The acceleration and the,, establishment of viscos and viscoelastic media between two parallel plates acted upon by an abruptly applied constant pressure gradient is investigated. The Shvedov-Bingham model is used for the viscoplastic medium and a solution is sought in the form of layered fl~Β’ with a time-dependent thickness of the solid kernel. The problem is reduced to a one-dimensional heatconduction equation fo:r a rod of variable length, where the expansion law is not known in advance and an additional non-linear boundary condition is used to determine it. Two versions of the approximate solution are given. The first is a general modified Slezkin-Targ method, by means of which the problem of the motion of the medium acted upon by the specified time-varying pressure gradient is reduced to solving an ordinary differential equation. The second is an asymptotic method of solving the problem for short and long values of the dimensionless time. The time dependences of the flow rate and the boundary of the solid kernel are constructed for various rations of the pressure drop to the shear stress limit. The time taken for the flow rate and the boundary of the solid kernel to bece,me established are determined as function of this ratio.