Точные решения задачи о течении вязкой жидкости в цилиндрической области с меняющимся радиусом
In the frameworks of a class of exact solutions of the Navier–Stokes equations with linear dependence of part the speed components on one spatial variablethe axisymmetricalnonselfsimilar ﬂows of viscous ﬂuid in the cylindrical area which radius changes over the time under some law calculated during the solution are considered. The problem is reduced to two-parametrical dynamic system. The qualitative and numerical analysis of the system allowed to allocate three areas on the phase plane corresponding to various limit sizes of a pipe radius: radius of a pipe and stream velocity tend to inﬁnity for ﬁnite time, the area of a cross section of the cylinder tend to zero during a ﬁnite time span, radius of the tube inﬁnitely long time approaches to a constant value, and the ﬂow tend to the state of rest. For a case of ideal ﬂuid ﬂow the solution of the problem is obtained in the closed form and satisfying the slip condition.