Evolution and breaking of liquid film flowing on a vertical cylinder

An amplitude equation is derived, which describes the evolution of a disturbed film interface H(τ,Z,Y) flowing down an infinite vertical cylindrical column. Using a new approach, which accounts for fast spatial changes, the nonlinear evolution of the interface is shown to be governed by H_τ + βHH_z + αH_zz + γ∇²{N [ (1/ω²)H + ∇²H ]} = 0, where ω is the normalized cylinder radius and α, β, and γ are constants, ∇ ≡ (∂_Z, ∂_Y), and N= [ 1 + ∈⁴(∇H)²] ^-3/2. It is shown numerically that for some linearly unstable equilibria the evolving waves break in a finite time.