Axial instability of a free-surface front in a partially filled horizontal rotating cylinder

We investigate the axial instability of the free-surface front of a viscous fluid in a horizontal cylinder rotating about its longitudinal axis. A simplified model equation for the evolution of the free surface is derived and includes the effects of gravity, capillarity, inertia, and viscosity. This equation is solved numerically to determine the base state with no axial variation, and a numerical linear stability analysis is carried out to examine the onset of unstable axial modes. Various computational results are presented for the wavelength of the axial instability. Inertia is found to play an important role in the onset of the instability and the wavelength of the instability λ satisfies the power law $λ∼γ1/3,$ where γ is surface tension. Finally some numerical simulations of the simplified evolution equation are presented to show that they can capture the steady shark-teeth patterns observed in recent experiments [R. E. Johnson, in Engineering Science, Fluid Dynamics: A Symposium to Honor T. Y. Wu (World Scientific, Singapore, 1990), pp. 435–449; S. T. Thoroddsen and L. Mahadevan, “Experimental studies of the instabilities in a partially filled horizontal rotating cylinder,” Exp. Fluids 23, 1 (1997)].