Equilibrium forms of a liquid on the interior surface of a rotating cylinder and their stability

The problem of the viscous liquid, partially filling the cylindrical cavity of a finite length, which rotates with a constant angular velocity, is considered (the rimming flow). It is assumed that the gravity force is absent. If the wetting angle is equal to π /2, equilibrium forms with cylindrical free surface exist (trivial forms of equilibrium). There exist values of the Weber number (bifurcation values) at which the nontrivial forms branch off from the trivial form of equilibrium. Graphics of the axisymmetric equilibrium forms of the liquid are constructed numerically. With the Weber number growth, the nontrivial forms increasingly deviate from the cylindrical equilibrium state. With a further increase in the values of the Weber number the topology of equilibrium forms changes, as the form contacts the cavity wall (when the liquid layer is thin) or its axis (when the liquid layer is thick).The stability of equilibrium states is investigated using Lagrange’s stability theorem and its converse. The cylindrical state is unstable when the values of the Weber numbers are less than the bifurcation value or slightly exceed the bifurcation value. The value of the Weber number, for which the trivial form of equilibrium becomes stable, is obtained. Nontrivial forms of equilibrium are unstable for all admissible values of the Weber number.