Langevin analysis of dumbbells with hydrodynamic interaction

Summary In this communication we consider the hydrodynamic interaction problem in dumbbell kinetic theory using the traditional Langevin analysis. The pre-averaged Oseen tensor is avoided in the formulation. It is shown that the behaviour of the dumbbell is qualitatively different from that predicted by models employing the pre-averaged Oseen tensor; the most important result is that the predicted non-Newtonian viscosity in a steady shearing flow now occurs at the experimentally observed order of magnitude of shear rate. This contrasts with the constant viscosity predicted by the pre-averaged calculation.

---

Zusammenfassung In dieser Veröffentlichung wird das Problem der hydrodynamischen Wechselwirkung in der kinetischen Theorie der Hanteln mit der traditionellen Langevin-Analyse behandelt. Bei dieser Methode wird die Vorab-Mittelung des Oseen-Tensors vermieden. Es wird gezeigt, daß das Verhalten der Hanteln von demjenigen qualitativ verschieden ist, das durch das Modell des vorab gemittelten Oseen-Tensors vorausgesagt wird. Das wichtigste Ergebnis ist, daß die so bestimmte nicht-newtonische Viskosität für die stationäre Scherströmung jetzt größenordnungsmäßig mit der experimentell beobachteten übereinstimmt, im Gegensatz zu der bei der Vorab-Mittelung erhaltenen konstanten Viskosität.

---

---

With 7 figures     

The constitutive equation of a dilute suspension of spherical microcapsules

This paper studies the rheological behavior of a dilute suspension of spherical microcapsules, i.e. spherical thin elastic membranes filled with an incompressible liquid. Previous results obtained for the motion of such capsules freely suspended in a simple shear flow are first extended to general linear shear flows. Then the stresslet term in the external flow is computed to order ?², where ?, assumed to be small with respect to unity, is the ratio of viscous to elastic forces acting on the particle. The resulting constitutive equation is of the viscoelastic type and is similar to the one obtained for liquid droplets. It predicts that the microcapsule suspension exhibits a shear dependent viscosity and normal stress effects. The exact dependency of these phenomena on the microscopic parameters of the suspension is explicitly provided by the model.     

A new approximate solution of nonlinear diffusion equation

In this paper,the same problem in ref.[1] is studied.The author’s solutionapproximately satisfies the whole fundamental equations (1.1)and(1.2)and the wholebound values conditions (1.3-1.5).But the Liu’s solution does not satisfy theequation of continuity(1.2).     

A constitutive equation for a non-Newtonian liquid

The K-BKZ constitutive equation for non-Newtonian liquids usually uses as invariants the coefficients of the characteristic equation of a certain finite strain tensor. We show that the use of the roots of this equation, the squares of the principal stretches, is more efficient. We use a product of the linear relaxation spectrum and a simple sum of powers of the principal stretches as the energy function in the K-BKZ theory. The predictions of this theory agree with Meissner's data to a good approximation over many decades.Dedicated to Professor Arthur S. Lodge on the occasion of his 70th birthday and his retirement from the University of Wisconsin.     

Kinetic models for dilute solutions of dumbbells in non-homogeneous flows revisited

We propose a two-fluid theory to model a dilute polymer solution assuming that it consists of two phases, polymer and solvent, with two distinct macroscopic velocities. The solvent phase velocity is governed by the macroscopic Navier–Stokes equations with the addition of a force term describing the interaction between the two phases. The polymer phase is described on the mesoscopic level using a dumbbell model and its macroscopic velocity is obtained through averaging. We start by writing down the full phase-space distribution function for the dumbbells and then obtain the inertialess limits for the Fokker–Planck equation and for the averaged friction force acting between the phases from a rigorous asymptotic analysis. The resulting equations are relevant to the modelling of strongly non-homogeneous flows, while the standard kinetic model is recovered in the locally homogeneous case.     

Numerical perturbation method for approximate solution of poisson's equation on a moderately deforming grid

In problems such as the computation of incompressible flows with moving boundaries, it may be necessary to solve Poisson's equation on a large sequence of related grids. In this paper the LU decomposition of the matrix A ₀ representing Poisson's equation discretized on one grid is used to efficiently obtain an approximate solution on a perturbation of that grid. Instead of doing an LU decomposition of the new matrix A , the RHS is perturbed by a Taylor expansion of A ^?1 about A ₀. Each term in the resulting series requires one ‘backsolve’ using the original LU . Tests using Laplace's equation on a square/rectangle deformation look promising; three and seven correction terms for deformations of 20% and 40% respectively yielded better than 1% accuracy. As another test, Poisson's equation was solved in an ellipse (fully developed flow in a duct) of aspect ratio 2/3 by perturbing about a circle; one correction term yielded better than 1% accuracy. Envisioned applications other than the computation of unsteady incompressible flow include: three-dimensional parabolic problems in tubes of varying cross-section, use of ‘elimination’ techniques other than LU decomposition, and the solution of PDEs other than Poisson's equation.     

A model of dilute polymer solutions with hydrodynamic interaction and finite extensibility. II. Shear flows

The results from numerical calculations for steady shear, the start-up and cessation of steady shear, and the stress relaxation after a step shear stain are discussed in detail for a bead-spring chain model with consistently averaged hydrodynamic interaction between the beads and consistently averaged finite extensible springs. Calculations are made for a large range of spring stiffnesses 10 ⩽ b ⩽ ∞ and a hydrodynamic interaction strength h^* = 0.15 a value which has been estimated from experimental results. This model is found to satisfy the Hassager-Bird and the Lodge-Meissner relations.     

A simplified frequency equation and its approximate solution of a beam with an incipient crack from a wave perspective

This paper presents a simplified frequency equation and its approximate solution to predict the modal frequencies of a beam with an incipient crack. The physical implication of the simplified frequency equation is fully described from a wave perspective for the cracked beam with arbitrary support conditions. The approximate solution of the proposed frequency equation is derived from a wave perspective as well. The asymptotic equivalence is demonstrated between the approximate solution and that obtained by the first order perturbation method as the mode number increases. The validity of the proposed approach is demonstrated through comparison to both numerical results from finite element analysis and experimental data.     

A differential equation for approximate wall distance

A partial differential equation to compute the distance from a surface is derived and solved numerically. The benefit of such a formulation especially in combination with turbulence models is shown. The details of the formulation as well as several examples demonstrating the influence of its parameters are presented. The proposed formulation has computational advantages and can be favourably incorporated into one‐ and two‐equation turbulence models like e.g. the Spalart–Allmaras, the Secundov or Menter's SST model. Copyright © 2002 John Wiley & Sons, Ltd.     

A constitutive equation of elastic-visco-plasticity for polymers

The rheological behavior and a constitutive relationship of elastic-visco-plasticity for polymers are investigated in this paper. Several sets of experiments have been carried out to determine the material constants and to test the validity of constitutive formulation. It is shown that the theoretical profiles are in good agreement with experimental results. The rheological characteristic and the strain-rate effect of model are analyzed by computer simulation.The project is supported by the National Natural Science Foundation of China     

WAVELET APPROXIMATE INERTIAL MANIFOLD AND NUMERICAL SOLUTION OF BURGERS'''' EQUATION

ＩｎｔｒｏｄｕｃｔｉｏｎＢｕｒｇｅｒｓ’ｅｑｕａｔｉｏｎｉｓｏｎｅｏｆｖｅｒｙｉｍｐｏｒｔａｎｔｆｌｕｉｄｅｑｕａｔｉｏｎｓ.Ｉｔｓｄｙｎａｍｉｃａｌｂｅｈａｖｉｏｒｉｓｒａｔｈｅｒｔｈａｎｃｏｍｐｌｅｘｉｔｙｏｗｉｎｇｔｏｔｈｅｎｏｎｌｉｎｅａｒｔｅｒｍ .Ｔｅｍａｎｅｔａｌ.[1] ｓｔｕｄｉｅｄｔｈｅｅｘｉｓｔｅｎｃｅｏｆａｇｌｏｂａｌａｔｔｒａｃｔｏｒ,ｉｎｅｒｔｉａｌｍａｎｉｆｏｌｄａｎｄａｐｐｒｏｘｉｍａｔｅｉｎｅｒｔｉａｌｍａｎｉｆｏｌｄａｓｓｏｃｉａｔｅｄｗｉｔｈｄｙｎａｍｉｃａｌｓｙｓ…     

The effect of the hydrodynamic interaction on the rheological properties of Hookean dumbbell suspensions in steady state shear flow

The diffusion equation for the configurational distribution function of Hookean dumbbell suspensions with the hydrodynamic interaction (HI) was solved, in terms of Galerkin’s method, in steady state shear flow; and viscosity,first and second normal-stress coefficients and molecular stretching were then calculated. The results indicate that the HI included in a microscopic model of molecules gives rise to a significant effect on the macroscopic properties of Hookean dumbbell suspensions. For example, the viscosity and the first normal stress coefficient, decreasing as shear rate increases, are no longer constant; the second normal-stress coefficient, being negative with small absolute value and shear-rate dependent, is no longer zero; and an additional stretching of dumbbells is yielded by the HI. The viscosity function and the first normal-stress coefficient calculated from this method are in agreement with those predicted from the self-consistent average method qualitatively, while the negative second normal-stress coefficient from the former seems to be more reasonable than the positive one from the latter.     

A differential constitutive equation for polymer melts

A modification of the Giesekus constitutive equation is derived by incorporating (approximately, via the Peterlin approximation) the finite extensibility of polymer molecules into dumbbell kinetic theory along with the anisotropic hydrodynamic drag suggested by Giesekus. The constitutive equation that is obtained retains much of the simplicity of Giesekus' constitutive equation, but it involves terms that are cubic in the stress as well as those that are quadratic. It is shown that the constitutive equation quantitatively describes the steady elongational viscosity of the IUPAC polymer melt A (including the strain softening of the melt), but it cannot describe the elongational and shear viscosities simultaneously. It is also shown that the constitutive equation satisfies the Lodge-Meissner relation for shear strains less than unity.     

水泥砂浆的一个热粘弹性率型损伤本构模型

利用SHPB实验系统及自行研制的混凝土类材料快速高温加热设备,对水泥砂浆试件进行了不同 温度(20~600℃)和3种冲击速度下的实验,得到了不同温度和冲击速度下水泥砂浆试件的应力应变关系曲 线。基于ZWT粘弹性本构模型,并且考虑高温下水泥砂浆损伤演化规律都服从Weibull分布,提出了一个水 泥砂浆的热粘弹性率型损伤本构模型。通过数据拟合,获得了本构模型的相关参数,结果表明:理论预测和实 验结果吻合良好。