Core-annular flows are very important for oil companies. In general, the crude is transported through water lubricated pipes, and usually instabilities are present. Applied mathematicians are very interested in the study of these instabilities and ways how to affect them. In this work, we present several ways how to do so, and consider that the system is axisymmetric. Also, we study two situations, one with constant pressure gradient at the beginning of the problem and, the other one, without pressure gradient. One way to affect the instability is by adding insoluble surfactants at the interface between the two fluids, distributed uniformly and non-uniformly throughout the interface. Using an asymptotic analysis, a coupled system of two nonlinear integro-differential equations is derived for each case. Implementing a pseudo-spectral method applied to the system of the first situation, an extensive number of numerical experiments are explored. The numerical results show a rich behavior of the solution as the model parameters vary. The possible long time behavior, of the numerical solutions, includes stationary and travelling wave attractors, as well as time-periodic and chaotic attractors. On the other hand, we also consider a moving coaxial rod in the direction of the tube axis. A full problem is derived carrying out an asymptotic solution of the problem when the annulus is thin with respect to the core thickness.

Key words: core annular film flow, surfactant, interface, asymptotic, pseudo-spectral.