Elastoviscoplastic fluid flow in non-circular tubes: Transversal field and interplay of elasticity and plasticity

M.F. Letelier, C. Barrera, D.A. Siginer, A. González

Research output: Contribution to journalArticle

Abstract

An analytical study of elastoviscoplastic fluid flow in tubes of non-circular cross section is presented. The constitutive structure of the fluid is described by a linear frame invariant combination of the Phan-Thien−Tanner model of viscoelastic fluids and the Bingham model of plastic fluids. Non-circular tube cross sections are modeled by the shape factor method a one-to-one mapping of the circular base contour into a wide spectrum family of arbitrary tube contours. Field variables are expanded into asymptotic series in terms of the elasticity measure, the Weissenberg number We, coupled with an asymptotic expansion in terms of the geometrical mapping parameter ε leading to a set of hierarchical momentum balance equations which are solved successively up to and including the third order in We when the secondary field appears for the first time. The computational algorithm developed is applied to the study of the non-rectilinear flow in tubes with triangular and square cross sections. We find that the presence of the yield stress dampens the intensity of the purely viscoelastic vortices, the higher the yield stress the lower the intensity of the vortices in the cross-section, and the further away the vortices are from the center of the cross section as compared to the purely viscoelastic vortices. The results also evidence that viscoelasticity increases the axial flow for given viscoplastic conditions and pressure drop, and consequently increases the rate of flow, a phenomenon that may find applications in optimizing material transportation. © 2017 Elsevier Inc.
LanguageEnglish
Pages768-781
Number of pages14
JournalApplied Mathematical Modelling
Volume54
DOIs
Publication statusPublished - 2018

Fingerprint

Plasticity
Fluid Flow
Elasticity
Flow of fluids
Tube
Vortex flow
Cross section
Vortex
Yield Stress
Fluids
Yield stress
Axial flow
Viscoelasticity
Fluid
Asymptotic series
Viscoelastic Fluid
Pressure drop
Pressure Drop
Balance Equations
Computational Algorithm

Keywords

  • Elastoviscoplasticity
  • Interplay of elasticity and plasticity
  • Non-circular cross-section
  • Transversal field
  • Tube flow
  • Elasticity
  • Mapping
  • Materials handling
  • Non Newtonian flow
  • Pipe flow
  • Plasticity
  • Tubes (components)
  • Viscoelasticity
  • Vortex flow
  • Yield stress
  • Computational algorithm
  • Elasto-viscoplasticity
  • Material transportation
  • Momentum balance equations
  • Square cross section
  • Vis-coelastic fluids
  • Flow of fluids

Cite this

Elastoviscoplastic fluid flow in non-circular tubes: Transversal field and interplay of elasticity and plasticity. / Letelier, M.F.; Barrera, C.; Siginer, D.A.; González, A.

In: Applied Mathematical Modelling, Vol. 54, 2018, p. 768-781.

Research output: Contribution to journalArticle

@article{efe76efaf84b49398b04fcab04d0e8e9,
title = "Elastoviscoplastic fluid flow in non-circular tubes: Transversal field and interplay of elasticity and plasticity",
abstract = "An analytical study of elastoviscoplastic fluid flow in tubes of non-circular cross section is presented. The constitutive structure of the fluid is described by a linear frame invariant combination of the Phan-Thien−Tanner model of viscoelastic fluids and the Bingham model of plastic fluids. Non-circular tube cross sections are modeled by the shape factor method a one-to-one mapping of the circular base contour into a wide spectrum family of arbitrary tube contours. Field variables are expanded into asymptotic series in terms of the elasticity measure, the Weissenberg number We, coupled with an asymptotic expansion in terms of the geometrical mapping parameter ε leading to a set of hierarchical momentum balance equations which are solved successively up to and including the third order in We when the secondary field appears for the first time. The computational algorithm developed is applied to the study of the non-rectilinear flow in tubes with triangular and square cross sections. We find that the presence of the yield stress dampens the intensity of the purely viscoelastic vortices, the higher the yield stress the lower the intensity of the vortices in the cross-section, and the further away the vortices are from the center of the cross section as compared to the purely viscoelastic vortices. The results also evidence that viscoelasticity increases the axial flow for given viscoplastic conditions and pressure drop, and consequently increases the rate of flow, a phenomenon that may find applications in optimizing material transportation. {\circledC} 2017 Elsevier Inc.",
keywords = "Elastoviscoplasticity, Interplay of elasticity and plasticity, Non-circular cross-section, Transversal field, Tube flow, Elasticity, Mapping, Materials handling, Non Newtonian flow, Pipe flow, Plasticity, Tubes (components), Viscoelasticity, Vortex flow, Yield stress, Computational algorithm, Elasto-viscoplasticity, Material transportation, Momentum balance equations, Square cross section, Vis-coelastic fluids, Flow of fluids",
author = "M.F. Letelier and C. Barrera and D.A. Siginer and A. Gonz{\'a}lez",
note = "Export Date: 10 April 2018 CODEN: AMMOD Correspondence Address: Siginer, D.A.; Centro de Investigaci{\'o}n en Creatividad y Educaci{\'o}n Superior & Departamento de Ingenier{\'i}a Mec{\'a}nica, Universidad de Santiago de Chile and concurrently at the Botswana International University of Science and Technology, Palapye, Botswana, Chile; email: dennis.siginer@usach.cl Funding details: DICYT, Departamento de Investigaciones Cient{\'i}ficas y Tecnol{\'o}gicas, Universidad de Santiago de Chile Funding details: 1130346, FONDECYT, Fondo Nacional de Desarrollo Cient{\'i}fico y Tecnol{\'o}gico Funding details: Usach, Universidad de Santiago de Chile Funding text: The authors gratefully acknowledge the financial support of FONDECYT through grant 1130346 , and the support of DICYT of the Universidad de Santiago de Chile. References: Bingham, E.C., Fluidity and Plasticity (1922), Mc Graw-Hill; Herschel, W.H., Bulkley, T., Measurement of consistency as applied to rubber-benzene solutions (1926) Am. Soc. Test Proc, 26, pp. 621-633; Walton, I.C., Bittleston, S.H., The axial flow of a Bingham plastic in a narrow eccentric annulus (1991) J. Fluid Mech., 222, pp. 39-60; Wachs, A., Numerical simulation of steady Bingham flow through an eccentric annular cross-section by distributed Lagrange multiplier fictitious domain and augmented Lagrangian methods (2007) J. Non-Newton. Fluid Mech, 142, pp. 183-198; Beverly, C.R., Tanner, R.I., Numerical analysis of extrudate swell in viscoelastic materials with yield stress (1989) J. Rheol., 33, p. 989; Moyers-Gonzalez, M.A., Frigaard, I.A., Numerical solution of duct flows of multiple visco-plastic fluids (2004) J. Non-Newton. Fluid Mech., 122, pp. 227-241; Huilgol, R.R., You, Z., Application of the augmented Lagrangian method to steady pipe flows of Bingham, Casson and Herschel-Bulkley fluids (2005) J. Non-Newton. Fluid Mech, 128, pp. 126-143; Saramito, P., Roquet, N., An adaptive finite element method for viscoplastic fluid flows in pipes (2001) Comput. Methods Appl. Mech. Eng., 190, pp. 5391-5412; Roquet, N., Saramito, P., An adaptive finite element method for viscoplastic flows in a square pipe with stick-slip at the wall (2008) J. Non-Newton. Fluid Mech, 155, pp. 101-115; Letelier, M.F., Siginer, D.A., On the flow of a class of viscoinelastic-viscoplastic fluids in tubes of non-circular contour (2007) Int. J. Eng. Sci., 45, pp. 873-881; Fraggedakis, D., Dimakopoulos, Y., Tsamopoulos, J., Yielding the yield stress analysis: a thorough comparison of recently proposed elasto-visco-plastic (EVP) fluid models (2016) J. Non-Newton. Fluid Mech, 238, pp. 170-188; Oldroyd, J.G., On the formulation of rheological equations of states (1950) Proc. R. Soc. London, Ser. A, 200, pp. 523-541; Saramito, P., A new constitutive equation for elastoviscoplastic fluid flows (2007) J. Non-Newton. Fluid Mech., 145, pp. 1-14; Saramito, P., A new elastoviscoplastic model based on the Herschel−Bulkley viscoplasticity (2009) J. Non-Newton. Fluid Mech., 158, pp. 154-161; de Souza Mendes, P.R., Dimensionless non-Newtonian fluid mechanics (2007) J. Non-Newton. Fluid Mech, 147 (1-2), pp. 109-116; Cheddadi, I., Saramito, P., A new operator splitting algorithm for elastoviscoplastic flow problems (2013) J. Non-Newt. Fluid Mech., 202, pp. 13-21; Nassar, B., Souza Mendes, P.R., Naccache, M.F., Flow of elasto-viscoplastic liquids through an axisymmetric expansion-contraction (2011) J. Non-Newton. Fluid Mech., 166, pp. 386-394; Siginer, D.A., Developments in Tube Flow of Complex Fluids (2015), Springer Inc New York; Siginer, D.A., Isothermal tube flow of non-linear viscoelastic fluids, part II: transversal field (2011) Int. J. Eng. Sci., 49 (6), pp. 443-465; Siginer, D.A., Letelier, M.F., Laminar flow of non-linear viscoelastic fluids in straight tubes of arbitrary contour, Int. J. Heat Mass Transf. 54 (2011) 2188-2202 updated by the editor for errors introduced in printing and reappeared in the same journal, Int. J. Heat Mass Transf., 55 (2012) 2731‐2745; Ericksen, J.L., Over determination of the speed in rectilinear motion of non-Newtonian fluids (1956) Quart. Appl. Math., 14, pp. 319-321; Green, A.E., Rivlin, R.S., Steady flow of non-Newtonian fluids through tubes (1956) Quart. Appl. Math., 14, pp. 299-308; Langlois, W.E., Rivlin, R.S., Slow steady-state flow of viscoelastic fluids through non-circular tubes (1963) Rend. Math., 22, pp. 169-185; Xue, S., Phan-Thien, N., Tanner, R.I., Numerical study of secondary flows of viscoelastic fluid in straight pipes by an implicit finite method (1995) J. Non-Newton. Fluid Mech., 59 (2-3), pp. 191-213; Oliver, D.R., Non-Newtonian heat transfer: an interesting effect observed in non-circular tubes (1969) Trans. Inst. Chem. Eng., 47, p. T18; Gao, S.X., Hartnett, J.P., Heat transfer behavior of Reiner-Rivlin fluids in rectangular ducts (1996) Int. J. Heat Mass Transf., 39, pp. 1317-1324; Siginer, D.A., Letelier, M.F., Heat transfer asymptote in laminar flow of non-linear viscoelastic fluids in straight non-circular tubes (2010) Int. J. Eng. Sci., 48, pp. 1544-1562; Letelier, M.F., Barrera, C., Siginer, D.A., Analytical solution of the Graetz problem for non-linear viscoelastic fluids (2017) Int. J. Therm. Sci., 111, pp. 369-378; Pinho, F.T., Oliveira, P.J., Analysis of forced convection in pipes and channels with the simplified Phan−Thien−Tanner fluid (2000) Int. J. Heat Mass Transf., 43, pp. 2273-2287; Letelier, M.F., Siginer, D.A., Gonz{\'a}lez, A., Elasto-viscoplastic fluid flow in tubes of arbitrary cross-section (2017) Appl. Math. Model., 46, pp. 572-580; Johnson, M., Segalman, D., A model for viscoelastic fluid behavior which allows non-affine deformation (1977) J. Non-Newton. Fluid Mech., 2, pp. 255-270; Letelier, M.F., Barrera, C., Siginer, D.A., On the physics of viscoplastic fluid flow in non-circular tubes (2017) Int. J. Non-Linear Mech, 88, pp. 1-10; Slater, L.J., Confluent Hypergeometric Functions (1960), pp. 503-515. , Cambridge University Press; L{\'o}pez-Aguilar, J.E., Webster, M.F., Tamaddon-Jahromi, H.R., Numerical modelling of thixotropic and viscoelastoplastic materials in complex flow (2015) Rheol. Acta, 54, p. 307; Fraggedakis, D., Dimakopoulos, Y., Tsamopoulos, J., Yielding the yield-stress analysis: a study focused on the effects of elasticity on the settling of a single spherical particle in simple yield-stress fluids (2016) Soft matter, 12 (24), pp. 5378-5401",
year = "2018",
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language = "English",
volume = "54",
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}

TY - JOUR

T1 - Elastoviscoplastic fluid flow in non-circular tubes: Transversal field and interplay of elasticity and plasticity

AU - Letelier, M.F.

AU - Barrera, C.

AU - Siginer, D.A.

AU - González, A.

N1 - Export Date: 10 April 2018 CODEN: AMMOD Correspondence Address: Siginer, D.A.; Centro de Investigación en Creatividad y Educación Superior & Departamento de Ingeniería Mecánica, Universidad de Santiago de Chile and concurrently at the Botswana International University of Science and Technology, Palapye, Botswana, Chile; email: dennis.siginer@usach.cl Funding details: DICYT, Departamento de Investigaciones Científicas y Tecnológicas, Universidad de Santiago de Chile Funding details: 1130346, FONDECYT, Fondo Nacional de Desarrollo Científico y Tecnológico Funding details: Usach, Universidad de Santiago de Chile Funding text: The authors gratefully acknowledge the financial support of FONDECYT through grant 1130346 , and the support of DICYT of the Universidad de Santiago de Chile. References: Bingham, E.C., Fluidity and Plasticity (1922), Mc Graw-Hill; Herschel, W.H., Bulkley, T., Measurement of consistency as applied to rubber-benzene solutions (1926) Am. Soc. Test Proc, 26, pp. 621-633; Walton, I.C., Bittleston, S.H., The axial flow of a Bingham plastic in a narrow eccentric annulus (1991) J. Fluid Mech., 222, pp. 39-60; Wachs, A., Numerical simulation of steady Bingham flow through an eccentric annular cross-section by distributed Lagrange multiplier fictitious domain and augmented Lagrangian methods (2007) J. Non-Newton. Fluid Mech, 142, pp. 183-198; Beverly, C.R., Tanner, R.I., Numerical analysis of extrudate swell in viscoelastic materials with yield stress (1989) J. Rheol., 33, p. 989; Moyers-Gonzalez, M.A., Frigaard, I.A., Numerical solution of duct flows of multiple visco-plastic fluids (2004) J. Non-Newton. Fluid Mech., 122, pp. 227-241; Huilgol, R.R., You, Z., Application of the augmented Lagrangian method to steady pipe flows of Bingham, Casson and Herschel-Bulkley fluids (2005) J. Non-Newton. Fluid Mech, 128, pp. 126-143; Saramito, P., Roquet, N., An adaptive finite element method for viscoplastic fluid flows in pipes (2001) Comput. Methods Appl. Mech. Eng., 190, pp. 5391-5412; Roquet, N., Saramito, P., An adaptive finite element method for viscoplastic flows in a square pipe with stick-slip at the wall (2008) J. Non-Newton. Fluid Mech, 155, pp. 101-115; Letelier, M.F., Siginer, D.A., On the flow of a class of viscoinelastic-viscoplastic fluids in tubes of non-circular contour (2007) Int. J. Eng. Sci., 45, pp. 873-881; Fraggedakis, D., Dimakopoulos, Y., Tsamopoulos, J., Yielding the yield stress analysis: a thorough comparison of recently proposed elasto-visco-plastic (EVP) fluid models (2016) J. Non-Newton. Fluid Mech, 238, pp. 170-188; Oldroyd, J.G., On the formulation of rheological equations of states (1950) Proc. R. Soc. London, Ser. A, 200, pp. 523-541; Saramito, P., A new constitutive equation for elastoviscoplastic fluid flows (2007) J. Non-Newton. Fluid Mech., 145, pp. 1-14; Saramito, P., A new elastoviscoplastic model based on the Herschel−Bulkley viscoplasticity (2009) J. Non-Newton. Fluid Mech., 158, pp. 154-161; de Souza Mendes, P.R., Dimensionless non-Newtonian fluid mechanics (2007) J. Non-Newton. Fluid Mech, 147 (1-2), pp. 109-116; Cheddadi, I., Saramito, P., A new operator splitting algorithm for elastoviscoplastic flow problems (2013) J. Non-Newt. Fluid Mech., 202, pp. 13-21; Nassar, B., Souza Mendes, P.R., Naccache, M.F., Flow of elasto-viscoplastic liquids through an axisymmetric expansion-contraction (2011) J. Non-Newton. Fluid Mech., 166, pp. 386-394; Siginer, D.A., Developments in Tube Flow of Complex Fluids (2015), Springer Inc New York; Siginer, D.A., Isothermal tube flow of non-linear viscoelastic fluids, part II: transversal field (2011) Int. J. Eng. Sci., 49 (6), pp. 443-465; Siginer, D.A., Letelier, M.F., Laminar flow of non-linear viscoelastic fluids in straight tubes of arbitrary contour, Int. J. Heat Mass Transf. 54 (2011) 2188-2202 updated by the editor for errors introduced in printing and reappeared in the same journal, Int. J. Heat Mass Transf., 55 (2012) 2731‐2745; Ericksen, J.L., Over determination of the speed in rectilinear motion of non-Newtonian fluids (1956) Quart. Appl. Math., 14, pp. 319-321; Green, A.E., Rivlin, R.S., Steady flow of non-Newtonian fluids through tubes (1956) Quart. Appl. Math., 14, pp. 299-308; Langlois, W.E., Rivlin, R.S., Slow steady-state flow of viscoelastic fluids through non-circular tubes (1963) Rend. Math., 22, pp. 169-185; Xue, S., Phan-Thien, N., Tanner, R.I., Numerical study of secondary flows of viscoelastic fluid in straight pipes by an implicit finite method (1995) J. Non-Newton. Fluid Mech., 59 (2-3), pp. 191-213; Oliver, D.R., Non-Newtonian heat transfer: an interesting effect observed in non-circular tubes (1969) Trans. Inst. Chem. Eng., 47, p. T18; Gao, S.X., Hartnett, J.P., Heat transfer behavior of Reiner-Rivlin fluids in rectangular ducts (1996) Int. J. Heat Mass Transf., 39, pp. 1317-1324; Siginer, D.A., Letelier, M.F., Heat transfer asymptote in laminar flow of non-linear viscoelastic fluids in straight non-circular tubes (2010) Int. J. Eng. Sci., 48, pp. 1544-1562; Letelier, M.F., Barrera, C., Siginer, D.A., Analytical solution of the Graetz problem for non-linear viscoelastic fluids (2017) Int. J. Therm. Sci., 111, pp. 369-378; Pinho, F.T., Oliveira, P.J., Analysis of forced convection in pipes and channels with the simplified Phan−Thien−Tanner fluid (2000) Int. J. Heat Mass Transf., 43, pp. 2273-2287; Letelier, M.F., Siginer, D.A., González, A., Elasto-viscoplastic fluid flow in tubes of arbitrary cross-section (2017) Appl. Math. Model., 46, pp. 572-580; Johnson, M., Segalman, D., A model for viscoelastic fluid behavior which allows non-affine deformation (1977) J. Non-Newton. Fluid Mech., 2, pp. 255-270; Letelier, M.F., Barrera, C., Siginer, D.A., On the physics of viscoplastic fluid flow in non-circular tubes (2017) Int. J. Non-Linear Mech, 88, pp. 1-10; Slater, L.J., Confluent Hypergeometric Functions (1960), pp. 503-515. , Cambridge University Press; López-Aguilar, J.E., Webster, M.F., Tamaddon-Jahromi, H.R., Numerical modelling of thixotropic and viscoelastoplastic materials in complex flow (2015) Rheol. Acta, 54, p. 307; Fraggedakis, D., Dimakopoulos, Y., Tsamopoulos, J., Yielding the yield-stress analysis: a study focused on the effects of elasticity on the settling of a single spherical particle in simple yield-stress fluids (2016) Soft matter, 12 (24), pp. 5378-5401

PY - 2018

Y1 - 2018

N2 - An analytical study of elastoviscoplastic fluid flow in tubes of non-circular cross section is presented. The constitutive structure of the fluid is described by a linear frame invariant combination of the Phan-Thien−Tanner model of viscoelastic fluids and the Bingham model of plastic fluids. Non-circular tube cross sections are modeled by the shape factor method a one-to-one mapping of the circular base contour into a wide spectrum family of arbitrary tube contours. Field variables are expanded into asymptotic series in terms of the elasticity measure, the Weissenberg number We, coupled with an asymptotic expansion in terms of the geometrical mapping parameter ε leading to a set of hierarchical momentum balance equations which are solved successively up to and including the third order in We when the secondary field appears for the first time. The computational algorithm developed is applied to the study of the non-rectilinear flow in tubes with triangular and square cross sections. We find that the presence of the yield stress dampens the intensity of the purely viscoelastic vortices, the higher the yield stress the lower the intensity of the vortices in the cross-section, and the further away the vortices are from the center of the cross section as compared to the purely viscoelastic vortices. The results also evidence that viscoelasticity increases the axial flow for given viscoplastic conditions and pressure drop, and consequently increases the rate of flow, a phenomenon that may find applications in optimizing material transportation. © 2017 Elsevier Inc.

AB - An analytical study of elastoviscoplastic fluid flow in tubes of non-circular cross section is presented. The constitutive structure of the fluid is described by a linear frame invariant combination of the Phan-Thien−Tanner model of viscoelastic fluids and the Bingham model of plastic fluids. Non-circular tube cross sections are modeled by the shape factor method a one-to-one mapping of the circular base contour into a wide spectrum family of arbitrary tube contours. Field variables are expanded into asymptotic series in terms of the elasticity measure, the Weissenberg number We, coupled with an asymptotic expansion in terms of the geometrical mapping parameter ε leading to a set of hierarchical momentum balance equations which are solved successively up to and including the third order in We when the secondary field appears for the first time. The computational algorithm developed is applied to the study of the non-rectilinear flow in tubes with triangular and square cross sections. We find that the presence of the yield stress dampens the intensity of the purely viscoelastic vortices, the higher the yield stress the lower the intensity of the vortices in the cross-section, and the further away the vortices are from the center of the cross section as compared to the purely viscoelastic vortices. The results also evidence that viscoelasticity increases the axial flow for given viscoplastic conditions and pressure drop, and consequently increases the rate of flow, a phenomenon that may find applications in optimizing material transportation. © 2017 Elsevier Inc.

KW - Elastoviscoplasticity

KW - Interplay of elasticity and plasticity

KW - Non-circular cross-section

KW - Transversal field

KW - Tube flow

KW - Elasticity

KW - Mapping

KW - Materials handling

KW - Non Newtonian flow

KW - Pipe flow

KW - Plasticity

KW - Tubes (components)

KW - Viscoelasticity

KW - Vortex flow

KW - Yield stress

KW - Computational algorithm

KW - Elasto-viscoplasticity

KW - Material transportation

KW - Momentum balance equations

KW - Square cross section

KW - Vis-coelastic fluids

KW - Flow of fluids

U2 - 10.1016/j.apm.2017.10.008

DO - 10.1016/j.apm.2017.10.008

M3 - Article

VL - 54

SP - 768

EP - 781

JO - Applied Mathematical Modelling

T2 - Applied Mathematical Modelling

JF - Applied Mathematical Modelling

SN - 0307-904X

ER -