Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-18T20:51:02.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Polymer stretch in dilute fixed beds of fibres or spheres

Published online by Cambridge University Press:  26 April 2006

Eric S. G. Shaqfeh  and
Donald L. Koch
Eric S. G. Shaqfeh
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, CA 94305-5025, USA
Donald L. Koch
Affiliation:
School of Chemical Engineering, Cornell University, Ithaca, NY 14853, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

A theory is developed to describe the conformation change of polymers in flow through dilute, random fixed beds of spheres or fibres. The method of averaged equations is used to analyse the effect of the stochastic velocity fluctuations on polymer conformation via an approach similar to that used in our previous analysis of particle orientation in flow through these beds (Shaqfeh & Koch 1988a, b). The polymers are treated as passive tracers, i.e. the polymeric stress in the fluid is neglected in calculating the stochastic flow field. Simple dumbbell models (either linear or FENE) are used to model the polymer conformation change. In all cases we find that the long-range interactions provide the largest contribution (in the limit of vanishingly small bed volume fraction) to an evolution equation for the probability density of conformation. These interactions create a conformation-dependent diffusivity in such an equation. Solutions for the second moment of the distribution demonstrate that there is a critical pore-size Deborah number beyond which the radius of gyration of a linear dumbbell will grow indefinitely and that of the FENE dumbbell will grow to a large fraction of its maximum extensibility. This behaviour is shown to be related to the development of ‘algebraic tails’ in the distribution function. The physical reasons for this critical condition are examined and its dependence on bed structure is analysed. These results are shown to be equivalent to those which we derive by the consideration of a polymer in a class of anisotropic Gaussian flow fields. Thus, our results are explicitly related to recent work regarding polymer stretch in model turbulent flows. Finally, the effect of close interactions and their modification of our previous results is discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 244 , November 1992 , pp. 17 - 54
Copyright
© 1992 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bird R. B., Curtiss C. F., Armstrong, R. C. & Hassager O. 1978 Dynamics of Polymeric Liquids: Vol. 2 Kinetic Theory. John Wiley and Sons.
Chilcott, M. D. & Rallison J. M. 1988 Creeping flow of dilute polymer solutions past cylinders and spheres. J. Non-Newtonian Fluid Mech. 29, 381.Google Scholar
Chmielewski C, Petty, C. A. & Jayaraman K. 1990 Crossflow of elastic liquids through arrays of cylinders. J. Non-Newtonian Fluid Mech. 35, 309.Google Scholar
Dieber, J. A. & Schowalter W. R. 1979 Flow through tubes with sinusoidal variations in diameter. AIChE J. 25, 638.Google Scholar
Durst F., Haas, R. & Kaczmar B. U. 1981 Flows of hydrolysed polyacrylamide solutions in porous media under various solvent conditions. J. Appl. Polymer Sci. 26, 3125.Google Scholar
Frattini P. L., Shaqfeh E. S. G., Levy, J. L. & Koch D. L. 1991 Observations of axisymmetric tracer particle orientation during flow through a dilute fixed bed of fibers Phys. Fluids A 3, 2516.Google Scholar
Harlen O. G. 1990 High-Deborah-number flow of a dilute polymer solution past a sphere falling along the axis of a cylindrical tube. J. Non-Newtonian Fluid Mech. 37, 157.Google Scholar
Harlen O. G., Rallison, J. M. & Chilcott M. D. 1990 High-Deborah-number flows of dilute polymer solutions. J. Non-Newtonian Fluid Mech. 34, 319.Google Scholar
Hinch E. J. 1977 An averaged-equation approach to particle interactions in a fluid suspension. J. Fluid Mech. 83, 695.Google Scholar
Howells I. D. 1974 Drag due to the motion of a Newtonian fluid through a sparse random array of small fixed objects. J. Fluid Mech. 64, 449.Google Scholar
Jackson, G. W. & James D. F. 1986 The permeability of fibrous porous media. Can. J. Chem. Engng 64, 364.Google Scholar
James, D. F. & McClaren D. R. 1975 The laminar flow of dilute polymer solutions through porous media. J. Fluid Mech. 70, 733.Google Scholar
James D. F., Phan-Thien N., Khan M. M. K., Beris, A. N. & Pilitsis S. 1990 Flow of test fluid M1 in corrugated tubes. J. Non-Newtonian Fluid Mech. 35, 405.Google Scholar
Jensen R. 1981 A functional integral approach to classical statistical dynamics. J. Statist. Phys. 25, 183.Google Scholar
Jhon M. S., Sekhon, G. & Armstrong R. 1987 The response of polymer molecules in a flow. In Advances in Chemical Physics (ed. I. Prigogine & S. A. Rice), pp. 153–211. John Wiley and Sons.
Kaser, F. & Keller R. 1980 Flow of dilute polymers through porous media. J. Engng Mech. 3, 525.Google Scholar
Koch, D. L. & Brady J. F. 1985 Dispersion in fixed beds. J. Fluid Mech. 154, 399.Google Scholar
Koch, D. L. & Brady, J. F. 1986 The effective diffusivity of fibrous media. AIChE J. 32, 575.Google Scholar
Koch, D. L. & Shaqfeh E. S. G. 1990 Polymer stretch in fixed beds. Paper given at Particulate and Multiphase Flows Sessions, Annual AIChE Meeting, November.
Koch, D. L. & Shaqfeh. E. S. G. 1992 Averaged-equation and diagrammatic approximations to the average concentration of a tracer dispersed by a Gaussian, random velocity field. Phys. Fluids A (to appear 5/92).Google Scholar
Kraichnan R. H. 1959 The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech. 5, 497.Google Scholar
Kraichnan R. H. 1970 Diffusion by a random velocity field. Phys. Fluids 13, 22.Google Scholar
Larson R. G. 1988 Constitutive Equations for Polymer Melts and Solutions. Butterworths.
Larson R. G. 1990 The unravelling of a polymer chain in a strong extensional flow. Rheol. Acta 29, 371.Google Scholar
Lumley J. L. 1972 On the solution of equations describing small deformation. Symp. Math. 9, 315.Google Scholar
Lundgren T. S. 1981 Turbulent pair dispersion and scalar diffusion. J. Fluid Mech. 111, 27.Google Scholar
Nollert, M. U. & Olbricht W. L. 1985 Macromolecular deformation in periodic extensional flows. Rheol. Acta 24, 3.Google Scholar
Olbricht W. L., Rallison, J. M. & Leal L. G. 1982 Strong flow criteria based on microstructure deformation. J. Non-Newtonian Fluid Mech. 10, 201.Google Scholar
Phan-Thien, N. & Khan M. M. K. 1987 Flow of an Oldroyd-type fluid through a sinusoidally corrugated tube. J. Non-Newtonian Fluid Mech. 24, 203.Google Scholar
Pilitsis, S. & Beris A. N. 1989 Calculations of steady-state viscoelastic flow in an undulating tube. J. Non-Newtonian Fluid Mech. 31, 231.Google Scholar
Prud'homme, R. K. & Hoagland D. A. 1983 Orientation of rigid macromolecules during hydrodynamic chromatography separation. Sep. Sci. Technol. 18, 121.Google Scholar
Rallison, J. M. & Hinch E. J. 1988 Do we understand the physics in the constitutive equation? J. Non-Newtonian Fluid Mech. 29, 37.Google Scholar
Roberts P. H. 1961 Analytical theory of turbulent diffusion. J. Fluid Mech. 11, 257.Google Scholar
Saffman P. G. 1959 A theory of dispersion in porous media. J. Fluid Mech. 6, 321.Google Scholar
Shaqfeh, E. S. G. & Koch D. L. 1988a The effect of hydrodynamic interactions on the orientation of axisymmetric particles flowing through a fixed bed of spheres or fibers. Phys. Fluids 31, 728.Google Scholar
Shaqfeh, E. S. G. & Koch D. L. 1988b The combined effects of hydrodynamic interactions and Brownian motion on the orientation of particles flowing through fixed beds. Phys. Fluids 31, 2769.Google Scholar
Shaqfeh, E. S. G. & Koch D. L. 1990 Orientational dispersion of fibers in extensional flows Phys. Fluids A 2, 1077.Google Scholar
Spielman, L. & Goren S. L. 1968 Model for predicting pressure drop and filtration efficiency in fibrous media. Environ. Sci. Tech. 2, 279.Google Scholar
Tanner R. I. 1985 Engineering Rheology. Oxford University Press.
Van Kampen N. G. 1981 Stochastic Processes in Physics and Chemistry. Elsevier.
Williams, E. W. & Javadpour S. H. 1980 The flow of an elastico-viscous liquid in an axisymmetric pipe of slowly varying cross section. J. Non-Newtonian Fluid Mech. 7, 171.Google Scholar
Zick, A. & Homsy G. M. 1984 Numerical simulation of the flow of an Oldroyd fluid through a periodically constricted tube. In Proc. IX Int. Congr. Rheology (ed. G.-R. Mena & R. Nafaile). Universidad Nacional Autonoma de Mexico.