Free surface flows of polymer solutions with models based on the conformation tensor

Abstract

A computational method is presented for analyzing free surface flows of polymer solutions with the conformation tensor. It combines methods of computing Newtonian free surface flows [J. Comp. Phys. 99 (1992) 39; V.F. deAlmeida, Gas–Liquid Counterflow Through Constricted Passages, Ph.D. thesis, University of Minnesota, Minneapolis, MN, 1995 (Available from UMI, Ann Arbor, MI, order number 9615160); J. Comp. Phys. 125 (1996) 83] and viscoelastic flows [J. Non-Newtonian Fluid Mech. 60 (1995) 27; J. Non-Newtonian Fluid Mech. 59 (1995) 215]. Modifications are introduced to compute a traceless velocity gradient, to impose inflow boundary conditions on the conformation tensor that are independent of the specific model adopted, and to include traction boundary conditions at free surfaces and open boundaries. A new method is presented for deriving and coding the entries of the analytical Jacobian for Newton’s method by keeping the derivatives of the finite element weighted residual equations with respect to the finite element basis functions in their natural vector and tensor forms, and then by mapping such vectors and tensors into the elemental Jacobian matrix. A new definition of extensional and shear flow is presented that is based on projecting the rate of strain tensor onto the principal basis defined by the conformation tensor. The method is validated with two benchmark problems: flow around a cylinder in a channel, and flow under the downstream section of a slot or knife coater. Regions of molecular stretch—determined by monitoring the eigenvalues of the conformation tensor—and molecular extension and shear rate—determined by projecting the rate of strain dyadic onto the eigenvectors of the conformation tensor—are shown in the flow around a cylinder of an Oldroyd-B liquid. The free surface coating flow between a moving rigid boundary and a parallel static solid boundary from which a free surface detaches is analyzed with several models of dilute and semidilute solutions of polymer of varying degree of stiffness based on the conformation tensor approach [J. Non-Newtonian Fluid Mech. 23 (1987) 271; A.N. Beris, B.J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, 1st ed., Oxford University Press, Oxford, 1994; J. Rheol. 38 (1994) 769; M. Pasquali, Polymer Molecules in Free Surface Coating Flows, Ph.D. thesis, University of Minnesota, Minneapolis, MN, 2000 (Available from UMI, Ann Arbor, MI, order number 9963019)]. Although the boundary conditions at the static contact line introduce a singularity, that singularity does not affect the computation of flows at high Weissenberg number when a recirculation is present under the static boundary. In this case, a steep layer of molecular stretch develops under the free surface downstream of the stagnation point. Here the polymer aligns with its principal stretching axis parallel to the free surface. When the recirculation is absent, the singularity at the contact line strongly affects the computed velocity gradient, and the computations fail at moderate Weissenberg number irrespective of the polymer model and of whether the polymer is affecting the flow or not.