We consider a fluid film on the inner walls of a capillary. The film surrounds another fluid in the core. It is known that the capillary instability, driven by the surface tension at the fluid-fluid interface, breaks up the film if it is primarily stagnant. In contrast, as we show, a primary flow, in a certain range of parameters, can keep the linearly unstable film from rupturing. This is a result of the nonlinear low-level saturation of the interface instability. This saturation is due to the coordinated action of the destabilizing factors, the shear of flow, and the surface tension at the interface. The resulting state of the interface is, in general, chaotic oscillations, with their amplitude being much less than the unperturbed film thickness. The approximate equation of interface evolution is derived. The saturation mechanism is explained. The characteristic scales of the developed oscillations are found, and the parameter range of the theory applicability is discussed.