Zabolotskaya-Khokhlov equations in both cartesian and cylindrical coordinates, are presented. These are obtained using the Direct Method, which was originally developed by Clarkson & Kruskal (1989) to study symmetry reductions of the Boussinesq equation, and which involves no group theoretic techniques. In particular, we derive exact solutions of these Zabolotskaya-Khokhlov equations expressible in terms of elementary functions, Weierstrass elliptic and zeta functions, Weber-Hermite functions and Airy functions. Additionally, it is shown that some previously known solutions of these equations actually arise from non-classical symmetries.