From molecular models to the solution of flow problems

Abstract

One of the goals in polymer fluid dynamics is to use kinetic theory to derive a constitutive equation for a polymeric liquid starting from a molecular model, to use the constitutive equation to solve flow problems, and finally to use kinetic theory to describe molecular stretching and orientation. We illustrate this procedure by using a finitely extensible, nonlinear, elastic dumbbell with the Peterlin approximation (FENE-P dumbbell) as a crude model of a polymer molecule and then derive the constitutive equation. A number of new results for the FENE-P dumbbell are given which include an analytic expression for the elongational viscosity, numerical calculations of the stress growth after the start-up of elongational flow, and the third-order fluid constants. The technique of "model matching" is illustrated for squeezing flow between circular disks, the Weissenberg rod-climbing effect, and the torque on a rotating sphere. Conclusions can be drawn about the average extension and alignment of the dumbbell in these three flow fields. (5) where Qo is the maximum extension. Equation 4 cannot f(Q2) = H/[1 -(Q/QO)~I

Viscoelastic effects on higher order statistics and on coherent structures in turbulent channel flow

“…16 The same trend has also been observed in the presence of viscoelasticity. 3,8,9 In our companion work, 3 we reported first order statistic results for a series of viscoelastic DNS in a channel flow obtained with the FENE-P constitutive equation 18 which is a kinetic theory-based, legitimate model for dilute polymer solutions. 19 The results were corresponding to fixed values for the molecular extensibility parameter L = 30 and the zero shear-rate solvent viscosity ratio ␤ 0 = 0.9.…”

Accurate asymptotic formulas for the transient PDF of a FENE dumbbell in suddenly started uniaxial extension followed by relaxation

“…Loops in the stress-extension plane are observed in transient elongation, because different PDFs in the stretching versus relaxing phases can be characterized by the same value of the second moment, while leading to different stresses [21,24,27,29]. At least a second-order closure, such as the L closure of Lielens et al [11,12], is required to capture this memory effect.…”

“…Note the appearance of an O(ln ) term, proportional to the homogeneous solution P [I] 0 (η, T ), between the ord( −1 ) and ord(1) terms in the perturbation expansion (27).…”

Asymptotic basis of the L closure for finitely extensible dumbbells in suddenly started uniaxial extension

“…The beads represent two centers of hydrodynamic friction which can be pulled in different directions by the flow field. The resistance to unraveling and stretching of a real coiled macromolecule is primarily of entropic origin, for which the spring law represents an enthalpic substitute, and should be finitely extensible to prevent unbounded stretching of the dumbbell in strong flows [1,24,26,28]. The state of the dumbbell at any instant is described in statistical terms, by a probability density function defined on the space of internal (conformational) degrees of freedom: orientation and length.…”

“…Systematic selection of state variables based upon a maximum entropy principle [5] and applications to liquid crystalline polymers [6] have been developed more recently. Hysteresis loops in the stress-extension plane are observed in transient elongation, because different PDF's in the stretching versus relaxing phases can be characterized by the same value of the second moment, while leading to different stresses [26,28,20,16]. At least a second-order closure is required to capture this feature.…”