On the stability of shallow rivulets

We examine the linear stability of a capillary rivulet under the assumption that it is shallow enough to be described by the lubrication approximation. It is shown that rivulets on a sloping plate are stable regardless of their parameters, whereas rivulets on the underside of a plate can be either stable or unstable, depending on their widths and the plate's slope. For the case of a horizontal plate, sufficiently narrow rivulets are shown to be stable and sufficiently wide ones unstable, with the threshold width being pi/2 (sigma/g rho)(1/2) (rho and sigma are the liquid's density and surface tension, g is the acceleration due to gravity). It is also shown that, even though the plate's slope induces in a rivulet a sheared flow (which would normally be viewed as a source of instability) - in the present problem, it is a stabilizing factor. The corresponding stability criterion involving the rivulet's width and the plate's slope is computed, and it is demonstrated that, if the latter is sufficiently strong, all rivulets are stable regardless of their-widths.