Modeling of the waves on a surface of a cylindrical configuration of magnetic fluid, surrounding a long porous core

A mathematical model of waves propagation and instability on a surface of a cylindrical column of magnetic fluid of an infinite length, surrounding a coaxial infinite long porous core of the round section is constructed and studied. The conditions are found under which the disturbances of the surface of fluid column become unstable and result in its fragmentation into a chain of connected droplets. The presence of a surface tension is taken into account. The gravity is neglected. The axis of the porous cylinder coincides with the axis of a coaxial solenoid, generating an uniform magnetic field. The problem is solved in a cylindrical coordinate system ( r, θ, z ), in which the fluid column is at rest. The z -axis is directed along the axis of the solenoid. The equations of the motion of magnetic fluid inside and outside of the porous medium and the equations for the magnetic field are written. The boundary conditions for hydrodynamic and magnetic values on discontinuity surfaces are formulated. The disturbances of magnetic field (according to wave propagation) is searched inside and outside of the porous medium, as well as in the air clearance of the solenoid. The full solution of a boundary value problem for hydrodynamic and magnetic values is found. The numerical analysis of the dispersion equation, describing wave propagation on surface is done. The different special cases are considered. The conditions are found under which the disturbances of the surface of fluid column become stable (wave damping) or become unstable (which result to the disturbances growth and the fragmentation of the cylinder into a chain of droplets). It is shown that the size of droplets appearanced in fragmentation process increases with the growth of the magnetic field i.e. magnetic field has a stabilizing influence upon the fragmentation of the fluid column.