Herschel-Bulkley Fluid Flows Through a Sudden Axisymmetric Expansion via Galerkin Least-Squares Methodology

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1 Herscel-Bulkley Fluid Flows Troug a Sudden Axisymmetric Expansion via Galerkin Least-Squares Metodology Fernando Macado, Flávia Zinani, Sérgio Frey Laboratory of Applied and Computational Fluid Mecanics (LAMAC) - Mecanical Engineering Department - UFRGS Rua Sarmento Leite, , Porto Alegre, RS frey@mecanica.ufrgs.br 1. Introduction Among non-newtonians fluids, a class called viscoplastic fluids represents a group wic as a special interest for researcers, due to possessing several applications in different sections in engineering. Suc fluids seem to beave as rigid solids wen subjected to sear stress smaller tan a yield stress, and wen submitted to greater levels of stress, tey beave as viscous fluids, usually presenting viscosity sear-tinning. In fact, tese fluids beave as extremely viscous materials wen subjected to low stress, and suffer an abrupt cange in beavior wen te stress attains te yield stress, beginning to beave as mild viscous fluids [2]. Examples of viscoplastic materials are molten cocolate, polymer melts and solutions, parmaceutical products, among oters. In te literature, many models ave been proposed in order to caracterize viscoplasticity, usually in te form of a nonsmoot function for te sear-stress. Tese models may be written in terms of a viscosity function, wic is ten employed beneat a generalized Newtonian constitutive equation. Te Herscel-Bulkley model is one of te most employed viscosity functions. In numerical approximations, its regularized versions are usually employed instead of te classical model, in order to use a smoot viscosity function for te wole flow domain (see, for instance, [1], [4] and [6]). Te main goal of te present work is to employ a Galerkin least-squares formulation (see [11] and references terein) to approximate te flow dynamics of a Herscel-Bulkley fluid troug an axisymmetric 4:1 expansion. Wit tis purpose, we employ a Papanastasiou [9] regularized version of te Herscel-Bulkley equation. We study te arisen of unyielded zones of two kinds: stagnation zones in te corner of te expansion and rigid body motion zones around te symmetry axis bot upstream and downstream from expansion. We range parameters as te yield stress and te powerlaw index and perform a sensibility analysis of ow tese parameters affect te topology of te yield surface and te flow dynamics. 2. Mecanical Modeling Te mecanical modeling tat we concern is constituted of te laws of mass and momentum balance, and a constitutive equation for te stress [1]. Te form of te stress tensor T is given by a constitutive equation. Here we assume tat T obeys te Generalized Newtonian Liquid (GNL) model, [3], T= pi+ 2 η( γ ) D (1) were p is te ydrostatic pressure, I te unit tensor, D te strain rate tensor, η te viscosity function wic is dependent of γ te magnitude of D ( γ =(trd 2 ) 1/2 ) [8]. For Newtonian fluids, η is a constant viscosity function and it is denoted by µ. All fluids wose beavior differ from te ideal Newtonian one are called non Newtonian. Probably te most important feature of non Newtonian fluids is te fact tat tey present a sear-rate dependent viscosity [3]. Te viscosity functional dependence on te sear rate allows us to classify variable viscosity fluids as pseudoplastic, viscoplastic or dilatant. Tere are many constitutive models tat aim to describe viscoplastic beavior. Tey are usually formulated as non-smoot viscosity functions, and tey are built in order to accommodate empirical flow curves of sear stress versus sear rate. As examples of viscoplastic models we ave te Bingam plastic, te Herscel-Bulkley and te Casson models [3]. Te Herscel-Bulkley equation is one of te most employed viscoplastic model. It employs tree reological parameters: te yield stress τ, te consistency index and te power-law index n. Te Herscel-Bulkley model for te sear stress may be stated as: n τ = τ + γ if τ > τ γ = if τ τ (2)

2 were τ is te sear stress and γ te sear rate in viscometric flow. If n=1 tis function reduces to te classical Bingam plastic equation. In numerical approximations, Eq. (2) poses te difficulty of possessing a discontinuous caracter. Tis difficulty may be overcome employing a regularization of te model. Te most common regularization tecniques are te bi-viscosity approximation [6] and te Papanastasiou`s regularization [9]. Te latter is te one employed in te present paper, in te form of a continuous viscosity function, wic is valid for te wole stress range (bot below and above τ ): τ γ ( ( )) n 1 η( γ) = γ + 1 exp αγ (3) were α is te regularization parameter. As α grows, Papanastasiou`s regularized model mimics te original model Herscel-Bulkley model, as it may be observed in te curve τ versus γ, sowed in te Fig.1, Figure 1. Sear stress as a function of sear rate as predicted by Eq.(3). 3. Finite Element Approximation In tis section, a Galerkin leastsquares (GLS) formulation for isocoric flows of generalized Newtonian fluids is presented. Te boundary value problem defined by te laws of mass conservation and momentum balance [1] for a generalized Newtonian liquid (Eq.(1)), plus te boundary conditions of prescribed velocity and forces are given by: divu = in ρ[grad uu ] = grad p + div(2 η( γ) D) + b in u= ug on Γ g ( pi+ 2 ηγ ( ) D) n= t on Γ (4) were R Ν=2 is te problem domain and Γ te boundary of, wit Γ g te portion of Γ were Diriclet conditions are imposed and Γ te portion of Γ were Neumann conditions are imposed. Te usual approximation spaces for fluid dynamics were employed to define te finite element subspaces for velocity (V and V g ) and pressure (P ) fields [5], over a finite element partition of te problem domain parameterized by a caracteristic mes size, P = { p C ( ) L ( ) p R ( ), 2 V = { v H ( ) v R ( ), } 1 N N k V v v 1 N N g = { H ( ) Rk( ), l, v= ug on Γg} (5) were R l and R k are polynomial functional spaces of degree l and k, respectively [5]. Based on te finite spaces defined by Eq. (5), a Galerkin least-squares formulation for te non linear boundary value problem defined by Eq. (4) may be stated as: Find te pair (u, p ) V g x P suc as: B( u : u, p ; v, q) = F( v: u, q), ( v, q) V P (6) were }

3 [ ] B( u: u, p; v: u, q) = ρ grad u u vd+ + 2 ηγ ( ) Du ( ) Dv ( ) d pdiv vd qdivud+ ( ρ[grad uu ] grad p 2div ( η( γ) Du ( ))) ((Re) τ ρ([grad vu ]) (2div ( ) ( ) grad q))) d ( ηγ Dv) + + F( v, q) = f vd+ t vdγ+ C f ( τ(re )( ρ[grad v] u ( ηγ Dv) 2div ( ) ( ) grad q)) d Γ (7) (8) wit te stability parameter τ being te same as in [11] τ(re ) = ξ(re ) 2 u p Re, Re < 1 wit ξ (Re ) = 1,Re 1 ρ u p and Re = 2 ηγ ( ) (9) were stands for te -element size, Re te grid Reynolds number and te. p te p- norm on R N. Te solution of Eq. (6)-(8) is performed via a incremental quasi Newton metod for te solution of te non-linear system [11]. 4. Numerical Results Te geometry studied erein is an axisymmetric 4:1 abrupt expansion. Te inlet cannel as lengt L and radius R, te outlet cannel as lengt L 1 and radius R 1. Te ratios L /R = 3 and L 1 /R = 75 were employed to assure fully development of te flow. We adopted te conditions of non-slip and impermeability at te walls, a orizontal parabolic velocity profile at te flow inlet, wit mean velocity u =1m/s, and free traction at te flow outlet. Te problem statement is depicted in Fig. 2. After a mes-dependency study, we ave cosen a mes wit 16,236 nodes and 15,825 bilinear elements Q1/Q1. In order to enance te numerical results for stagnation zones and vortex formation, te mes is igly refined near te expansion corner (Fig. 2 ). Figure 2. Flow troug a sudden expansion: Problem statement; mes employed: 16,236 nodes and 15,825 elements. Te flow of viscoplastic fluids possesses two distinct flow dynamics regime: yielded and unyielded zones. Te unyielded zones occur were te stress is smaller tan te fluid s yield stress. In suc regions, te strain rates tend to zero. Te remaining regions are called yielded zones. In tese regions te fluid flows as a viscous material. Te limit between yielded and unyielded zones is called yield surface. In te case of a Papanastasiou regularized model, its position is determined troug te von Mises criteria, i.e., te stress is post-processed from te approximated velocity field and compared wit te yield limit of te fluid. In te following figures, we investigate of te flow structure in terms of yielded and unyielded zones. Te unyielded zones are plotted in black, and te yielded zones in wite. We study te influence of te yield stress limit and te power-law index, for a fixed negligible Reynolds number, at te yielded surface location. In te case of te flow troug an expansion, two distinct unyielded zones tend to arise: one at te expansion corner, due to very low velocity values and gradients, and oter along te symmetry axis, were strain rates and stresses are sufficiently small to form a plug-flow. Te dimensionless pressure, radius, Reynolds and Herscel-Bulkley numbers are defined as:

4 * p p ρu D 2 p =, Re =, 2 n 1 ρ u L u D 2 τ x2 Hb =, r = n u D D 2 (1) In Fig. 3, it is possible to examine te influence of te Herscel-Bulkley number, Hb, in te formation of unyielded zones at te expansion corner and on te symmetry axis. Te Hb was varied from Hb=.1 to Hb=1 and te power-law index was taken as n=.37. (c) (d) Figure 4.Velocity profiles: fully developed axial velocity at x 1 =1D ; Pressure drop along te symmetry axis. In Fig. 5 we investigate ow te decrease in te power-law index, n, from 1 to.2, affects te formation of te unyielded zones in te flow. Tese results of Fig. 5 correspond to Hb=.1. Figure 3. Yielded and unyielded regions, for n=.37: Hb=.1; Hb=1; (c) Hb=1; (d) Hb=2; (e) Hb=1. It is possible to perceive tat as Hb increases, te size of te unyielded zones increase, due to te increasing yield stress. As below te yield stress te strain rates tend to zero, te unyielded zones form a plug-flow region along te symmetry axis. Te plugflow may be evidenced if we plot te axial velocity u*=u 1 /u versus te tube radius, r=x 2 /D in a fully developed flow location, as in Fig. 4, for x 1 =1D. In Fig. 4 we plot te pressure drop (log(p * )) along te symmetry axis for te various fluids investigated. It is clear tat te formation of te unyielded plugflow increases te pressure drop considerably, due to te ig resistance to flow tat it causes. (e) (c) (d) (e) Figure 5. Yielded and unyielded regions, for Hb =.1: n=1; n=.8; (c) n=.6; (d) n=.37; (e) n=.2. We observe tat, as n decreases, te size of te unyielded zones also decreases and are pused downstream farter from te expansion. Tese results are in agreement wit

5 Jay et al [6]. Tis beavior is due to fluid s sear-tinning, wic goes against te viscosity increase tendency in te low sear rate range, preventing te fluid s plasticization. In Fig. 6 we present a comparison of our results for n=.37 and Hb=1, Hb=1. Te dots correspond to Jay et al. results [6] and te filled lines to ours. In te corner, te yield surfaces are quite similar, but in te plug-flow along te symmetry axis, tey look quite different. Our results tend to predict a muc greater unyielded region. As tis work is still in progress, a more detailed analysis will be performed, in order to find out about tese discrepancies, but attempting to te fact tat te mes employed erein is muc more refined tan tat of Jay et. al (21). Figure. 6. Comparison wit results of Jay et. al, 21, for n=.37 and Hb=1, Hb=1. 5. Acknowledgements Te autor F. Macado tanks te agency CAPES/Brazil for te master s grant. Te autor F. Zinani tanks MCT/CNPq (Proc /26-) for te post doctoral grant. Te autor and S. Frey tanks MCT/CNPq grant No. 5747/ We also acknowledge MCT/CNPq for te financial support provided by projects of Proc /24-7 and Proc / References [2] Barnes, H.A., 1999, Te Yield Stress a Review or παντα ρει Everyting Flows?. Journal of Non-Newtonian Fluid Mecanics, 81, pp [3] Bird R. B, Armstrong R. C., Hassager, O., Dynamics of polymeric liquids, Vol. 1, Fluid Dynamics. Jon Wiley and Sons, New York, pp [4] Burgos, G., Alexandrou, A. N., and Entov, V., "On te determination of yield surfaces in Herscel-Bulkley fluids", J. of Reology, vol. 43 (3), pp [5] Ciarlet, P.G., Te Finite Element Metod for Elliptic Problems. Nort Holland, Amsterdam. [6] Jay, P., Magnin, A., and Piau, J. M., 21. "Viscoplastic Fluid Flow Troug a Sudden Axisymmetric Expansion ", AICE Journal, vol. 47, pp [7] Macedo, A., P, Aplicações de Métodos de Elementos Finitos Totalmente Estabilizados - GLS à Simulação Numérica de Escoamentos Laminar e Turbulentos. Dissertação de Mestrado, Universidade de Brasília, Brasília. [8] Morgan,., Periaux, J., and Tomasset, F., Analysis os laminar flow over a Backward Facing Step. A GAMM- Worksop, Friedr. Vieweg & Son. [9] Papanastasiou, T. C., "Flows of Materials wit Yield", J. of Reology, vol. 31 (5), pp [1]Slattery, J.C., Advanced transport penomena, Cambridge University Press, Cambridge. [11]Zinani, F.; Frey, S., 26. Galerkin Least-Squares Finite Element Approximations for Isocoric Flows of Viscoplastic Liquids. Journal of Fluids Engineering - Transactions of te Asme, Estados Unidos, v. 128, n. 4, p , Responsibility Notice Te autors are te only responsible for te printed material included in tis paper. [1] Alexandrou, A. N., McGilvreay, T. M., and Burgos, G., 21. "Steady Herscel- Bulkley fluid in tree-dimensional expansions", J. Non-Newt. Fluid Mec., vol. 1, pp