Abstract
Laminar flow of a viscoelastic fluid obeying the linear simplified Phan-Thien/Tanner model (LPTT) is numerically studied in a planar channel partially obstructed by a cosinusoidal constriction. Based on published data (Tammadon-Jahromi et al., 2011) there is no excess pressure drop for this particular fluid when flowing through an orifice-plate. Numerical results obtained using OpenFoam software at a typically low Reynolds number suggest that there exists a strong competition between the fluid’s strain-hardening/shear-thinning behavior on the one side with its first normal-stress difference in extension, on the other side, in controlling the pressure drop caused by the presence of the constriction. It is shown that, an excess-pressure-drop (epd) can correctly be predicted provided that use is made of a proper (inelastic) baseline in the definition of the “epd”. At moderate Reynolds numbers a flow-reversal is predicted to occur at the lee side of the constriction ruling out this technique as an extensional rheometer. It is argued that such vortices can be very useful in high-throughput microfluidic systems for mixing enhancement. To reduce the excessive pressure drop experienced by the fluid when working at high Reynolds numbers, it is shown that the Deborah number of the flow should be increased.
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References
Aguayo, J.P., H.R. Tamaddon-Jahromi, and M.F. Webster, 2008, Excess pressure-drop estimation in contraction and expansion flows for constant shear-viscosity, extension strain-hardening fluids. J. Non-Newton. Fluid Mech. 153, 157–176.
Alves, M.A., F.T. Pinho, and P.J. Oliveira, 2001, Study of steady pipe and channel flows of a single-mode Phan-Thien-Tanner fluid. J. Non-Newton. Fluid Mech. 101, 55–76.
Anderson, H.I., R. Halden, and T. Glomsaker, 2000, Effects of surface irregularities on flow resistance in differently shaped arterial stenoses. J. Biomech. 33, 1257–1262.
Azaiez, J., R. Guénette, and A. Ait-Kadi, 1996, Numerical simulation of viscoelastic flows through a planar contraction. J. Non-Newton. Fluid Mech. 62, 253–277.
Binding, D.M., P.M. Phillips, and T.N. Phillips, 2006, Contraction/expansion flows: The pressure drop and related issues. J. Non-Newton. Fluid Mech. 137, 31–38.
Bird, R.B., R.C. Armstrong, and O. Hassager, 1987, Dynamics of Polymeric Liquids Vol. 1: Fluid Mechanics, 2nd ed., John Wiley and Sons Inc., New York.
Cheng, R.T.S., 1972, Numerical solution of the Navier-Stokes equations by the finite element method. Phys. Fluids 15, 2098–2105.
Cogswell, F.N., 1972, Converging flow of polymer melts in extrusion dies. Polym. Eng. Sci. 12, 64–73.
Favero, J.L., A.R. Secchi, N.S.M. Cardozo, and H. Jasak, 2010, Viscoelastic flow analysis using the software OpenFOAM and differential constitutive equations, J. Non-Newton. Fluid Mech. 165, 1625–1636.
Fernandes, C., V. Vukčević, T. Uroić, R. Simoes, O.S. Carneiro, H. Jasak, and J.M. Nóbrega, 2019, A coupled finite volume flow solver for the solution of incompressible viscoelastic flows. J. Non-Newton. Fluid Mech. 265, 99–115.
Giddens, D.P., C.K. Zarins, and S. Glagov, 1993, The role of fluid mechanics in the localization and detection of atherosclerosis. J. Biomech. Eng. 115, 588–594.
Grillet, A.M., A.C.B. Bogaerds, G.W.M. Peters, F.P.T. Baaijens, and M. Bulters, 2002, Numerical analysis of flow mark surface defects in injection molding flow. J. Rheol. 46, 651–669.
Harten, A., 1983, High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357–393.
James, D.F., 2009, Boger fluids. Annu. Rev. Fluid Mech. 41, 129–142.
James, D.F., G.M. Chandler, and S.J. Armor, 1990, A converging channel rheometer for the measurement of extensional viscosity. J. Non-Newton. Fluid Mech. 35, 421–443.
Larson, R.G., 1988, Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston.
Lee, H.S. and S.J. Muller, 2017, A differential pressure extensional rheometer on a chip with fully developed elongational flow. J. Rheol. 61, 1049–1059.
Lee, J.W., D. Kim, and Y. Kwon, 2002, Mathematical characteristics of the pom-pom model. Rheol. Acta 41, 223–231.
Lopez-Aguilar J.E., M.F. Webster, H.R. Tamaddon-Jahromi, O. Manero, D.M. Binding, and K. Walters, 2017, On the use of continuous spectrum and discrete-mode differential models to predict contraction-flow pressure drops for Boger fluids, Phys. Fluids 29, 121613.
Magda, J.J., J. Lou, S.G. Baek, and K.L. DeVries, 1991, Second normal stress difference of a Boger fluid. Polymer 32, 2000–2009.
Mahapatra, T.R., G.C. Layek, and M.K. Maiti, 2002, Unsteady laminar separated flow through constricted channel. Int. J. Non-Linear Mech. 37, 171–186.
Marrucci, G., F. Greco, and G. Ianniruberto, 2001, Integral and differential constitutive equations for entangled polymers with simple versions of CCR and force balance on entanglements. Rheol. Acta 40, 98–103.
Ngamaramvaranggul, V. and M.F. Webster, 2002, Simulation of pressure-tooling wire-coating flow with Phan-Thien/Tanner models. Int. J. Numer. Methods Fluids 38, 677–710.
Nyström, M., H.R. Tamaddon-Jahromi, M. Stading, and M.F. Webster, 2016, Extracting extensional properties through excess pressure drop estimation in axisymmetric contraction and expansion flows for constant shear viscosity, extension strain-hardening fluids. Rheol. Acta 55, 373–396.
Ober, T.J., S.J. Haward, C.J. Pipe, J. Soulages, and G.H. McKinley, 2013, Microfluidic extensional rheometry using a hyperbolic contraction geometry. Rheol. Acta 52, 529–546.
Oliveira, P.J. and F.T. Pinho, 1999, Analytical solution for fully developed channel and pipe flow of Phan-Thien-Tanner fluids. J. Fluid Mech. 387, 271–280.
Perera, M.G.N. and K. Walters, 1977, Long range memory effects in flows involving abrupt changes in geometry: Part 2: The expansion/contraction/expansion problem. J. Non-Newton. Fluid Mech. 2, 191–204.
Perez-Camacho, M., J.E. Lopez-Aguilar, F. Calderas, O. Manero, and M.F. Webster, 2015, Pressure-drop and kinematics of viscoelastic flow through an axisymmetric contraction-expansion geometry with various contraction-ratios. J. Non-Newton. Fluid Mech. 222, 260–271.
Peters, G.W.M., J.F.M. Schoonen, F.P.T. Baaijens, and H.E.H. Meijer, 1999, On the performance of enhanced constitutive models for polymer melts in a cross-slot flow. J. Non-Newton. Fluid Mech. 82, 387–427.
Phan-Thien, N. and R.I. Tanner, 1977, A new constitutive equation derived from network theory. J. Non-Newton. Fluid Mech. 2, 353–365.
Phan-Thien, N., 1978, A nonlinear network viscoelastic model. J. Rheol. 22, 259–283.
Pimenta, F. and M.A. Alve, 2017, Stabilization of an open-source finite-volume solver for viscoelastic fluid flows. J. Non-Newton. Fluid Mech. 239, 85–104.
Poole, R.J., F.T. Pinho, M.A. Alves, and P.J. Oliveira, 2009, The effect of expansion ratio for creeping expansion flows of UCM fluids. J. Non-Newton. Fluid Mech. 163, 35–44.
Rodd, L.E., D. Lee, K.H. Ahn, and J.J. Cooper-White, 2010, The importance of downstream events in microfluidic viscoelastic entry flows: Consequences of increasing the constriction length. J. Non-Newton. Fluid Mech. 165, 1189–1203.
Rodd, L.E., T.P. Scott, D.V. Boger, J.J. Cooper-White, and G.H. McKinley, 2005, The inertio-elastic planar entry flow of low-viscosity elastic fluids in micro-fabricated geometries. J. Non-Newton. Fluid Mech. 129, 1–22.
Saramito, P., 1995, Efficient simulation of nonlinear viscoelastic fluid flows. J. Non-Newton. Fluid Mech. 60, 199–223.
Sousa, P.C., F.T. Pinho, M.S.N. Oliveira, and M.A. Alves, 2011, Extensional flow of blood analog solutions in microfluidic devices, Biomicrofluidics 5, 014108.
Tamaddon-Jahromi, H.R., I.E. Garduno, J.E. Lopez-Aguilar, and M.F. Webster, 2016, Predicting large experimental excess pressure drops for Boger fluids in contraction-expansion flow. J. Non-Newton. Fluid Mech. 230, 43–67.
Tamaddon-Jahromi, H.R., M.F. Webster, and P.R. Williams, 2011, Excess pressure drop and drag calculations for strain-hardening fluids with mild shear-thinning: Contraction and falling sphere problems. J. Non-Newton. Fluid Mech. 166, 939–950.
Walters, K., H.R. Tamaddon-Jahromi, M.F. Webster, M.F. Tome, and S. McKee, 2009, The competing roles of extensional viscosity and normal stress differences in complex flows of elastic liquids. Korea-Aust. Rheol. J. 21, 225–233.
Wang, J. and D.F. James, 2011, Lubricated extensional flow of viscoelastic fluids in a convergent microchannel. J. Rheol. 55, 1103–1126.
Wapperom, P. and R. Keunings, 2000, Simulation of linear polymer melts in transient complex flow. J. Non-Newton. Fluid Mech. 95, 67–83.
Wapperom, P. and R. Keunings, 2001, Numerical simulation of branched polymer melts in transient complex flow using pompom models. J. Non-Newton. Fluid Mech. 97, 267–281.
White, J.L. and A.B. Metzner, 1963, Development of constitutive equations for polymeric melts and solutions. J. Appl. Polym. Sci. 7, 1867–1889.
Xue, S.C., N. Phan-Thien, and R.I. Tanner, 1998, Three dimensional numerical simulations of viscoelastic flows through planar contractions. J. Non-Newton. Fluid Mech. 74, 195–245.
Zhang, Y., Y. Zhao, D. Chen, K. Wang, Y. Wei, Y. Xu, C. Huang, J. Wang, and J. Chen, 2019, Crossing constriction channel-based microfluidic cytometry capable of electrically phenotyping large populations of single cells. Analyst 144, 1008–1015.
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Rezaee, T., Esmaeili, M., Bazargan, S. et al. Predicting the excess pressure drop incurred by LPTT fluids in flow through a planar constricted channel. Korea-Aust. Rheol. J. 31, 149–166 (2019). https://doi.org/10.1007/s13367-019-0016-3
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DOI: https://doi.org/10.1007/s13367-019-0016-3