Dynamics of surface waves of a ferrofluid film

Abstract

This work aims to study the dynamical properties of nonlinear magnetized surface waves propagating on the free surface of a ferrofluid liquid film flowing down a vertical cylinder. The fluid film is coated by a motionless light gas and the system is assumed to be stressed by a radial magnetic field. The non-dimensional governing equations together with the appropriate boundary conditions are formulated using long wave approximation, and then we obtained the equations of wave evolution by means of the weighted residual integral boundary layer (WRIBL). According to the linear instability analysis, the dispersion relation is determined in the case of the entire domain, as well as in the limiting case of small wave number. Consequently the conditions of stability for the linear waves are determined. Numerical illustrations are achieved in order to demonstrate the effects of the various physical variables on the stability criteria of these waves. In the nonlinear case, the evolution equations are transformed to new ones that referred to a moving frame of coordinates that travels at the same celerity of the waves. Such equations are converted to a third-order dynamical system of the surface deflection and their derivatives. The fixed points of that system have been assigned and thus their behavior is studied with variation in the values of the wave celerity and the radius of the cylinder. In addition, the case of solitary waves is investigated in detail. Then, the response of the pattern of behavior of these waves are examined as a result of changes in the values of some parameters such as the magnetic number, radius of the internal cylinder and the Reynolds number.