Propagation of perturbations in a thin layer of a fluid with viscosity stratification

We consider a non-linear system of equations describing motion of a viscosity-layered fluid with a free surface in a long-wave approximation. In a semi-Lagrangian coordinate system we rewrite the governing equations in a integro-differencial form for which the necessary and sufficient hyperbolicity conditions are stated. An approximation for the integro-differential model in a form of finite-dimensional system of differential conservation laws with a right part is suggested. A modeling of propagation of nonlinear perturbations in a fluid with viscosity stratification was performed. In particular a problem about the evolution of a more viscous fluid column in a less viscous fluid during the passage of wave disturbances is considered.