Abstract: Two-dimensional unsteady stagnation-point flow of a viscoelastic fluid is studied assuming that it obeys the upper-convected Maxwell (UCM) model. The solutions of constitutive equations are found under the assumption that the components of the extra stress tensor are polynomials in the spatial variable along a rigid wall. The class of solutions for unsteady flows in a neighbourhood of the front or rear stagnation point on a plane boundary is considered, and the range of possible behaviors is revealed depending on the initial stage (initial data) and on whether the pressure gradient is an accelerating or decelerating function of time. The velocity and stress tensor component profiles are obtained by numerical integration of the system of nonlinear ordinary differential equations. The solutions of the equations exhibit finite-time singularities depending on the initial data and the type of dependence of pressure gradient on time. The aim of the present research is to consider the behavior of the solution in the problem on an unsteady flow of a viscoelastic medium near a critical point. A stationary solution of the problem of a flow directed to a critical point along a rigid boundary is not allowed [1]. At the same time, non-steady solutions are possible that can blow up in finite time or degenerate to a state of rest. In this paper, an attempt is made to analyze what causes such behavior of solutions. The solution of the initial–boundary value problem can blow up in finite time both because of the shape of the initial data and because of a specially chosen dependence of the pressure gradient on time. The problem with a weak discontinuity in the initial data, where a viscoelastic fluid flows onto a layer of medium at rest, is considered. Zones with steady-state values of the stress tensor compone

Cite: Moshkin N.P.
Non-stationary flow of a Viscoelastic Fluid near a Critical Point
Modern Achievements in Symmetries of Differential Equations (Symmetry 2022) 13-16 Dec 2022