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.
Highlights
It was found [1] that addition of a small number of polymers to water drastically decreases the friction drag
Various aspects of the dynamics of aqueous solutions of polymers were discussed in a special issue of the “Processes” journal [11]
The problem of steady motion of a polymer solution in a tube coincides with the classical problem of hydrodynamics of a viscous fluid for determining the only nonzero velocity component w μ ∆w = − A, ( x, y) ∈ S; w = 0, ( x, y) ∈ ∂S, where μ = ρν is the dynamic viscosity
Summary
It was found [1] that addition of a small number of polymers to water drastically decreases the friction drag. One more possible modification of the model of the model of motion of diluted aqueous solutions of polymers is to introduce an objective derivative of the tensor D [14]. In this case, the last derivative e in the right-hand side of Equation (5) is replaced by the expression 2Div(dD/dt. The present paper contains new examples of exact solutions for both of these models Their analysis shows that the qualitative properties of solutions described by mentioned models may differ from the solutions of the Navier-Stokes equations even at moderate Reynolds numbers. Together with the above-mentioned models, there are alternative models in dynamics of dilute polymer solutions, the so called Oldroyd-B model; see monograph [32] and the references presented there
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