A mechanism for linear instability in two-dimensional rimming flow
In rimming flow, a thin film of viscous liquid coats the inside of a cylinder whose axis is horizontal and which is rotating with constant angular velocity. It has been established experimentally that such flows are often unstable with a variety of secondary flow regimes having been observed experimentally [15]. We use a lubrication approximation extended to the first order in the dimensionless film thickness (including the small effects of the variation of the film pressure across its thickness and the surface tension) and study the stability of the steady solutions to two-dimensional disturbances. The modified evolution equation is found to have both asymptotically stable and unstable solutions arising from the pressure terms. Surface tension effects place a restriction on the critical wave number when instability occurs: in many cases, surface tension prevents instability. 1. The physical problem. The occurrence of instabilities in the flow of a thin film of viscous liquid placed in a cylindrical tube of radius R whose axis is horizontal and which is rotating with constant angular velocity u> (Fig. 1) has attracted considerable interest in the scientific literature [1]—[4]. Such flows are sometimes referred to as rimming flows. The liquid is entrained on the cylinder wall and, given suitable wetting properties, a continuous film can be obtained. Previous authors have investigated this problem both experimentally and theoretically and some progress has been made. In many cases, it is found experimentally that the flow appears to be unstable to both twoand threedimensional disturbances [15]. One well-known example of instability arises when the liquid, instead of settling into a configuration that is uniform in the axial direction, settles into a number of wet and (relatively) dry areas (or rings) along the length of the cylinder. So striking are these features that the term "hygrocysts" [1] has been coined to describe this apparently stable secondary state. Visually, the effect vaguely resembles the well-known results obtained by G. I. Taylor for the flow of a viscous liquid in the annular region between two rotating cylinders (see, e.g., [3]). Flows of this type obviously have industrial applications, a typical example being the process of coating the inside of cylindrical fluorescent light bulbs. Such a bulb consists of Received May 22, 2000. 2000 Mathematics Subject Classification. Primary 76D07, 76E17, 76D08, 76B45.