We show for the first time that only one similarity solution describes break-up of viscous liquid threads is linearly stable, while other solutions are unstable. Complex eigenvalues lead to presence oscillations close break-up.

Although there are 2339 pentomino tilings of the 6 x 10 rectangle, many pairs of solutions are similar. We show that the solutions can be divided into 911 equivalence classes by the similarity transformations: rotate or reflect a subset of the pieces interchange two congruent subsets rearrange two pieces Early steps of the computation utilize the standard "polar" representation of solutions to ...

In this paper, new classes of symmetries for partial differential equations (PDE) which can be written in a conserved form are introduced. These new symmetries called nonclassical potential symmetries, are neither potential symmetries nor nonclassical symmetries. Some of these symmetries are carried out for the Burgers equation ut + uux − uxx = 0. (1) by studying the nonclassical symmetries of ...

We present two classes of exact solutions of the Navier–Stokes equations, which describe steady vortex structures with two-dimensional symmetry in an infinite fluid. The first is a class of similarity solutions obtained by conformal mapping of the Burgers vortex sheet to produce wavy sheets, stars, flowers and other vorticity patterns. The second is a class of non-similarity solutions obtained ...