In this paper, we present a phase-field method for Rayleigh instability on a fibre. Unlike a liquid col-
umn, the evolutionary dynamics of a liquid layer on a fibre depends on the boundary condition at the
solid-liquid interface. We use a Navier–Stokes–Cahn–Hilliard system to model axisymmetric immiscible
and incompressible two-phase flow with surface tension on a fibre. We solve the Navier–Stokes equation
using a projection method and the Cahn–Hilliard equation using a nonlinearly stable splitting method.
We present computational experiments with various thicknesses of liquid thread and fibre. The numer-
ical results indicate that the size of the satellite droplet decreases as the thicknesses of the thread and
fibre increase.