Unsteady Flows of a Maxwell Viscoelastic Fluid near a Critical Point with a Countercurrent at the Initial Moment

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.