Flagella beating in complex fluids are significantly influenced by viscoelastic stresses. Relevant examples include the ciliary transport of respiratory airway mucus and the motion of spermatozoa in the mucus-filled female reproductive tract. We consider the simplest model of such propulsion and transport in a complex fluid, a waving sheet of small amplitude free to move in a polymeric fluid with a single relaxation time. We show that, compared to self-propulsion in a Newtonian fluid occurring at a velocity UN, the sheet swims (or transports fluid) with velocity U∕UN=(1+De2ηs∕η)∕(1+De2), where ηs is the viscosity of the Newtonian solvent, η is the zero-shear-rate viscosity of the polymeric fluid, and De is the Deborah number for the wave motion, product of the wave frequency by the fluid relaxation time. Similar expressions are derived for the rate of work of the sheet and the mechanical efficiency of the motion. These results are shown to be independent of the particular nonlinear constitutive equations ...

Propulsion in a viscoelastic fluid

A new method for numerical calculation of viscoelastic flow based on simulation of molecular models of polymers is presented. The CONNFFES-SIT ( C alculation o f N on- N ewtonian F low: F inite E lements and S tochastic S imulation T echnique) approach directly combines standard finite element methods as currently used in the calculation of viscoelastic flow with stochastic simulations of polymer dynamics and thus obviates the need for a rheological constitutive equation to describe the fluid. The stresses are obtained from the molecular configurations rather than from constitutive equations. As an illustration of the method, the time development of plane Couette flow is studied for the upper-convected Maxwell, Oldroyd-B, FENE-P and FENE fluids. For the upper-convected Maxwell, Oldroyd-B and FENE-P models comparisons with analytical and existing numerical solutions are presented. Significant deviations between the behavior of the FENE-P and FENE models for the start-up of plane Couette flow are found.

Calculation of viscoelastic flow using molecular models: the connffessit approach

Introduction.- The Source of Examples.- Invariance Equation in the Differential Form.- Film Extension of the Dynamics: Slowness as Stability.- Entropy, Quasi-Equilibrium and Projector Field.- Newton Method with Incomplete Linearization.- Quasi-chemical Representation.- Hydrodynamics from Grad's Equations: Exact Solutions.- Relaxation Methods.- Method of Invariant Grids.- Method of Natural Projector.- Geometry of Irreversibility: The Film of Nonequilibrium States.- Slow Invariant Manifolds for Open Systems.- Estimation of Dimension of Attractors.- Accuracy Estimation and Post-Processing.- Conclusion.

http://www.math.le.ac.uk/people/ag153/homepage/Beginning.pdf

Invariant Manifolds for Physical and Chemical Kinetics

It is well known that the drag in a turbulent flow of a polymer solution is significantly reduced compared to Newtonian flow. Here we consider this phenomenon by means of a direct numerical simulation of a turbulent channel flow. The polymers are modelled as elastic dumbbells using the FENE-P model. In the computations the polymer model is solved simultaneously with the flow equations, i.e. the polymers are deformed by the flow and in their turn influence the flow structures by exerting a polymer stress. We have studied the results of varying the polymer parameters, such as the maximum extension, the elasticity and the concentration. For the case of highly extensible polymers the results of our simulations are very close to the maximum drag reduction or Virk (1975) asymptote. Our simulation results show that at approximately maximum drag reduction the slope of the mean velocity profile is increased compared to the standard logarithmic profile in turbulent wall flows. For the r.m.s. of the streamwise velocity fluctuations we find initially an increase in magnitude which near maximum drag reduction changes to a decrease. For the velocity fluctuations in the spanwise and wall-normal directions we find a continuous decrease as a function of drag reduction. The Reynolds shear stress is strongly reduced, especially near the wall, and this is compensated by a polymer stress, which at maximum drag reduction amounts to about 40% of the total stress. These results have been compared with LDV experiments of Ptasinski et al. (2001) and the agreement, both qualitatively and quantitatively, is in most cases very good. In addition we have performed an analysis of the turbulent kinetic energy budgets. The main result is a reduction of energy transfer from the streamwise direction, where the production of turbulent kinetic energy takes place, to the other directions. A substantial part of the energy production by the mean flow is transferred directly into elastic energy of the polymers. The turbulent velocity fluctuations also contribute energy to the polymers. The elastic energy of the polymers is subsequently dissipated by polymer relaxation. We have also computed the various contributions to the pressure fluctuations and identified how these change as a function of drag reduction. Finally, we discuss some cross-correlations and various length scales. These simulation results are explained here by two mechanisms. First, as suggested by Lumley (1969) the polymers damp the cross-stream or wall-normal velocity fluctuations and suppress the bursting in the buffer layer. Secondly, the ‘shear sheltering’ mechanism acts to amplify the streamwise fluctuations in the thickened buffer layer, while reducing and decoupling the motions within and above this layer. The expression for the substantial reduction in the wall drag derived by considering the long time scales of the nonlinear fluctuations of this damped shear layer, is shown to be consistent with the experimental data of Virk et al. (1967) and Virk (1975).

/pdf/turbulent-channel-flow-near-maximum-drag-reduction-4thv5xatr2.pdf

Turbulent channel flow near maximum drag reduction: simulations, experiments and mechanisms

Within the last decade, fluorescence microscopy of single molecules of DNA in a plethora of flow fields has allowed an unprecedented examination of the dynamics of polymers in flow. As a result, new principles (e.g. “molecular individualism”) have been developed regarding these dynamics and old debates (e.g. conformational hysteresis of polymers in extensional flow) have received a fresh airing. The coupling of the microscopy, employing a spectrum of possible DNA molecules, fragments, and concatemers with dynamic simulation (e.g. Brownian dynamics) has created a tremendous opportunity to both revisit unanswered questions in the field of polymer solution dynamics (particularly in the nonlinear flow regime), as well as to examine new areas surrounding the engineering of flow dynamics of complex fluids in microfluidic devices. The foundation for future studies has been laid by careful experimentation and computer simulation of DNA dynamics in bulk or locally linear flows of many types. Future studies involving collective effects, complex flows, flow-induced reaction, adsorption and so forth are just beginning and again the promise is to develop a new understanding of these polymer processes by examining DNA “one molecule at a time.”

The dynamics of single-molecule DNA in flow

A complete rheological equation of state for dilute polymer solutions is obtained by modelling the polymer molecules as bead—spring chains, in which the springs are finitely extensible. Hydrodynamic interactions among the beads are accounted for approximately by using the equilibrium-averaged Oseen tensor as in the Zimm theory. The rheological equation of state thus obtained may be regarded as an extension of the Lodge—Wu equation. Consequently, it incorporates all the good features of the Zimm theory, but it can describe nonlinear viscoelastic phenomena as well, such as monotone decreasing shear-rate-dependent viscosity. Comparisons with experimental data on shear flows are encouraging.

R.B. Bird

Papers

Polymer solution rheology based on a finitely extensible bead—spring chain model