Predicting numerically the large increases in extra pressure drop when boger fluids flow through

ABSTRACT

Recent numerical studies on pressure-drops in contraction flows have introduced a variety of constitutive models to compare and contrast the competing influences of extensional vis-cosity, normal stress and shear-thinning. Early work on pressure-drops employed the constant viscosity Oldroyd-B and Upper Convected Max- well (UCM) models to represent the behavior of so-called Boger fluids in axisymmetric contrac-tion flows, in (unsuccessful) attempts to predict the very large enhancements that were ob-served experimentally. In more recent studies, other constitutive models have been employed to interpret observed behavior and some pro-gress has been made, although finding a (re-spectable) model to describe observed contrac-tion-flow behavior, even qualitatively, has been frustratingly difficult. With this in mind, the present study discusses the ability of a well- known FENE type model (the so-called FENE- CR model) to describe observed behavior. For various reasons, an axisymmetric (4:1:4) con-traction/expansion geometry, with rounded corners, is singled out for special attention, and a new hybrid finite element/volume algo-rithm is utilized to conduct the modeling, which reflects an incremental pressure-correction time-stepping structure. New to this algo-rithmic formulation are techniques in time discretization, discrete treatment of pressure terms, and compatible stress/velocity-gradient representation. We shall argue that the current simulations for the FENE-CR model have re-sulted in a major improvement in the sort-for agreement between theory and experiment in this important bench-mark problem.

Recent numerical studies on pressure-drops in contraction flows have introduced a variety of constitutive models to compare and contrast the competing influences of extensional vis-cosity, normal stress and shear-thinning. Early work on pressure-drops employed the constant viscosity Oldroyd-B and Upper Convected Max- well (UCM) models to represent the behavior of so-called Boger fluids in axisymmetric contrac-tion flows, in (unsuccessful) attempts to predict the very large enhancements that were ob-served experimentally. In more recent studies, other constitutive models have been employed to interpret observed behavior and some pro-gress has been made, although finding a (re-spectable) model to describe observed contrac-tion-flow behavior, even qualitatively, has been frustratingly difficult. With this in mind, the present study discusses the ability of a well- known FENE type model (the so-called FENE- CR model) to describe observed behavior. For various reasons, an axisymmetric (4:1:4) con-traction/expansion geometry, with rounded corners, is singled out for special attention, and a new hybrid finite element/volume algo-rithm is utilized to conduct the modeling, which reflects an incremental pressure-correction time-stepping structure. New to this algo-rithmic formulation are techniques in time discretization, discrete treatment of pressure terms, and compatible stress/velocity-gradient representation. We shall argue that the current simulations for the FENE-CR model have re-sulted in a major improvement in the sort-for agreement between theory and experiment in this important bench-mark problem.

Cite this paper

Tamaddon-Jahromi, H. , Webster, M. and Walters, K. (2010) Predicting numerically the large increases in extra pressure drop when boger fluids flow through.*Natural Science*, **2**, 1-11. doi: 10.4236/ns.2010.21001.

Tamaddon-Jahromi, H. , Webster, M. and Walters, K. (2010) Predicting numerically the large increases in extra pressure drop when boger fluids flow through.

References

[1] Debbaut, B. and Crochet, M.J. (1988) Extensional effects in complex flows. J Non-Newton Fluid Mech, 30, 169-184.

[2] Debbaut, B., Crochet, M.J., Barnes, H.A. and Walters, K. (1988) Extensional effects in inelastic liquids. Xth Inter. Congress on Rheology, Sydney, 291-293.

[3] Binding, D.M. (1991) Further considerations of axisym-metric contraction flows. J Non-Newton Fluid Mech, 41, 27-42.

[4] Nigen, S. and Walters, K. (2002) Viscoelastic contraction flows: comparison of axisymmetric and planar configu-rations. J Non-Newton Fluid Mech , 102, 343-359.

[5] Rothstein, J.P. and McKinley, G.H. (2001) The axisym-metric contraction-expansion: The role of extensional rheology on vortex growth dynamics and the enhanced pressure drop. J Non-Newton Fluid Mech, 98, 33-63.

[6] Phillips, T.N. and Williams, A.J. (2002) Comparison of creeping and inertial flow of an Oldroyd B fluid through planar and axisymmetric contractions. J Non-Newton Fluid Mech, 108, 25-47.

[7] Aboubacar, M., Matallah, H., Tamaddon-Jahromi, H.R. and Webster, M.F. (2002) Numerical prediction of exten-sional flows in contraction geometries: Hybrid finite vol-ume/element method. J Non-Newton Fluid Mech, 104, 125-164.

[8] Walters, K. and Webster, M.F. (2003) The distinctive CFD challenges of computational rheology. Inter J for Numer Meth in Fluids, 43, 577-596.

[9] Alves, M.A., Oliveira, P.J. and Pinho, F.T. (2004) On the effect of contraction ratio in viscoelastic flow through ab- rupt contractions. J Non-Newton Fluid Mech, 122, 117-130.

[10] Aguayo, J.P., Tamaddon-Jahromi, H.R. and Webster, M.F. (2008) Excess pressure-drop estimation in contraction flows for strain-hardening fluids. J Non-Newton Fluid Mech, 153, 186-205.

[11] Walters, K., Webster, M.F. and Tamaddon-Jahromi, H.R. (2009a) The numerical simulation of some contraction flows of highly elastic liquids and their impact on the relevance of the Couette correction in extensional rheol-ogy. Chem Eng Sci, 64, 4632-4639.

[12] Walters, K., Webster, M.F. and Tamaddon-Jahromi, H.R. (2009b) The White-Metzner model then and now. Pro-ceedings of the 25th Annual Meeting of the PPS meeting, Goa, India, 02, 1-14.

[13] Walters, K., Tamaddon-Jahromi, H.R., Webster, M.F., Tomé, M.F. and McKee, S. (2010) The competing roles of extensional viscosity and normal stress differences in complex flows of elastic liquids. Korean-Australian Journal (to be published), 2010.

[14] White, J.L. and Metzner, A.B. (1963) Development of constitutive equations for polymeric melts and solutions. J Appl Polym Sci, 7, 1867-1889.

[15] Szabo, P., Rallison, J.M. and Hinch, E.J. (1997) Start-up of flow of a FENE-fluid through 4:1:4 constrictions in a tube. J Non-Newton Fluid Mech, 72, 73-86.

[16] Chilcott, M.D. and Rallison, J.M. (1988) Creeping flow of dilute polymer solutions past cylinders and spheres. J Non-Newton Fluid Mech, 29, 381-432.

[17] Barnes, H.A., Hutton, J.F. and Walters, K. (1989) An introduction to rheology. Elsevier, Amsterdam.

[18] Jackson, K.P., Walters, K. and Williams, R.W. (1984) A rheometrical study of Boger fluids. J Non-Newton Fluid Mech, 14, 173-188.

[19] Tanner, R.I. and Walters, K. (1998) Rheology: An his-torical perspective, Elsevier Science & Technology, Netherlands.

[20] Binding, D.M., Phillips, P.M. and Phillips, T.N. (2006) Contraction/expansion flows: The pressure drop and re-lated issues. J Non-Newton Fluid Mech, 137, 31-38.

[21] Walters, K., Webster, M.F. and Tamaddon-Jahromi, H.R. (2008) Experimental and computational aspects of some contraction flows of highly elastic liquids and their im-pact on the relevance of the Couette correction in exten-sional rheology. Proc. 2nd Southern African Conference on Rheology (SASOR 2), 1-6.

[22] Oldroyd, J. G. (1950) On the formulation of rheological equations of state. Proc. Roy .Soc., A200, 523-541.

[23] Boger, D.V. and Walters, K. (1993) Rheological phe-nomena in focus, Elsevier Science Publishers.

[24] Webster, M.F., Tamaddon-Jahromi, H.R. and Aboubacar, M. (2005) Time-dependent algorithms for viscoelastic flow: Finite element/volume schemes. Numer Meth Par Diff Equ, 21, 272-296.

[25] Matallah, H., Townsend, P. and Webster, M.F. (1998) Recovery and stress-splitting schemes for viscoelastic flows. J Non-Newton Fluid Mech, 75, 139-166.

[26] Donea, J. (1984) A Taylor-Galerking method for convective transport problems. Int J Numer Methods Eng, 20, 101-119.

[27] Zienkiewicz, O.C., Morgan, K, Peraire, J., Vandati, M. and L?hner R. (1985) Finite elements for compressible gas flow and similar systems. Seventh International Conference on Computational Methods of Applied Sci-ence and Engineering.

[28] Wapperom, P. and Webster, M.F. (1999) Simulation for viscoelastic flow by a finite volume/element method. Comput Meth Appl Mech Eng, 180, 281-304.

[29] Wapperom, P. and Webster, M.F. (1998) A second-order hybrid finite-element/volume method for viscoelastic flows. J Non-Newton Fluid Mech, 79, 405-431.

[30] Ghosh, I., Lak, Joo Y., McKinley, G.H., Brown, R.A. and Armstrong, R.C. (2002) A new model for dilute polymer solutions in flows. J Rheol, 46(5), 1057-1089.

[1] Debbaut, B. and Crochet, M.J. (1988) Extensional effects in complex flows. J Non-Newton Fluid Mech, 30, 169-184.

[2] Debbaut, B., Crochet, M.J., Barnes, H.A. and Walters, K. (1988) Extensional effects in inelastic liquids. Xth Inter. Congress on Rheology, Sydney, 291-293.

[3] Binding, D.M. (1991) Further considerations of axisym-metric contraction flows. J Non-Newton Fluid Mech, 41, 27-42.

[4] Nigen, S. and Walters, K. (2002) Viscoelastic contraction flows: comparison of axisymmetric and planar configu-rations. J Non-Newton Fluid Mech , 102, 343-359.

[5] Rothstein, J.P. and McKinley, G.H. (2001) The axisym-metric contraction-expansion: The role of extensional rheology on vortex growth dynamics and the enhanced pressure drop. J Non-Newton Fluid Mech, 98, 33-63.

[6] Phillips, T.N. and Williams, A.J. (2002) Comparison of creeping and inertial flow of an Oldroyd B fluid through planar and axisymmetric contractions. J Non-Newton Fluid Mech, 108, 25-47.

[7] Aboubacar, M., Matallah, H., Tamaddon-Jahromi, H.R. and Webster, M.F. (2002) Numerical prediction of exten-sional flows in contraction geometries: Hybrid finite vol-ume/element method. J Non-Newton Fluid Mech, 104, 125-164.

[8] Walters, K. and Webster, M.F. (2003) The distinctive CFD challenges of computational rheology. Inter J for Numer Meth in Fluids, 43, 577-596.

[9] Alves, M.A., Oliveira, P.J. and Pinho, F.T. (2004) On the effect of contraction ratio in viscoelastic flow through ab- rupt contractions. J Non-Newton Fluid Mech, 122, 117-130.

[10] Aguayo, J.P., Tamaddon-Jahromi, H.R. and Webster, M.F. (2008) Excess pressure-drop estimation in contraction flows for strain-hardening fluids. J Non-Newton Fluid Mech, 153, 186-205.

[11] Walters, K., Webster, M.F. and Tamaddon-Jahromi, H.R. (2009a) The numerical simulation of some contraction flows of highly elastic liquids and their impact on the relevance of the Couette correction in extensional rheol-ogy. Chem Eng Sci, 64, 4632-4639.

[12] Walters, K., Webster, M.F. and Tamaddon-Jahromi, H.R. (2009b) The White-Metzner model then and now. Pro-ceedings of the 25th Annual Meeting of the PPS meeting, Goa, India, 02, 1-14.

[13] Walters, K., Tamaddon-Jahromi, H.R., Webster, M.F., Tomé, M.F. and McKee, S. (2010) The competing roles of extensional viscosity and normal stress differences in complex flows of elastic liquids. Korean-Australian Journal (to be published), 2010.

[14] White, J.L. and Metzner, A.B. (1963) Development of constitutive equations for polymeric melts and solutions. J Appl Polym Sci, 7, 1867-1889.

[15] Szabo, P., Rallison, J.M. and Hinch, E.J. (1997) Start-up of flow of a FENE-fluid through 4:1:4 constrictions in a tube. J Non-Newton Fluid Mech, 72, 73-86.

[16] Chilcott, M.D. and Rallison, J.M. (1988) Creeping flow of dilute polymer solutions past cylinders and spheres. J Non-Newton Fluid Mech, 29, 381-432.

[17] Barnes, H.A., Hutton, J.F. and Walters, K. (1989) An introduction to rheology. Elsevier, Amsterdam.

[18] Jackson, K.P., Walters, K. and Williams, R.W. (1984) A rheometrical study of Boger fluids. J Non-Newton Fluid Mech, 14, 173-188.

[19] Tanner, R.I. and Walters, K. (1998) Rheology: An his-torical perspective, Elsevier Science & Technology, Netherlands.

[20] Binding, D.M., Phillips, P.M. and Phillips, T.N. (2006) Contraction/expansion flows: The pressure drop and re-lated issues. J Non-Newton Fluid Mech, 137, 31-38.

[21] Walters, K., Webster, M.F. and Tamaddon-Jahromi, H.R. (2008) Experimental and computational aspects of some contraction flows of highly elastic liquids and their im-pact on the relevance of the Couette correction in exten-sional rheology. Proc. 2nd Southern African Conference on Rheology (SASOR 2), 1-6.

[22] Oldroyd, J. G. (1950) On the formulation of rheological equations of state. Proc. Roy .Soc., A200, 523-541.

[23] Boger, D.V. and Walters, K. (1993) Rheological phe-nomena in focus, Elsevier Science Publishers.

[24] Webster, M.F., Tamaddon-Jahromi, H.R. and Aboubacar, M. (2005) Time-dependent algorithms for viscoelastic flow: Finite element/volume schemes. Numer Meth Par Diff Equ, 21, 272-296.

[25] Matallah, H., Townsend, P. and Webster, M.F. (1998) Recovery and stress-splitting schemes for viscoelastic flows. J Non-Newton Fluid Mech, 75, 139-166.

[26] Donea, J. (1984) A Taylor-Galerking method for convective transport problems. Int J Numer Methods Eng, 20, 101-119.

[27] Zienkiewicz, O.C., Morgan, K, Peraire, J., Vandati, M. and L?hner R. (1985) Finite elements for compressible gas flow and similar systems. Seventh International Conference on Computational Methods of Applied Sci-ence and Engineering.

[28] Wapperom, P. and Webster, M.F. (1999) Simulation for viscoelastic flow by a finite volume/element method. Comput Meth Appl Mech Eng, 180, 281-304.

[29] Wapperom, P. and Webster, M.F. (1998) A second-order hybrid finite-element/volume method for viscoelastic flows. J Non-Newton Fluid Mech, 79, 405-431.

[30] Ghosh, I., Lak, Joo Y., McKinley, G.H., Brown, R.A. and Armstrong, R.C. (2002) A new model for dilute polymer solutions in flows. J Rheol, 46(5), 1057-1089.