In this paper, we present a phase-field method for Rayleigh instability on a fibre. Unlike a liquid column, 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 numerical results indicate that the size of the satellite droplet decreases as the thicknesses of the thread and fibre increase. (C) 2018 Elsevier Ltd. All rights reserved.