Rectilinear Vortex Thread in a Radially Nonhomogeneous Bingham Solid

We study an initial boundary value problem of axially symmetric one-dimensional unsteady shear in the viscoplastic space (a Bingham solid) initiated by a rectilinear vortex thread located along the symmetry axis. The force intensity of the thread is represented by a given monotone piecewise continuous function of time. The density and the dynamical viscosity of the medium are constant, and the yield point is a given piecewise continuous function of radius. We find similar and quasisimilar expressions for the tangent stress and for the rotating component of the velocity both in viscoplastic shear domains and in rigid zones. We show that the vortex thread with time-bounded force intensity may generate a viscoplastic shear only inside a cylinder of certain radius. If the thread intensity growth linearly with time, then the radius of the shear domain grows proportionally to \(\sqrt t\).