Abstract
Numerical simulations are presented for flows of inelastic non-Newtonian fluids through periodic arrays of aligned cylinders, for cases in which fluid inertia can be neglected. The truncated power-law fluid model is used to define the relationship between the viscous stress and the rate-of-strain tensor. The flow is shown to be dominated by shear effects, not extension. Numerical simulation results are presented for the drag coefficient of the cylinders and the velocity variance components, and are compared against asymptotically valid analytical results. Square and hexagonal arrays are considered, both for crossflow in the plane perpendicular to the alignment vector of the cylinders (flows along the axes of the array as well as off-axis flows), and for flow along the cylinders. It is shown that the observed strong dependence of the drag coefficient on the power-law index (through which the stress tensor is related to the rate-of-strain tensor) can be described at all solid area fractions by scaling the drag on a cylinder with appropriate velocity and length scales. The velocity variance components show only a weak dependence on the power-law index. The results for steady-state drag and velocity variances are used in an approximate theory for flows accelerated from rest. The numerical simulation data for unsteady flows agree very well with the approximate theory.
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A.S. Sangani and A. Acrivos, Slow flow past periodic arrays of cylinders with application to heat transfer. Int. J. Multiphase Flow 8 (1982) 193–206.
J.E. Drummond and M.I. Tahir, Laminar viscous flow through regular arrays of parallel solid cylinders. Int. J. Multiphase Flow 10 (1984) 515–540.
A.S. Sangani and C. Yao, Transport processes in random arrays of cylinders. II. Viscous flow. Phys. Fluids 31 (1988) 2435–2444.
D.L. Koch and A.J.C. Ladd, Moderate Reynolds number flows through periodic and random arrays of aligned cylinders. J. Fluid Mech. 349 (1997) 31–66.
C. Chmielewski, C.A. Petty and K. Jayaraman, Crossflow of elastic liquids through arrays of cylinders. J. Non-Newt. Fluid Mech. 35 (1990) 309–325.
J. Vorwerk and P.O. Brunn, Porous medium flow of the fluid A1: effects of shear and elongation. J. Non-Newt. Fluid Mech. 41 (1991) 119–131.
L. Skartsis, B. Khomami and J.L. Kardos, Polymeric flow through fibrous media. J. Rheol. 36 (1992) 589–620.
C. Chmielewski, and K. Jayaraman, Elastic instability in crossflow of polymer solutions through periodic arrays of cylinders. J. Non-Newt. Fluid Mech. 48 (1993) 285–301.
B. Khomami and L.D. Moreno, Stability of viscoelastic flow around periodic arrays of cylinders. Rheol. Acta 36 (1997) 367–383.
G.K. Batchelor, The stress generated in a non-dilute suspension of elongated particles by pure straining motion. J. Fluid Mech. 46 (1971) 813–829.
K.K. Talwar and B. Khomami, Flow of viscoelastic fluids past periodic square arrays of cylinders: inertial and shear thinning viscosity and elasticity effects. J. Non-Newt. Fluid Mech. 57 (1995) 177–202.
A.W. Liu, D.E. Bornside, R.C. Armstrong and R.A. Brown, Viscoelastic flow of polymer solutions around a periodic, linear array of cylinders: comparisons of predictions for microstructure and flow fields. J. Non-Newt. Fluid Mech. 77 (1998) 153–190.
R.B. Bird, R.C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids. Vol. 1: Fluid Mechanics. New York: John Wiley (1987) 649 pp.
P.D.M. Spelt, T. Selerland, C.J. Lawrence and P.D. Lee, Flows of inelastic non-Newtonian fluids through arrays of aligned cylinders. Part 2. Inertial effects for square arrays. Submitted to J. Eng. Math. (2005) 81–97.
R.I. Tanner, Stokes paradox for power-law flow around a cylinder. J. Non-Newt. Fluid Mech. 50 (1993) 217–224.
M.V. Bruschke and S.G. Advani, Flow of generalized Newtonian fluids across a periodic array of cylinders. J. Rheol. 37 (1993) 479–498.
S.J.D. D’Alessio, and J.P. Pascal, Steady flow of a power-law fluid past a cylinder. Acta Mech. 117 (1996) 87–100.
M. Vijaysri, R.P. Chhabra and V. Eswaran, Power-law fluid flow across an array of infinite circular cylinders: a numerical study. J. Non-Newt. Fluid Mech. 87 (1999) 263–282.
C.B. Shah and Y.C. Yortsos, Aspects of flow of power-law fluids in porous media. AIChE J. 41 (1995) 1099–1112.
J.R.A. Pearson and P.M.J. Tardy, Models for flow of non-Newtonian and complex fluids through porous media. J. Non-Newt. Fluid Mech. 102 (2002) 447–473.
E.J. Hinch, An averaged-equation approach to particle interactions in a fluid suspension. J. Fluid Mech. 83 (1977) 695–720.
R.J. Hill, D.L. Koch and A.J.C. Ladd, The first effects of fluid inertia on flows in ordered and random arrays of spheres. J. Fluid Mech. 448 (2001) 213–241.
A.J.C. Ladd, Hydrodynamic transport coefficients of random dispersions of hard spheres.J. Chem. Phys. 93 (1990) 3484–3494.
G. Mo, and A.S. Sangani, A method for computing Stokes flow interactions among spherical objects and its application to spherical drops and porous particles. Phys. Fluids 6 (1994) 1637–1652.
N. Martys, D.P. Bentz and E.J. Garboczi, Computer simulation study of the effective viscosity in Brinkman’s equation. Phys. Fluids 6 (1994) 1434–1439.
S.K. Gupte, S.K. and S.G. Advani, Flow near the permeable boundary of an aligned fibre preform: an experimental investigation using laser doppler anemometry. Polym. Comp. 18 (1997) 114–124.
C.D. Tsakiroglou, A methodology for the derivation of non-Darcian models for the flow of generalized Newtonian fluids in porous media. J. Non-Newt. Fluid Mech. 105 (2002) 79–110.
W.B. Russel, D.A. Saville and W.R. Schowalter, Colloidal Dispersions. Cambridge: Cambridge University Press (1989) 525 pp.
Y. Zang, R.L. Street and J.R. Koseff, A non-staggered grid, fractional step method for time-dependent incompressible Navier-Stokes equations in curvilinear coordinates. J. Comp. Phys. 114 (1994) 18–33.
Y. Zang and R.L. Street, A composite multigrid method for calculating unsteady incompressible flows in geometrically complex domains. Int. J. Num. Meth. Fluids 20 (1995) 341–361.
L.L. Yuan, R.L. Street and J.H. Ferziger, Large-eddy simulations of a round jet in crossflow. J. Fluid Mech. 379 (1999) 71–104.
H.F. Bulthuis, Dynamics of Bubbly Flows. Ph.D. thesis, University of Twente, the Netherlands (1997) 132 pp.
A. Iafrati, A. Di Mascio and E.F. Campana, A level set technique applied to unsteady free surface flows. Int. J. Num. Meth. Fluids 35 (2001) 281–297.
J.K. Woods, P.D.M. Spelt, T. Selerland, C.J. Lawrence and P.D. Lee, Creeping flows of power-law fluids through periodic arrays of elliptical cylinders. J. Non-Newt. Fluid Mech. 111 (2003) 211–228.
T.A.K. Sadiq, S.G. Advani and R.S. Parnas, Experimental investigation of transverse flow through aligned cylinders. Int. J. Multiphase Flow 21 (1995) 755–774.
D. Adams and K.J. Bell, Fluid friction and heat transfer for flow of sodium carboxy methylcellulose solutions across banks of tubes. Chem. Eng. Prog. Symp. Ser. 64 (1968) 133–145.
D.L. Koch, R.J. Hill and A.S. Sangani, Brinkman screening and the covariance of the fluid velocity in fixed beds. Phys. Fluids 10 (1998) 3035–3037.
H. Hasimoto, On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5 (1959) 317–328.
P.R. Schunk and L.E. Scriven, Constitutive equation for modeling mixed extension and shear in polymer solution processing. J. Rheol. 34 (1990) 1085–1119.
A.M. Wunderlich, P.O. Brunn and F. Durst, Flow of dilute polyacrylamide solutions through a sudden planar contraction. J. Non-Newt. Fluid Mech. 28 (1988) 267–285.
E. Ryssel and P.O. Brunn, Flow of a quasi-Newtonian fluid through a planar contraction. J. Non-Newt. Fluid Mech. 85 (1999) 11–27.
R. Mei, Flow due to an oscillating sphere and an expression for unsteady drag on the sphere at finite Reynolds number. J. Fluid Mech. 270 (1994) 133–174.
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Spelt, P.D.M., Selerland, T., Lawrence, C.J. et al. Flows of inelastic non-Newtonian fluids through arrays of aligned cylinders. Part 1. Creeping flows. J Eng Math 51, 57–80 (2005). https://doi.org/10.1007/s10665-004-5783-1
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DOI: https://doi.org/10.1007/s10665-004-5783-1