Viscoelastic Fluid Flow

In general, as discussed in Chapter 5, the total stress tensor is defined by 

For the viscoelastic stress, we can use differential or integral constitutive models (see Chapter 2). For differential models we have the general form 

In order to solve viscoelastic problems, we must select the most convenient model for the stress and then proceed to develop the finite element formulation. Doue to the excess in non-linearity and coupling of the viscoelastic momentum equations, three distinct Galerkin formulations are used for the governing equations, i.e., we use different shape functions for the viscoelastic stress, the velocity and the pressure 

The next step will be to formulate the Galerkin-weighted residual for each of the governing equations 

Applying the Green-Gauss Theorem (9.1.2) to the residual of the momentum eqn. (9.165) we get 

---

Keunings, R., 1989. Simulation of viscoelastic fluid flow. Tn Tucker, C. L. HI (ed.), Computer Modeling for Polymer Proces.sing, Chapter 9, Hanser Publishers, Munich, pp. 403-469. 

Normal Stress (Weissenberg Effect). Many viscoelastic fluids flow in a direction normal (perpendicular) to the direction of shear stress in steady-state shear (21,90). Examples of the effect include flour dough climbing up a beater, polymer solutions climbing up the inner cylinder in a concentric cylinder viscometer, and paints forcing apart the cone and plate of a cone—plate viscometer. The normal stress effect has been put to practical use in certain screwless extmders designed in a cone—plate or plate—plate configuration, where the polymer enters at the periphery and exits at the axis. 

M. Kawahara and N. Takeuchi. Mixed finite element method for analysis of viscoelastic fluid flow. Comput. Fluids., 5 33, 1977. 

Viscoelastic fluids have elastic properties in addition to their viscous properties. When under shear, such fluids exhibit a normal stress in addition to a shear stress. For example, if a vertical rod is partly immersed and rotated in a non-viscoelastic liquid the rod s rotation will create a centrifugal force that drives liquid outwards toward the container walls, as shown in Figure 6.16(a). If, on the other hand, the liquid is viscoelastic then as the liquid is sheared about the rod s axis of rotation, a stress normal to the plane of rotation is created which tends to draw fluid in towards the centre. At some rotational speed, the normal force will exceed the centrifugal force and liquid is drawn towards and up along the rod see Figure 6.16(b). This is called the Weissenberg effect. Viscoelastic fluids flow when stress is applied, but some of their deformation is recovered when the stress is removed [381]. 

Fig. 13.33 Fractional coverage predicted by simulations (solid circles) in comparison with the experiments of Taylor (62) (open diamond) and Hyzyak and Koelling (67). [Reprinted by permission from V. Gauri and K. W. Koelling, Gas-assisted Displacement of Viscoelastic Fluids Flow Dynamics at the Bubble Front, J. Non-Newt. Fluid Meek, 83, 183-203 (1999).]...

V. Gauri and K. W. Koelling, Gas-assisted Displacement of Viscoelastic Fluids Flow Dynamics at the Bubble Front, J. Non-Newt. Fluid Mech., 83, 183-203 (1999). 

Figure 3.11 shows the difference between a viscous and a viscoelastic fluid flowing out of a die. The silicone oil (case A) is only slightly different in diameter compared to the one of the die, while the emerging strand of PEO solution (case B) is considerably wider. 

As a matter of fact we do think that a better understanding of the mathematical properties of the models for viscoelastic fluid flows is fundamental in order to select a good constitutive equation, and to develop and implement numerical codes of practical use. 

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. 

In the context of viscoelastic fluid flows, numerical analysis has been performed for differential models only, and for the following types of approximations finite element methods for steady flows, finite differences in time and finite element methods in space for unsteady flows. Finite element methods are the most popular ones in numerical simulations, but some other methods like finite differences, finite volume approximations, or spectral methods are also used. 

Mathematical and numerical analyses of differential models for viscoelastic fluid flows are highly challenging domains, which stiU need a lot of effort. However, significant progress heis been made during the last decade, and mathematical results have shown to be quite useful for the modelling. 

D. Saadri, Finite element approximation of viscoelastic fluid flow existence of approximate solutions and error bounds. Continuous approximation of the stress, SIAM J. Numer. Anal., 31 (1994) 362-377. 

A. Bahar, J. Baranger and D. Sandri, Quadrilateral finite element approximation of viscoelastic fluid flow. Rapport de I equipe d cinalyse numerique Lyon-Saint-Etienne 162 (1993), submitted. 

IIOJ. Baranger and D. Sandri, Finite element method for the approximation of viscoelastic fluid flow with a differential constitutive law. First European Computational Fluid Dynamics Conference, Bruxelles, 1992, C. Hirsch (ed.), Elsevier, Amsterdam, 1993, 1021-1025. 

J<. Najib and D. Sandri, On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow, Num. Math., 72 (1995) 223-238. 

R. Keunings, Simulation of Viscoelastic fluid flow, in Computer Modeling fra- Polymer Processing, C.L. Tucker HI (Ed.), Hanser Verlag (1989) p. 403. 

These materials exhibit both viscous and elastic properties. In a purely Hookean elastic solid, the stress corresponding to a given strain is independent of time, whereas for viscoelastic substances the stress will gradually dissipate. In contrast to purely viscous liquids, on the other hand, viscoelastic fluids flow when subjected to stress, but part of their deformation is gradually recovered upon removal of the stress. 

M. Kostic, Heat Transfer and Hydrodynamics of Water and Viscoelastic Fluid Flow in a Rectangular Duct, Ph.D. thesis, University of Illinois at Chicago, 1984. 

H. Usui, Transport Phenomena in Viscoelastic Fluid Flow, Ph.D. thesis, Kyoto University, Kyoto, Japan, 1974. 

Karnis, Goldsmith, and Mason (K5, K5d) performed experiments on the radial migration of rigid spheres suspended in a viscoelastic fluid flowing through a circular tube. Migration towards the axis was observed under conditions where inertial effects would normally be expected to be nil. The observed radial motion is thus apparently attributable to the non-Newtonian properties of the fluid. The unperturbed velocity profile is very flat over the central portion of the tube. Particles placed in this region neither rotated nor moved radially. Experiments are also reported for rods and disks (G9b, K5d). 

Chaotk Mixing Based on Viscoelasticity, Fig. 3 Consecutive snapshots of viscoelastic fluids flow at Q = 12ml/h. Each fluorescent tracer s snapshots, (a) and (b) or (c) and (d), were collected at 50 ms interval. Viscoelastic instability (whipping) upstream and... 

In 1977, Lucy [1] and Gingold and Monaghan [2] independently introduced the so-called smoothed particle hydrodynamics method which is one of the oldest meshless methods. The method was originally developed for astrophysi-cal studies such as formation of asteroids and the evolution of galaxies and has now become a standard tool in this field. In recent years, SPH has been extensively used for various fluid flow problems including both compressible and incompressible flow regimes. The method has been recently used for simulation of the generalized non-Newtonian and viscoelastic fluid flow problems. The SPH method has been also... 

Mizushina, T., and H. Usui, "Reduction of Eddy Diffusivity for Momentum and Heat in Viscoelastic Fluid Flow in a Circular Tube", Phys. Fluids Suppl., S 100 (1977)... 

Lee, H.S., "Turbulence Measurements and Modeling of Viscoelastic Fluid Flow in a Rectangular Open Channel", Ph.D. Thesis, Mechanical Engineering Dept., State University of New York, Stony Brook, N.Y. (1982)... 

Mizushina, T. Usui, H. Reduction of eddy diffusion for momentum and heat in viscoelastic fluid flow in a circular tube. Phys. Fluids 20 (1977) S100-S108. 

Abel, M. S. Khan, S. K. Prasad, K. V. (2002). Study of viscoelastic fluid flow and heat transfer over stretching sheet with variable viscosity, Int.. Non-Linear Mech. 37, pp. 81-88, ISSN 0020-7462. 

---