Simulation viscoelastic flow

The VOF-code has been extended to simulate viscoelastic flows. In the current investigation, the Oldroyd-B model is employed which can be derived either from the dilute solution theory with bead-spring dumbbell model or from the stress-... 

The outlined scheme is shown to yield stable solutions for non-zero Weissenberg number flows in a number of benchmark problems (Swarbric and Nassehi, 1992b). However, the extension of this scheme to more complex problems may involve modifications such as increasing of elemental subdivisions for stress calculations from 3 x 3 to 9 x 9 and/or the discretization of the stress field by biquadratic rather than bi-linear sub-elements. It should also be noted that satisfaction of the BB condition in viscoelastic flow simulations that use mixed formulations is not as clear as the case of purely viscous regimes. 

Differential Viscoelastic Models. Differential models have traditionally been the choice for describing the viscoelastic behavior of polymers when simulating complex flow systems. Many differential viscoelastic models can be described by the general form... 

The constants in eqn. (2.73) are defined in Table 2.7 for various viscoelastic models commonly used to simulate polymer flows. 

The complexity of viscoelastic flows requires a multidisciplinary approach including modelling, computational and mathematical aspects. In this chapter we will restrict ourselves to the latter and briefly review the state of the art on the most basic mathematical questions that can be raised on differential models of viscoelastic fluids. We want to emphasize the intimate connections that exist between the theoretical issues discussed here and the modelling of complex polymer flows (see Part III) and their numerical simulations (see Chapter II.3). 

Finally we conclude by presenting a few numerical schemes appropriate for simulating viscoelastic fluid flows, and we give some error estimates related to these schemes. 

So far only domains of the flow with smooth boundaries have been considered. However, re-entrant corners 2is in a sudden 4 1 contraction are well-known to give rise to numerical difficulties in the numerical simulation of viscoelastic flows. (See e.g., Chapter II.3.)... 

M.J. Crochet and V. Delvaux, Numerical simulation of inertial viscoelastic flows with change of type, in Nonlinear Evolution Equations That Change Type, B.L. Kejrfitz and M. Shearer (eds.), IMA Volumes in Mathematics and its Applications 27, Springer-Verlag, Berlin, 1991, 47-66. 

Severe difficulties have been encountered for several years in the numerical simulation of viscoelastic flow for differential constitutive equations. Let us now give a summary of the numerical problems previously presented. 

In the story of numerical flow simulation, the ability to predict observed and significant viscoelastic flow phenomena of polymer melts and solutions in an abrupt contraction has been unsuccessful for many years, in relation to the incomplete rheological characterization of materials, especially in elongation. The numerical treatments have often been confined to flow of elastic fluids with constant viscosity, described by differential constitutive equations as the Upper Convected Maxwell and Oldroyd-B models. Fortunately, the recent possibility to use real elastic fluids with constant viscosity, the so-called Boger fluids [10], has narrowed the gap between experimental observation and numerical prediction [11]. 

The correlation between rheology and thermodynamics is likely to prove a fruitful area for investigation in the future. Very little is as yet known about the detailed mechanisms of non-linear viscoelastic flows, such as those involved in large-amplitude oscillatory shear. Mesoscopic modelling will no doubt throw light on the role of defects in such flows. This is likely to involve both analytical models, and mesoscopic simulation techniques such as Lattice... 

Karra, S. Modeling electrospinning process and a numerical scheme using lattice Boltzmann method to simulate viscoelastic fluid flows, in Mechanical Engineering. Indian Institute of Technology Madras, Chennai, p. 60 (2007). 

Einstein A (1906) A new determination of molecular dimensions. Annu Physik 19 289-306. Corrections, ibid. (1911) 34 591-592. hr Ftirth R ( 1956) (ed) Investigations on the TheOTy of the Brownian Movement (Translated by Cowper AD). Dover, New York Ellero M, Tanner RI (2005) SPH simulations of transient viscoelastic flows at low Reynolds number. J Non-Newtonian Fluid Meeh 132 61-72 Ericksen JL (1960) Anistropic fluids. Arch Rat Mech Anal 4 231-237... 

Pang JN, Owens RG, Tacher L, Parriaux A (2006) A numerical study of the SPH method for simulating transient viscoelastic free surface flow. J Non-Newtonian Fluid Mech 139 68-84 Feng J, Leal LG (1997) Simulating complex flows of liquid crystalline polymer using the Doi theory. J Rheol 41 1317—1335... 

Huilgol RR, You Z (2006) On the importance of the pressure dependence of viscosity in steady non-isothermal shearing flows of compressible and incompressible fluids and in the isothermal fountain flow. J Non-Newtonian Fluid Mech 136 106-117 Hulsen MA, Van Heel APG, Van den Brule BHAA (1997) Simulation of viscoelastic flows using Brownian configuration fields. J Non-Newtonian Fluid Mech 70 79-101 Ingber MS, Mondy LA (1994) A numerical study of three-dimensional Jeffery orbits in shear flow. J Rheol 38 1829-1843... 

Hu, H. H. and D. D. Joseph, Numerical simulation of viscoelastic flow past a cylinder, J. Non-Newtonian Fluid Mech. 37 347-377 (1990). 

Townsend, P., A numerical simulation of Newtonian and viscoelastic flow past stationary and rotating cylinders, J. Non-Newtonian Fluid Mech. 6 219-243 (1980). 

To simulate viscoelastic two-phase flow problems, the VOF method has been extended to capture the rheological properties of the Oldroyd-B fluid. To alleviate the High Weissenberg Number Problem in the simulation of viscoelastic flow, stabilization approaches have been adapted and implemented in FS3D. The simulation results show that the viscoelastic effect is reflected in the oscillation process during the collision, and the elasticity restrains the deformation of the collision complex. 

In the simulation of viscoelastic flow, a significant numerical problem, the so-called High Weissenberg Number Problem (HWNP), often occurs with loss of convergence of numerical algorithms. In order to alleviate the problem, we have, as the first attempt, implemented the conformation tensor Positive Definiteness Preserving Scheme (PDFS) by Stewart et al. [28], and then adapted and implemented the Log-Conformation Representation (LCR) approach by Fattal and Kupferman [7] in the viscoelastic two-phase flow solver in FS3D. 

To simulate the viscoelastic flow, the Oldroyd-B model has been implemented in the VOF-code. Stabilization approaches, such as the Positive Definiteness Preserving Scheme and the Log-Conformation Representation approach have been adapted and implemented in the code to stabilize the simulations at high Weissenberg numbers. The collision of viscoelastic droplets behaves as an oscillation process. The amplitude of the oscillation increases and the oscillation frequency decreases when the Deborah number becomes larger. The phenomenon can be explained with the dilute solution theory with Hookean dumbbell models. An increase of the fluid relaxation time yields a decrease of the stiffness of the spring in the dumbbell and restrains the deformation of the droplets. In addition, with larger the viscosity ratio the collision process is more similar to the Newtonian one since the fluid has less portion of polymers. 

Fattal, R., Kupferman, R. (2005). Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation. Journal of Non-Newtonian Fluid Mechanics, 726(1), 22-21. 

Vaithianafhan, T. and Collins, LR. (2003) Numerical approach to simulating turbulent flow of a viscoelastic polymer solution. /. Comput. Phys.,... 

Figure 4.12 Viscoelastic simulation of flow patterns for the melts of Figure 4.11 using the K-BKZ integral model (Eq. (4.12)) [34].

Crochet, M.J. (1989) Numerical simulation of viscoelastic flow a review. Rubber Chem. Technol, 62, 426-455. 

Karra, S. Modeling Electrospinning Process and a Numerical Scheme Using Lattice Boltzmann Method to Simulate Viscoelastic Fluild Flows. ti Mechanical Engineering 2007, Indian Institute of Technology Madras Chennai, 60. 

Gotsis, A. D. 1987. Study of the Numerical Simulation of Viscoelastic Flow Effect of the Rheological Model and the Mesh (Ph.D. Thesis, Department of Chemical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA.)... 

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