# Prediction of onset of Taylor-Couette instability for shear-thinning fluids

- 480 Downloads
- 6 Citations

## Abstract

*Re*) in a Taylor-Couette flow for a shear-thinning fluid is discussed in this paper. Since the shear-thinning property causes spatial distribution of fluid viscosity in a Taylor-Couette flow reactor (TCFR), a method to determine

*Re*by using a numerical simulation is suggested. The effective viscosity (

*η*

_{eff}) in

*Re*was the average viscosity using a weight of dissipation function

where *N* is the total mesh number, *η* _{ i } (Pa·s) is the local viscosity, \( {\overset{\cdot }{\gamma}}_i \) (1/s) is the local shear-rate, and Δ*V* _{ i } (m^{3}) is the local volume for each cell. The critical Reynolds number, *Re* _{cr}, at which Taylor vortices start to appear, was almost the same value with the *Re* _{cr} obtained by a linear stability analysis for Newtonian fluids. Consequently, *Re* based on *η* _{eff} could be applicable to predict the occurrence of Taylor vortices for a shear-thinning fluid. In order to understand the relation between the rotational speed of the inner cylinder and the effective shear rate that resulted in *η* _{eff}, a correlation equation was constructed. Furthermore, the critical condition at which Taylor vortices appear was successfully predicted without further numerical simulation.

## Keywords

Taylor-Couette flow Shear-thinning fluid Effective Reynolds number Numerical simulation## References

- Alibenyahia B, Lemaitre C, Nouar C, Ait-Messaoudene N (2012) Revisiting the stability of circular Couette flow of shear-thinning fluids. J Non-Newtonian Fluid Mech 183-184:37–51CrossRefGoogle Scholar
- Azaditalab M, Houshmand A, Sedaghat A (2016) Numerical study on skin friction reduction of nanofluid flows in a Taylor-Couette system. Tribol Int 94:329–335CrossRefGoogle Scholar
- Baumert BM, Muller SJ (1995) Flow visualization of the elastic Taylor-Couette instability in Boger fluids. Rheol Acta 34:147–159CrossRefGoogle Scholar
- Bird RB, Armstrong RC, Hassager O (1987) Dynamics of polymeric liquids, vol. I, fluid dynamics, 2nd edn. John Wiley & Sons, Inc., U.S.A.Google Scholar
- Carreau PJ (1972) Rheological equations from molecular network theories. Trans Soc Rheol 16:99–127CrossRefGoogle Scholar
- Carreau PJ, Patterson J, Yap CY (1976) Mixing of viscoelastic fluids with helical-ribbon agitators. I—mixing time and flow patterns. Can J Chem Eng 54:135–142CrossRefGoogle Scholar
- Cho YI, Kensey KR (1991) Effects of the non-Newtonian viscosity of blood on flows in a diseased arterial vessel. Part 1: steady flows. Biorheology 28:241–262Google Scholar
- Coronado-Matutti O, Souza Mendes PR, Carvalho MS (2004) Instability of inelastic shear-thinning liquids in a Couette flow between concentric cylinders. J Fluids Eng 126:385–390CrossRefGoogle Scholar
- Di Prima RC, Swinney HL (1981) Instabilities and transition in flow between concentric rotating cylinders, in: Hydrodynamic instabilities and the transition to turbulence, Springer-Verlag: 139–180Google Scholar
- Dutta PK, Ray AK (2004) Experimental investigation of Taylor vortex photocatalytic reactor for water purification. Chem Eng Sci 59:5249–5259CrossRefGoogle Scholar
- Escudier MP, Gouldson IW, Jones DM (1995) Taylor vortices in Newtonian and shear-thinning liquids. Proc R Soc Lond A 449:155–176CrossRefGoogle Scholar
- Escudier MP, Poole RJ, Presti F, Dales C, Nouar C, Desaubry C, Graham L, Pullum L (2005) Observations of asymmetrical flow behavior in transitional pipe flow of yield-stress and other shear-thinning liquids. J Non-Newtonian Fluid Mech 127:143–155CrossRefGoogle Scholar
- Fontaine A, Guntzburger Y, Bertrand F, Fradette L, Heuzey MC (2013) Experimental investigation of the flow dynamics of rheologically complex fluids in a Maxblend impeller system using PIV. Chem Eng Res Des 91:7–17CrossRefGoogle Scholar
- Güzel B, Frigaard I, Martinez DM (2009) Predicting laminar-turbulent transition in Poiseuille pipe flow for non-Newtonian fluids. Chem Eng Sci 64:254–264CrossRefGoogle Scholar
- Hubacz R, Buczyńska M (2011) Starch gelatinization in Couette-Taylor flow apparatus. Chem Process Eng 32:267–279CrossRefGoogle Scholar
- Hubacz R, Ohmura N, Dluska E (2013) Intensification of processing using apparatus with Couette-Taylor flow. J Food Process Eng 36:774–785CrossRefGoogle Scholar
- Hwang JY, Yang KS (2004) Numerical study of Taylor-Couette flow with an axial flow. Comput Fluids 33:97–118CrossRefGoogle Scholar
- Jastrzębski M, Zaidani HA, Wroński S (1992) Stability of Couette flow of liquids with power law viscosity. Rheol Acta 31:264–273CrossRefGoogle Scholar
- Jenny M, Plaut E, Briard A (2015) Numerical study of subcritical Rayleigh-Bénard convection rolls in strongly shear-thinning Carreau fluids. J Non-Newtonian Fluid Mech 219:19–34CrossRefGoogle Scholar
- Kaminoyama M, Nishi K, Misumi R, Otani F (2011) A method for determining the representative apparent viscosity of highly viscous pseudoplastic liquids in a stirred vessel by numerical simulation. J Chem Eng Jpn 44:868–875CrossRefGoogle Scholar
- Kataoka K, Doi H, Hongo T, Futagawa M (1975) Ideal plug-flow properties of Taylor vortex flow. J Chem Eng Jpn 8:472–476CrossRefGoogle Scholar
- Kataoka K, Ohmura N, Kouzu M, Simamura Y, Okubo M (1995) Emulsion polymerization of styrene in a continuous Taylor vortex flow reactor. Chem Eng Sci 50:1409–1416CrossRefGoogle Scholar
- Kozicki W, Chou CH, Tiu C (1966) Non-Newtonian flow in ducts of arbitrary cross-sectional shape. Chem Eng Sci 21:665–679CrossRefGoogle Scholar
- Larson RG, Shaqfeh ESG, Muller SJ (1990) A purely elastic instability in Taylor-Couette flow. J Fluid Mech 218:573–600CrossRefGoogle Scholar
- Lee S, Lueqtow RM (2001) Rotating reverse osmosis: a dynamic model for flux and rejection. J Memb Sci 192:129–143CrossRefGoogle Scholar
- Lockett TJ, Richardson SM, Worraker WJ (1992) The stability of inelastic non-Newtonian fluids in Couette flow between concentric cylinders: a finite-element study. J Non-Newtonian Fluid Mech 43:165–177CrossRefGoogle Scholar
- Makino T, Kaise T, Sasaki K, Ohmura N, Kataoka K (2001) Isolated mixing region in a Taylor-vortex-flow reactor. Kagaku Kogaku Ronbunshu 27:566–573CrossRefGoogle Scholar
- Masuda H, Horie T, Hubacz R, Ohmura N (2013) Process intensification of continuous starch hydrolysis with a Couette-Taylor flow reactor. Chem Eng Res Des 91:2259–2264CrossRefGoogle Scholar
- Masuda H, Horie T, Hubacz R, Ohta M, Ohmura N (2015) Numerical analysis of the flow of fluids with complex rheological properties in a Couette-Taylor flow reactor. Theo Appl Mech Japan 63:25–32Google Scholar
- Matsumoto K, Ohta M, Iwata S (2015) Numerical analysis of flow dynamics of milk in a milk-filling process. Kagaku Kougaku Ronbunshu 41:1–10CrossRefGoogle Scholar
- Metzner AB, Otto RE (1957) Agitation of non-Newtonian fluids. AICHE J 3:3–10CrossRefGoogle Scholar
- Metzner AB, Reed JC (1955) Flow of non-Newtonian fluids-correlation of the laminar, transition and turbulent-flow regions. AICHE J 1:434–440CrossRefGoogle Scholar
- Michele J, Patzold R, Donis R (1977) Alignment and aggregation effects in suspensions of spheres in non-Newtonian media. Rheol Acta 16:316–321Google Scholar
- Muller SJ, Larson RG, Shaqfeh ESG (1989) A purely elastic transition in Taylor-Couette flow. Rheol Acta 28:499–503CrossRefGoogle Scholar
- Nemri M, Climent E, Charton S, Lanoë JY (2013) Experimental and numerical investigation on mixing and axial dispersion in Taylor–Couette flow patterns. Chem Eng Res Des 91:2346–2354CrossRefGoogle Scholar
- Niederkorn TC, Ottino JM (1994) Chaotic mixing of shear-thinning fluids. AICHE J 40:1782–1793CrossRefGoogle Scholar
- Orlowska M, Koutchma T, Kostrzynska M, Tang J, Defelice C (2014) Evaluation of mixing flow conditions to inactivate
*Escherichia coli*in opaque liquids using pilot-scale Taylor-Couette UV unit. J Food Eng 120:100–109CrossRefGoogle Scholar - Pakdel P, McKinley GH (1996) Elastic instability and curved streamlines. Phys Rev Lett 77:2459–2462CrossRefGoogle Scholar
- Parmentier EM (1978) A study of thermal convection in non-Newtonian fluids. J Fluid Mech 84:1–11CrossRefGoogle Scholar
- Patel VR, Ein-Mozaffari F, Upreti SR (2011) Effect of time delays in characterizing the continuous mixing of non-Newtonian fluids in stirred-tank reactors. Chem Eng Res Des 89:1919–1928CrossRefGoogle Scholar
- Poole RJ, Escudier MP (2004) Turbulent flow of viscoelastic liquids through an axisymmetric sudden expansion. J Non-Newtonian Fluid Mech 117:25–46CrossRefGoogle Scholar
- Ramezani M, Kong B, Gao X, Olsen MG, Vigil RD (2015) Experiment measurement of oxygen mass transfer and bubble size distribution in an air-water multiphase Taylor-Couette vortex bioreactor. Chem Eng J 279:286–296CrossRefGoogle Scholar
- Saeed S, Ein-Mozaffari F (2008) Using dynamic tests to study the continuous mixing of xanthan gum solutions. J Chem Technol Biotechnol 83:559–568CrossRefGoogle Scholar
- Sczechowski JG, Koval CA, Noble RD (1995) A Taylor vortex reactor for heterogeneous photocatalysis. Chem Eng Sci 50:3163–3173CrossRefGoogle Scholar
- Shaqfeh ESG, Muller SJ, Larson RG (1992) The effects of gap width and dilute solution properties on the viscoelastic Taylor-Couette instability. J Fluid Mech 235:285–317CrossRefGoogle Scholar
- Sinevic V, Kuboi R, Nienow AA (1986) Power numbers, Taylor numbers and Taylor vortices in viscous Newtonian and non-Newtonian fluids. Chem Eng Sci 41:2915–2923CrossRefGoogle Scholar
- Sobolík V, Izrar B, Lusseyran F, Skali S (2000) Interaction between the Ekman layer and Couette-Taylor instability. Int J Heat Mass Trans 43:381–4393CrossRefGoogle Scholar
- Sulivan TM, Middleman S (1986) Film thickness in blade coating of viscous and viscoelastic liquids. J Non-Newtonian Fluid Mech 21:13–38CrossRefGoogle Scholar
- Taylor GI (1923) Stability of a viscous liquid contained between two rotating cylinders. Phil Trans Roy Soc A 223:289–343CrossRefGoogle Scholar
- White JM, Muller SJ (2002) Experimental studies on the stability of Newtonian Taylor-Couette flow in the presence of viscous heating. J Fluid Mech 462:133–159CrossRefGoogle Scholar
- Yap CY, Patterson WI, Carreau PJ (1979) Mixing with helical ribbon agitators: part III non-Newtonian fluids. AICHE J 25:516–521CrossRefGoogle Scholar
- Yasuda K, Armstrong RC, Cohen RE (1981) Shear flow properties of concentrated solutions of linear and star branched polystyrenes. Rheol Acta 20:163–178CrossRefGoogle Scholar