The capillary instability of an infinite axisymmetric viscous liquid column in an immiscible medium is investigated. The process of disintegration is simulated numerically using a (second-order) finite-difference method applied to the ’vorticity-stream function’ formulation of the Navier-Stokes equations. These equations and corresponding boundary conditions are written in there detailed form including convective terms in Navier-Stokes equations and nonlinear terms in the mass and momentum conservation equations at the unknown interface. Then the evolution in time of a given cosinusoidal disturbance is studied subjected to the action of the nonlinear effects. In these conditions the formation of a satellite drop attached to the main drop is observed. In the case where the liquid column is submerged into a low density inviscid fluid, the basic characteristics of the column disintegration such as drops sizes and breakup time are in a good agreement with those calculated by previous authors. New results are obtained for the instability parameters of a liquid column surrounded by another viscous fluid.