In the work a mathematical model for the non-stationary motion of liquid homogeneous viscous mixtures through pipelines is constructed. The corresponding integral equations expressing the laws of conservation of mass, momentum and energy are deriving, from which, in turn, we get the corresponding differential equations. The formulation of the initial and boundary conditions is given; the necessary additional relations that close the corresponding system of differential equations are indicated. A numerical algorithm for solving such a system is proposed. The corresponding numerical examples are given.