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 l satisfies the power law
l;g1/3, where g 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!#.