Analysis of the Models of Motion of Aqueous Solutions of Polymers on the Basis of Their Exact Solutions

Abstract

Abstract:The qualitative properties of solutions of a hereditary model of motion of aqueous solutions of polymers, its modification in the limiting case of short relaxation times, and a similar second grade fluid model are studied. Unsteady shear flows are considered. In the first case, their properties are similar to those of motion of a usual viscous fluid. Other models can include weak discontinuities, which are retained in the course of fluid motion. Exact solutions are found by using the group analysis of the examined systems of equations. These solutions describe the fluid motion in a gap between coaxial rotating cylinders, the stagnation point flow, and the motion in a half-space induced by plane rotation (analog of the Karman vortex). The problem of motion of an aqueous solution of a polymer in a cylindrical tube under the action of a streamwise pressure gradient is considered. In this case, a flow with straight-line trajectories is possible (analog of the Hagen-Poiseuille flow). In contrast to the latter, however, the pressure in the flow considered here depends on all three spatial variables.

Exact Solutions of Boundary Layer Equations in Polymer Solutions

“…The corresponding nonstationary problem and the stationary solutions of the problem of the motion of a polymer solution near stagnation points were studied and investigated in [18][19][20]. This section provides three exact solutions of the nonstationary problem of Equation (2).…”

“…where v is the velocity vector, p is the pressure, ρ is the fluid density, ν is the kinematic viscosity and κ is the normalized relaxation viscosity [2],…”

“…Stationary solutions of system (1) were analyzed in [8,9]. The group properties of Equation (1) and the construction of their exact solutions were studied in [2,[10][11][12].…”

“…For g = 0, one can assume that g = 0 by virtue of an equivalence transformation, and the generators (3) are extended by the additional admitted generator ∂ t . The paper [13] also contains a number of exact solutions of system (2). The knowledge of an admitted Lie group allows for constructing invariant and partially invariant solutions.…”

“…The present paper is devoted to constructing invariant and partially invariant solutions of system (2). It is organized as follows.…”