Movies / General Introduction

Simulations of a dispersive-dissipative evolution equation

The movies are computer animations of simulations of a dispersive-dissipative evolution equation derived in the context of liquid film flow down an inclined plane. The dimensionless evolution equation for the film thickness deviation (from its mean), η, is (in a suitable reference frame moving in the downstream direction)

η_t + ηη_z + ∇²η_z − κη_yy + ε(η_zz + ∇⁴η) = 0,

where ∇² = ∂²/∂y² + ∂²/∂z².

Here y and z are, respectively, the spanwise and streamwise coordinates, t is time, and the subscripts indicate the partial derivatives. We used periodic boundary conditions on the extended spatial domains 0 ≤ y ≤ 2πp (with p ≫ 1) and 0 ≤ z ≤ 2πq (with q ≫ 1).

For κ = 0, the evolution equation reduces to a 2D generalization of the KdV equation for ε = 0 and a 2D generalization of the Kuramoto-Sivashinsky equation for ε → ∞. The movies have been grouped into three categories:

• Large dispersivity λ = 1/ε (ε ≪ 1).
• Small dispersivity (ε ≫ 1).
• Intermediate dispersivity.

The thickness of the film is indicated by the color: blue end of spectrum for larger thickness and the red end for smaller thickness.