液晶高分子各向异性粘弹性流体本构方程理论

将液晶高分子－各向异性流体的本构方程,建立在Oldroyd随体导数观点基础上。推广上随机Oldroyd B流体模型,提出共转OldroydB流体模型,同时将微观结构的影响通过宏观参数表示出来,使在宏观理论中包含微观结构的贡献,即引入取向物质函数,非线性各向异性黏度函数和各向异性松弛时间及推迟时间等,表征取向运动对黏度和松弛及推迟现象的影响,在此基础上开展了一类新的液晶高分子－Oldroyd型本构方程理论,由该类型本构方程得出的物质函数,液晶高分子流体的第一、第二法向应力差与实验结果一致,解释了液晶高分子溶液的第一、第二法向应力差的特殊流变学行为。     

Constitutive equation of co-rotational derivative type for anisotropic-viscoelastic fluid

A constitutive equation theory of Oldroyd fluid B type, i.e. the co-rotational derivative type, is developed for the anisotropic-viscoelastic fluid of liquid crystalline (LC) polymer. Analyzing the influence of the orientational motion on the material behavior and neglecting the influence, the constitutive equation is applied to a simple case for the hydrodynamic motion when the orientational contribution is neglected in it and the anisotropic relaxation, retardation times and anisotropic viscosities are introduced to describe the macroscopic behavior of the anisotropic LC polymer fluid. Using the equation for the shear flow of LC polymer fluid, the analytical expressions of the apparent viscosity and the normal stress differences are given which are in a good agreement with the experimental results of Baek et al. For the fiber spinning flow of the fluid, the analytical expression of the extensional viscosity is given. The project supported by the National Natural Science Foundation of China (19832050 and 10372100)     

Squeeze film flow of ideal elastic liquids

An exact solution is presented for the squeeze film flow of an Oldroyd B. fluid. The solution demonstrates that the flow kinematics is similar to the Newtonian (or Maxwellian) one. Theoretical predictions for constant velocity squeezing are compared to experimental observation for well characterized non-shear thinning elastic fluids. It is shown both theoretically and experimentally that the effect of elasticity in a constant velocity squeeze film flow is to always reduce the load relative to the inelastic (Newtonian) prediction and that this load reduction falls between the upper and lower asymptote prediction by the exact solution for the Oldroyd B fluid. The upper load asymptote is given by the Stefan solution for the viscosity of the polymer solution and the lower asymptote is given by the Stefan solution for the viscosity of the solvent. Experimental observations agree with the theoretical prediction for the Oldroyd B fluid at low shear rates where it is shown that the steady and dynamic flow properties of the test fluids used in the experimental program are well represented by the Oldroyd B constitutive equation. With the exception of the work of Lee et al. [6] for constant load squeezing of a Maxwell fluid, this work represents one of the few cases where experimental observation of large effects due to elasticity are indeed predicted with a constitutive equation which actually describes the steady and dynamic shear properties of the fluids used in the experimental program.     

Coupling of finite element method and discontinuous Galerkin method to simulate viscoelastic flows

In this paper, a numerical method, which is about the coupling of continuous and discontinuous Galerkin method based on the splitting scheme, is presented for the calculation of viscoelastic flows of the Oldroyd‐B fluid. The momentum equation is discretized in time by using the Adams‐Bashforth second‐order algorithm, and then decoupled via the splitting approach. Considering the Oldroyd‐B constitutive equation, the second‐order Runge‐Kutta approach is selected to complete the temporal discretization. As for the spatial discretizations, the fundamental purpose is to make the best of finite element method (FEM) and discontinuous Galerkin (DG) method to handle different types of equations. Specifically speaking, for the subequations, FEM is chosen to treat the Poisson and Helmholtz equations, and DG is employed to deal with the nonlinear convective term. In addition, because of the hyperbolic nature, DG is also utilized to discretize the Oldroyd‐B constitutive equation spatially. This coupled method avoids resorting to extra stabilization technique occurred in standard FEM framework even for moderately high values of Weissenberg number and also reduces the complexity compared with unified DG scheme. The Oldroyd‐B model is applied to investigate several typical and challenging benchmarks, such as the 4:1 planar contraction flow and the lid‐driven cavity flow, with a wide range of Weissenberg number to illustrate the feasibility, robustness, and validity of our coupled method.     

Alternative derivation of differential constitutive equations of the Oldroyd-B type

Now that almost 60 years have passed since the pioneering works of J.G. Oldroyd it seems appropriate as an homage to consider here constitutive equations that can be viewed as generalisations of the by now classical Oldroyd-B model. In this short communication we shall address heuristically the theme of differential constitutive models and will provide an alternative way of deriving a “modified FENE” equation (FENE-M) and inter-relating the PTT and FENE-P-like models.     

Natural convection of a viscoelastic fluid

The influence of elasticity and shear thinning viscosity on the temperature distribution and heat transfer in natural thermal convection is discussed. The numerical investigations are based on a four-parameter Oldroyd constitutive equation, which represents the typical fluid response of dilute solutions and melts. It was found that especially the second normal-stress difference affects the heat transfer mechanism.     

Flow of an inelastic Non-Newtonian fluid through an abrupt contraction: A recalculation

The axi-symmetric flows of Newtonian and Non-Newtonian fluids through a sudden one-to-four contraction are numerically simulated. As the Non-Newtonian constitutive equation the inelastic form of the four-constant Oldroyd implicit rheological model is used. Because of the Non-Newtonian viscosity, a single non-dimensional parameter cannot be found to characterize the equations of motion. Therefore in this work the equations of motion are solved using dimensional values, that is, with actual increase of the mass flow.     

Flow of an Oldroyd-fluid in a contracting channel with a moving boundary

The industrial process of coating flat surfaces with polymeric substances is numerically simulated by solving the full equations of motion for a flow through a contraction with a moving boundary. The four-constant Oldroyd constitutive equation is used to represent the viscoelastic fluid. Some adjustments to existing finite-difference methods are made in such a way as to avoid singular iterative matrices during the solution process. Results are presented for flow situations with Weissenberg numbers up to about three times larger than any previously published results for this problem.     

Surface tension driven oscillations of a bubble in a viscoelastic liquid

Small amplitude surface tension driven oscillations of a spherical bubble in a dilute polymer solution are considered. The rheological properties of the liquid are modelled by using a 3-constant constitutive equation of the Oldroyd type. The Laplace transform of the solution of the initial value problem is inverted numerically. As in the Newtonian fluid case, both a discrete and a continuous spectrum occurs. In addition to the non-dimensional parameters in the corresponding problem for a Newtonian fluid, the results depend on two other parameters: the ratio of the relaxation time of the polymer solution and the time scale of the flow (the Deborah number) and the product of the polymer concentration and the intrinsic viscosity. For small bubbles in an aqueous solution having a small relaxation time, significant additional damping is found even for dilute solutions.     

A rheological model for materials which support coexistent shear rates

In this work we study a version of the three constant differential-type Oldroyd constitutive relation which allows distinct objective time derivatives for the extra stress and the stretching. We integrate the constitutive equation and determine an equivalent history integral representation for this model for the general class of viscometric motions. For certain choices of the material parameters and initial conditions, we find that this model allows for the development of shear rate discontinuities in the flow domain as a steady viscometric flow is achieved. Correspondingly, we also give evidence that intense shear rate oscillations may occur during the transient period as an impulsively started viscometric flow in a channel tends to a steady state under a constant critical shear stress. This critical shear stress lies in an interval of values for which the material experiences the phenomenon of “flow yielding”. A qualitative comparison with experimental data is made for certain creams and greases. The material instabilities inherent in this constitutive theory for viscometric motions are suggestive of the instabilities that occur in many viscoelastic fluids such as sharkskin patterns, wavy fracture, and spurt flow.     

Transonic Euler computation in streamfunction co-ordinates

Numerical simulation by a finite element method is used to examine the problem of the rotating flow of a viscoelastic fluid in a cylindrical vessel agitated with a paddle impeller. The mathematical model consists of a viscoelastic constitutive equation of Oldroyd B type coupled to the hydrodynamic equations expressed in a rotating frame. This system is solved by using an unsteady approach for velocity, pressure and stress fields. For Reynolds numbers in the range 0.1–10, viscoelastic effects are taken into account up to a Deborah number De of 1.33 and viscoelasticity and inertia cross-effects are studied. Examining the velocity and stress fields as well as the power consumption, it is found that their evolutions are significantly different for low and moderate inertia. These results confirm the trends of experimental studies and show the specific contribution of elasticity without interference of the pseudoplastic character found in actual fluids.     

Spectral element method for viscoelastic flows in a planar contraction channel

A new algorithm, which combines the spectral element method with elastic viscous splitting stress (EVSS) method, has been developed for viscoelastic fluid flows in a planar contraction channel. The system of spectral element approximations to the velocity, pressure, extra stress and the rate of deformation variables is solved by a preconditioned conjugate gradient method based on the Uzawa iteration procedure. The numerical approach is implemented on a planar four‐to‐one contraction channel for a fluid governed by an Oldroyd‐B constitutive equation. The behaviour of the Oldroyd‐B fluids in the contraction channel is investigated with various Weissenberg numbers. It is shown that numerical solutions obtained here agree well with experimental measurements and other numerical predictions. Copyright © 2003 John Wiley & Sons, Ltd.     

Nonlinear waves in a viscous compressible fluid with relaxation

The propagation and stability of nonlinear waves in a viscous compressible fluid with relaxation that satisfies a Theological equation of state of Oldroyd type are investigated. An equation that describes the structure of the wave perturbations and its evolution is derived subject to the condition of balance of the nonlinear dissipative and relaxation effects, and its solutions of the solitary wave type are analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 31–35, May–June, 1993.     

Viscoelastic simulations of stick‐slip and die‐swell flows

Numerical solutions of viscoelastic flows are demonstrated for a time marching, semi‐implicit Taylor–Galerkin/pressure‐correction algorithm. Steady solutions are sought for free boundary problems involving combinations of die‐swell and stick‐slip conditions. Flows with and without drag flow are investigated comparatively, so that the influence of the additional component of the drag flow may be analysed effectively. The influence of die‐swell is considered that has application to various industrial processes, such as wire coating. Solutions for two‐dimensional axisymmetric flows with an Oldroyd‐B model are presented that compare favourably with the literature. The study advances our prior fixed domain formulation with this algorithm, into the realm of free‐surface viscoelastic flows. The work involves streamline‐upwind/Petrov–Galerkin weighting and velocity gradient recovery techniques that are applied upon the constitutive equation. Free surface solution reprojection and a new pressure‐drop/mass balance scheme are proposed. Copyright © 2001 John Wiley & Sons, Ltd.     

Viscoelastic flow past confined objects using a micro–macro approach

This paper is concerned with the numerical prediction of viscoelastic flow past a cylinder in a channel and a sphere in a cylinder using molecular-based models. The basis of the numerical method employed is a micro–macro model in which the polymer dynamics is described by the evolution of an ensemble of Brownian configuration fields. The spectral element method is used to discretize the equations in space. Comparisons are made between the macroscopic simulations based on the Oldroyd B constitutive model and microscopic simulations based on Hookean dumbbells, and excellent agreement is found. The micro–macro approach can be used to simulate models, such as the finitely extensible nonlinear elastic (FENE) dumbbell model, which do not possess a closed-form constitutive equation. Numerical simulations are performed for the FENE model. The influence of the model parameters on the flow is described and, in particular, the dependence of the drag as a function of the Weissenberg number.     

A finite volume approach for unsteady viscoelastic fluid flows

A finite volume, time‐marching for solving time‐dependent viscoelastic flow in two space dimensions for Oldroyd‐B and Phan Thien–Tanner fluids, is presented. A non‐uniform staggered grid system is used. The conservation and constitutive equations are solved using the finite volume method with an upwind scheme for the viscoelastic stresses and an hybrid scheme for the velocities. To calculate the pressure field, the semi‐implicit method for the pressure linked equation revised method is used. The discretized equations are solved sequentially, using the tridiagonal matrix algorithm solver with under‐relaxation. In both, the full approximation storage multigrid algorithm is used to speed up the convergence rate. Simulations of viscoelastic flows in four‐to‐one abrupt plane contraction are carried out. We will study the behaviour at the entrance corner of the four‐to‐one planar abrupt contraction. Using this solver, we show convergence up to a Weissenberg number We of 20 for the Oldroyd‐B model. No limiting Weissenberg number is observed even though a Phan Thien–Tanner model is used. Several numerical results are presented. Smooth and stable solutions are obtained for high Weissenberg number. Copyright © 2002 John Wiley & Sons, Ltd.     

率形式大变形弹塑性本构律中的超弹性条件

本文证明了若取客观应力率为 Kirchhoff 应力的 Oldroyd 导数,对于 Lame 参数λ、μ为常数的情况,率形式弹性本构律的可积条件为 λ=0。这显然表明在大变形情况下率形式弹塑性本构律与超弹性条件这两者之间在一般情况下并不协调。文中还讨论了几种弹性本构律可以近似用于大变形描述的情况。     

Theoretical analysis of non-uniform extensional flows in relation to processing rheology and predictions of some constitutive equations

In this paper, a characteristic equation involving the stream function, already given by one of the authors in a previous work for classifying axisymmetric incompressible flows, is re-considered. Non-uniform nearly extensional flows are derived as particular solutions from this equation. Using experimental data in the literature for polymer solutions and melts, it is proved that particular solutions of the characteristic equation lead to kinematics very close to those encountered in the fiber-spinning process. The kinematic equations satisfactorily correlating the fiber-spinning data are used in order to determine the ability of constitutive equations to predict realistic stresses in the flow domain. The rheological parameters of the fluids, obtained from experiments, are used for computation of differential and integral constitutive equations in the spinning conditions. Comparisons with the stress response of adequate constitutive equations are given and discussed.Also affiliated to: Université Joseph Fourier Grenoble I and Institut National Polytechnique de Grenoble, Associé au CNRS (URA 1510)     

Transient free‐surface flow of a viscoelastic fluid in a narrow channel

The interplay between inertia and elasticity is examined for transient free‐surface flow inside a narrow channel. The lubrication theory is extended for the flow of viscoelastic fluids of the Oldroyd‐B type (consisting of a Newtonian solvent and a polymeric solute). While the general formulation accounts for non‐linearities stemming from inertia effects in the momentum conservation equation, and the upper‐convected terms in the constitutive equation, only the front movement contributes to non‐linear coupling for a flow inside a straight channel. In this case, it is possible to implement a spectral representation in the depthwise direction for the velocity and stress. The evolution of the flow field is obtained locally, but the front movement is captured only in the mean sense. The influence of inertia, elasticity and viscosity ratio is examined for pressure‐induced flow. The front appears to progress monotonically with time. However, the velocity and stress exhibit typically a strong overshoot upon inception, accompanied by a plug‐flow behaviour in the channel core. The flow intensity eventually diminishes with time, tending asymptotically to Poiseuille conditions. For highly elastic liquids the front movement becomes oscillatory, experiencing strong deceleration periodically. A multiple‐scale solution is obtained for fluids with no inertia and small elasticity. Comparison with the exact (numerical) solution indicates a wide range of validity for the analytical result. Copyright © 2004 John Wiley & Sons, Ltd.     

Criteria for the appearance of recirculating and non-stationary regimes behind a cylinder in a viscoplastic fluid

The appearance of a recirculation zone and the formation of non-stationary vortices behind a cylinder in the unconfined flow of a Herschel–Bulkley fluid have been studied by numerical simulation. The Herschel–Bulkley constitutive equation was regularised by using the Papanastasiou model. Special attention was paid to determining the numerical parameters and comparing them to existing results. The influence of the Oldroyd number and power-law index on flow morphology and, in particular, on the unyielded zones was studied over a wide spectrum (0 ≤ Od ≤ 10) and (0.3 ≤ n ≤ 1.8). It was seen that the greater the Oldroyd number, the greater the critical Reynolds numbers and Strouhal number for the two flow regimes. The influence of the power-law index is more complex.