بررسی یک فرض تعادل عمومی برای توسعه مدل های ویسکوالاستیک جبری

The recent development of algebraic explicit stress models (AESM) for viscoelastic fluids rests upon a general equilibrium assumption, by invoking a slow variation condition on the evolution of the viscoelastic anisotropy tensor (the normalized de-viatoric part of the extra-stress tensor [G. Mompean, R.L. Thompson, P.R. Souza Mendes, A general transformation procedure for differential viscoelastic models, J. Non-Newtonian Fluid Mech. 111 (2003) 151–174]). This equilibrium assumption can take various forms depending on the general objective derivative which is used in the slow variation assumption. The purpose of the present paper is to assess the validity of the equilibrium hypothesis in different flow configurations. Viscometric flows (pure shear and pure elongation) are first considered to show that the Harnoy derivative [A. Harnoy, Stress relaxation effect in elastico-viscous lubricants in gears and rollers, J. Fluid Mech. 76(3) (1976) 501–517] is a suitable choice as an objective derivative that allows the algebraic models to retain the viscometric properties of the differential model from which they are derived. A creeping flow through a 4:1 planar contraction then serves as a benchmark for testing the equilibrium assumption in a flow exhibiting complex kinematics. Results of numerical simulations with the differential Oldroyd-B constitutive model allow to evaluate a posteriori the weight of extra-stress terms in different regions of the flow. Computations show that the equilibrium assumption making use of the Harnoy derivative is globally well verified. The assumption is exactly verified in flow regions of near-viscometric kinematics, whereas some departures are observed in the very near region of the corner entrance

The prediction of viscoelastic flows modeled with differential stress constitutive laws remains a numerical challenge in spite of the increasing speed and storage capacity of modern computers. For a three-dimensional flow, in addition to the momentum, mass and energy conservation equations, a set of six differential equations for the stress components has to be solved. The number of equations increases even more dramatically if multi-mode fluid models are to be considered. Starting from this simple fact, and motivated by the numerous industrial applications, the quest for simplified viscoelastic models has been a direction of research for years. The original approach towards algebraic models known as “ordered fluids” ([1], p. 299) is based upon Taylor series expansion of Rivlin–Ericksen kinematic tensors. Such ordered fluids possess instantaneous memory ([2], p. 76). In recent years, a different approach was proposed to derive algebraic viscoelastic models. The basic idea is to derive a unique differential equation for the first invariant (the trace) of the extra-stress tensor, and to obtain explicit expressions for the extra-stresses via a polynomial expansion [3]. This type of simplified model is named Algebraic Explicit Stress Model (AESM) and retains fluid memory through the transport equation for the trace of the extra-stress tensor. The transformation reduces the set of partial differential equations found in differential models into a single one, preserving their elasticity prediction capability. The reduction of the number of transport equations to be solved numerically must obviously result, especially for three-dimensional engineering applications, in a significant economy of computational time and storage memory. This technique was inspired by the analogy between viscoelastic fluids and turbulence closure mentioned by Rivlin [4]. It has been extensively employed by the turbulence modeling community since the pioneering work of Pope [5] to simplify the differential Reynolds stress equations. In the framework of viscoelastic fluids, the first attempt towards obtaining algebraic explicit stress models based on a transport equation for the trace was due to Mompean et al. [3] for the Oldroyd-B fluid. The same method was later extended to the PTT-fluid by Mompean [6]. A general transformation procedure (GTP) for viscoelastic differential constitutive equations was recently proposed by Mompean et al. [7]. This procedure was devised to simplify any differential stress model into an algebraic stress model having just one differential equation for the evolution of the trace of the extra-stress tensor. The procedure makes use of two kinematic tensors (see Refs. [8] and [9]), namely (i) Ω, the rate of rotation of the principal directions of the rate-of-deformation tensor S, and (ii) View the MathML sourceW¯=W−Ω. The latter, which is an objective quantity, is the relative rate-of-rotation tensor, and measures the rate of rotation of a fluid particle as seen by an observer fixed to the principal axes of S. This approach is inspired from the early work by Schunk and Scriven [10], Souza Mendes et al. [11] and Thompson et al. [12] who employed View the MathML sourceW¯ to propose algebraic constitutive models with enhanced capability for predicting rheological steady-state functions. To arrive at explicit expressions for the extra-stresses, an equilibrium assumption applied to the normalized deviatoric part of the extra-stress tensor must be made. This assumption makes use of a general objective derivative, invoking a slow variation condition on the advection of the normalized deviatoric part of the extra-stress tensor. The main purpose of the present paper is to derive the appropriate form of the equilibrium hypothesis for simple viscometric flows (pure shear and elongational), and assess its validity for viscoelastic complex flows. An analytical development is first conducted for viscometric flows, which permits to determine the correct form of the equilibrium assumption. For complex flows, it is necessary to solve the viscoelastic flow problem to check the equilibrium assumption a posteriori. The complex flow considered herein is a viscoelastic creeping flow through an abrupt planar 4:1 contraction. This study is carried out for the Oldroyd-B fluid model at Deborah numbers 1 and 2. The assessment of the equilibrium assumption in this geometry obviously requires a numerical solution to evaluate the weight of terms of the constitutive equation used. To solve the governing equations of the viscoelastic flow, a finite volume second-order accurate numerical method is employed and the elastic viscous split stress (EVSS) of Rajagopalan et al. [13] is implemented to improve stability. The paper is organized as follows. In Section 2 the governing equations are presented. The theoretical developments to assess the equilibrium assumption made to derive viscoelastic algebraic stress models are described in Section 3. Although valid for any type of differential constitutive equations (Oldroyd-B, Upper-Convected Maxwell, White-Metzner, Phan-Thien-Tanner models, etc.), we shall restrict ourselves in this work to the Oldroyd-B model. In Section 4, viscometric flows for which an analytical development is possible are first considered. The numerical simulations for the flow in the 4:1 contraction are then presented in Section 5. In this section, the geometrical configuration, the boundary conditions, the numerical method and the meshes used are first described. Results are then presented to validate the equilibrium hypothesis for this complex flow. Finally, the main conclusions are summarized in Section 6.

This work is a verification of various equilibrium assumptions which have been used in the recent past to derive algebraic stress models for viscoelastic liquids [21] and [7]. The study has been conducted for the differential Oldroyd-B model from which the algebraic models were derived. The Oldroyd-B model has been recast in the form equation(29) View the MathML source{τ}DbDt=Right-Hand-Side, Turn MathJax on where View the MathML sourceD/Dt is an objective derivative and b = Γ/{τ} is the viscoelastic anisotropy tensor (Eq. (11)) formed by taking the ratio of the deviatoric extra-stress tensor Γ to the trace of the extra-stress tensor, {τ}. Local equilibrium assumptions for b take the ad hoc form equation(30) View the MathML sourceDbDt≃0, Turn MathJax on which amounts to neglect the left-hand-side in Eq. (29). The most significant results of this study are: (i) An appropriate choice for the objective operator View the MathML sourceD/Dt to exactly verify equilibrium assumption (30) in steady shear and steady elongation flows is the Harnoy derivative [15]. A practical implication is that algebraic models derived from the equilibrium assumption making use of the Harnoy derivative will retain steady shear and elongation properties of the Oldroyd-B model. (ii) For the creeping flow in a 4:1 contraction, each term in Eq. (19) has been numerically evaluated taking the Harnoy derivative as the objective operator in the left-hand-side. As expected from the theoretical study, it is exactly verified in regions of the flow characterized by steady shear or elongation. In the critical region of the flow, the tip of the abrupt contraction induces sharp gradients in the terms of the right-hand-side of Eq. (29). These gradients counter-balance each other inducing a smoother behaviour of the left-hand-side. Yet, in this region the equilibrium assumption is not well verified, the left-hand-side of Eq. (19) being small but of same order of magnitude as term II. Algebraic models do lose some characteristics of differential type models where the equilibrium hypothesis is not exactly satisfied. However, algebraic models based upon equilibrium assumption (20) have been successfully used for flows other than viscometric (contraction flow and curved channel flow). In particular, for the contraction flow, the prediction of extra stresses by algebraic models across a critical section x1 = −0.4H is in good agreement with Oldroyd-B predictions (see Fig. 17 in Thais et al. [22]). This result is obtained in spite of the fact that the equilibrium assumption is not well verified across the critical section.