Bingham fluids: deformation and energy dissipation in triangular cross section tube flow

M. Letelier, C. Barrera, D. Siginer, J. Stockle, F. Godoy, C. Rosas

Research output: Contribution to journalArticle

Abstract

Flow of Bingham plastics through straight, long tubes with non-circular cross-sections is studied by means of an analytical method that allows to model a wide spectrum of tube geometries. Shear stress normal to equal velocity lines, velocity field and plug zones are explored, in particular in a tube with equilateral triangular cross-section for small values of the Bingham number Bi, and they are compared with corresponding numerical solutions. We show that a circular plug is present at the center of the triangular tube cross-section, consistent with numerical simulations as well as with previous results in the literature, if calculations up to and including first order in the shape perturbation parameter ϵ are taken into account. However with the inclusion of the second order terms in the algorithm this structure is no longer present and no plug zone is predicted for the same pressure drop. We find that in that case normal shear stresses are always greater than the yield stress of the fluid. As a result, the central region becomes a pseudo-plug since it presents small but non-vanishing relative deformations and does not move as a rigid core. The energy dissipation function for the Bingham fluid flow is written in terms of natural coordinates. Its distribution depends only on the normal shear stress at any point with Bingham number as a parameter. © 2017, Springer Science+Business Media B.V.
LanguageEnglish
Pages161-173
Number of pages13
JournalMeccanica
Volume53
Issue number1-2
DOIs
Publication statusPublished - 2018

Fingerprint

Pipe flow
plugs
Shear stress
Energy dissipation
dissipation
energy dissipation
shear stress
tubes
Fluids
fluids
cross sections
Pressure drop
Yield stress
Flow of fluids
pressure drop
Plastics
fluid flow
plastics
Geometry
velocity distribution

Keywords

  • Bingham
  • Non-circular
  • Tube flow
  • Viscoplastic
  • Bins
  • Deformation
  • Energy dissipation
  • Fluid dynamics
  • Perturbation techniques
  • Pipe flow
  • Shear flow
  • Shear stress
  • Tubes (components)
  • Velocity
  • Yield stress
  • Energy dissipation function
  • Non-circular cross-section
  • Perturbation parameters
  • Relative deformation
  • Triangular cross-sections
  • Flow of fluids

Cite this

Bingham fluids: deformation and energy dissipation in triangular cross section tube flow. / Letelier, M.; Barrera, C.; Siginer, D.; Stockle, J.; Godoy, F.; Rosas, C.

In: Meccanica, Vol. 53, No. 1-2, 2018, p. 161-173.

Research output: Contribution to journalArticle

Letelier, M, Barrera, C, Siginer, D, Stockle, J, Godoy, F & Rosas, C 2018, 'Bingham fluids: deformation and energy dissipation in triangular cross section tube flow' Meccanica, vol. 53, no. 1-2, pp. 161-173. https://doi.org/10.1007/s11012-017-0716-z
Letelier, M. ; Barrera, C. ; Siginer, D. ; Stockle, J. ; Godoy, F. ; Rosas, C. / Bingham fluids: deformation and energy dissipation in triangular cross section tube flow. In: Meccanica. 2018 ; Vol. 53, No. 1-2. pp. 161-173.
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note = "Export Date: 10 April 2018 CODEN: MECCB Correspondence Address: Letelier, M.; Universidad de Santiago de Chile, Alameda, Chile; email: mario.letelier@usach.cl References: Safronchik, A.I., Non-steady flows of a visco-plastic material between parallel walls (1959) J Appl Math Mech (PMM), 23, pp. 1314-1327; Safronchik, A.I., Rotation of a cylinder with a variable angular velocity in a visco-plastic medium, PMM (1959) J Appl Math Mech (PMM), 23, pp. 1504-1511; Safronchik, A.I., Unsteady flow of a visco-plastic material in a circular tube (1960) J Appl Math Mech (PMM), 24, pp. 200-207; Mosolov, P.P., Mjasnikov, V.P., Variational methods in the theory of viscous-plastic medium (1965) J Appl Math Mech (PMM), 29, pp. 468-492; Mosolov, P.P., Mjasnikov, V.P., On stagnant flow regions of a visco-plastic medium in pipes (1966) J Appl Math Mech (PMM), 30, p. 705719; Mosolov, P.P., Mjasnikov, V.P., On qualitative singularities of the flow of a visco-plastic medium in pipes (1967) J Appl Math Mech, 31, pp. 581-585; Huilgol, R.R., On kinetic conditions affecting the existence and non-existence of a moving yield surface in unsteady unidirectional flows of Bingham fluids (2004) J Non-Newton Fluid Mech, 123, pp. 215-221; Huilgol, R., On the description of the motion of the yield surface in unsteady shearing flows of a Bingham fluid as a jerk wave (2010) J Non-Newton Fluid Mech, 165, pp. 65-69; Glowinski, R., Sur l{\'e}coulement dun fluide de Bingham dans une conduite cylindrique (1974) J M{\'e}canique, 13, pp. 601-621; Sekimoto, K., An exact non-stationary solution of simple shear flow in a Bingham fluid (1991) J Non-Newton Fluid Mech, 39, pp. 107-113; Sekimoto, K., Motion of the yield surface in a Bingham fluid with a simple-shear geometry (1993) J Non-Newton Fluid Mech, 46, pp. 219-227; Duvaut, G., Lions, J.L., (1972) Les In{\'e}quations en m{\'e}chanique et en physique, , DuParis; 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; Fortin, M., (1972) Calcul num{\'e}rique des {\'e}coulements des fluids de Bingham et des fluides Newtoniens incompressibles par la m{\'e}thod des {\'e}l{\'e}ments finis, th{\'e}se, , Paris VI; Huilgol, R.R., Panizza, M.P., On the determination of the plug flow region in Bingham fluids through the application of variational inequalities (1995) J Non-Newton Fluid Mech, 58, pp. 207-217; Fortin, M., Glowinski, R., (1982) M{\'e}thodes de Lagrangian augment{\'e}. Applications {\'a} la r{\'e}solution num{\'e}rique de probl{\'e}mes aux limites, , M{\'e}thodes Math{\'e}matiques de lInformatique, Du; Glowinski, R., Le-Tallec, P., Augmented Lagrangian and operator-splitting methods in non-linear mechanics (1989) Stud Appl Math Soc Ind Appl Math; Walton, I., Bittleston, S., 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; Tang, G., Wang, S., Ye, P., Tao, W., Bingham fluid simulation with the incompressible lattice Boltzmann model (2011) J Non-Newton Fluid Mech, 166, pp. 145-151; Turan, O., Chakraborty, N., Poole, R., Laminar natural convection of Bingham fluids in a square enclosure with differentially heated side walls (2010) J Non-Newton Fluid Mech, 154, pp. 901-913; Akram, S., Nadeem, S., Hussain, A., Effects of heat and mass transfer on peristaltic flow of a Bingham fluid in the presence of inclined magnetic field and channel with different wave forms (2014) J Magn Magn Mater, 362, pp. 184-192; 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; Lipscomb, G., Denn, M., Flow of Bingham fluids in complex geometries (1984) J Non-Newton Fluid Mech, 14, pp. 337-346; Putz, A., Frigaard, I., Martinez, D.M., On the lubrication paradox and the use of regularisation methods for lubrication flows (2009) J Non-Newton Fluid Mech, 163, pp. 62-77; Letelier, M., Siginer, D., 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; Siginer, D., Letelier, M., 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; Barrera, C., Letelier, M., Siginer, D., Stockle, J., The Graetz problem in tubes of arbitrary cross section (2016) Acta Mech, 227, pp. 3239-3246; Siginer, D., Letelier, M., Laminar flow of non-linear viscoelastic fluids in straight tubes of arbitrary contour (2011) Int J Heat Mass Transf, 54, pp. 2188-2202; Letelier, M., Stockle, J., A shape-factor method for modelling parallel and axially-varying flow in tubes and channels of complex cross-section shapes (2011) Biomedical science, engineering and technology, INTECH, pp. 469-486. , In:, Croatia",
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TY - JOUR

T1 - Bingham fluids: deformation and energy dissipation in triangular cross section tube flow

AU - Letelier, M.

AU - Barrera, C.

AU - Siginer, D.

AU - Stockle, J.

AU - Godoy, F.

AU - Rosas, C.

N1 - Export Date: 10 April 2018 CODEN: MECCB Correspondence Address: Letelier, M.; Universidad de Santiago de Chile, Alameda, Chile; email: mario.letelier@usach.cl References: Safronchik, A.I., Non-steady flows of a visco-plastic material between parallel walls (1959) J Appl Math Mech (PMM), 23, pp. 1314-1327; Safronchik, A.I., Rotation of a cylinder with a variable angular velocity in a visco-plastic medium, PMM (1959) J Appl Math Mech (PMM), 23, pp. 1504-1511; Safronchik, A.I., Unsteady flow of a visco-plastic material in a circular tube (1960) J Appl Math Mech (PMM), 24, pp. 200-207; Mosolov, P.P., Mjasnikov, V.P., Variational methods in the theory of viscous-plastic medium (1965) J Appl Math Mech (PMM), 29, pp. 468-492; Mosolov, P.P., Mjasnikov, V.P., On stagnant flow regions of a visco-plastic medium in pipes (1966) J Appl Math Mech (PMM), 30, p. 705719; Mosolov, P.P., Mjasnikov, V.P., On qualitative singularities of the flow of a visco-plastic medium in pipes (1967) J Appl Math Mech, 31, pp. 581-585; Huilgol, R.R., On kinetic conditions affecting the existence and non-existence of a moving yield surface in unsteady unidirectional flows of Bingham fluids (2004) J Non-Newton Fluid Mech, 123, pp. 215-221; Huilgol, R., On the description of the motion of the yield surface in unsteady shearing flows of a Bingham fluid as a jerk wave (2010) J Non-Newton Fluid Mech, 165, pp. 65-69; Glowinski, R., Sur lécoulement dun fluide de Bingham dans une conduite cylindrique (1974) J Mécanique, 13, pp. 601-621; Sekimoto, K., An exact non-stationary solution of simple shear flow in a Bingham fluid (1991) J Non-Newton Fluid Mech, 39, pp. 107-113; Sekimoto, K., Motion of the yield surface in a Bingham fluid with a simple-shear geometry (1993) J Non-Newton Fluid Mech, 46, pp. 219-227; Duvaut, G., Lions, J.L., (1972) Les Inéquations en méchanique et en physique, , DuParis; 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; Fortin, M., (1972) Calcul numérique des écoulements des fluids de Bingham et des fluides Newtoniens incompressibles par la méthod des éléments finis, thése, , Paris VI; Huilgol, R.R., Panizza, M.P., On the determination of the plug flow region in Bingham fluids through the application of variational inequalities (1995) J Non-Newton Fluid Mech, 58, pp. 207-217; Fortin, M., Glowinski, R., (1982) Méthodes de Lagrangian augmenté. Applications á la résolution numérique de problémes aux limites, , Méthodes Mathématiques de lInformatique, Du; Glowinski, R., Le-Tallec, P., Augmented Lagrangian and operator-splitting methods in non-linear mechanics (1989) Stud Appl Math Soc Ind Appl Math; Walton, I., Bittleston, S., 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; Tang, G., Wang, S., Ye, P., Tao, W., Bingham fluid simulation with the incompressible lattice Boltzmann model (2011) J Non-Newton Fluid Mech, 166, pp. 145-151; Turan, O., Chakraborty, N., Poole, R., Laminar natural convection of Bingham fluids in a square enclosure with differentially heated side walls (2010) J Non-Newton Fluid Mech, 154, pp. 901-913; Akram, S., Nadeem, S., Hussain, A., Effects of heat and mass transfer on peristaltic flow of a Bingham fluid in the presence of inclined magnetic field and channel with different wave forms (2014) J Magn Magn Mater, 362, pp. 184-192; 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; Lipscomb, G., Denn, M., Flow of Bingham fluids in complex geometries (1984) J Non-Newton Fluid Mech, 14, pp. 337-346; Putz, A., Frigaard, I., Martinez, D.M., On the lubrication paradox and the use of regularisation methods for lubrication flows (2009) J Non-Newton Fluid Mech, 163, pp. 62-77; Letelier, M., Siginer, D., 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; Siginer, D., Letelier, M., 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; Barrera, C., Letelier, M., Siginer, D., Stockle, J., The Graetz problem in tubes of arbitrary cross section (2016) Acta Mech, 227, pp. 3239-3246; Siginer, D., Letelier, M., Laminar flow of non-linear viscoelastic fluids in straight tubes of arbitrary contour (2011) Int J Heat Mass Transf, 54, pp. 2188-2202; Letelier, M., Stockle, J., A shape-factor method for modelling parallel and axially-varying flow in tubes and channels of complex cross-section shapes (2011) Biomedical science, engineering and technology, INTECH, pp. 469-486. , In:, Croatia

PY - 2018

Y1 - 2018

N2 - Flow of Bingham plastics through straight, long tubes with non-circular cross-sections is studied by means of an analytical method that allows to model a wide spectrum of tube geometries. Shear stress normal to equal velocity lines, velocity field and plug zones are explored, in particular in a tube with equilateral triangular cross-section for small values of the Bingham number Bi, and they are compared with corresponding numerical solutions. We show that a circular plug is present at the center of the triangular tube cross-section, consistent with numerical simulations as well as with previous results in the literature, if calculations up to and including first order in the shape perturbation parameter ϵ are taken into account. However with the inclusion of the second order terms in the algorithm this structure is no longer present and no plug zone is predicted for the same pressure drop. We find that in that case normal shear stresses are always greater than the yield stress of the fluid. As a result, the central region becomes a pseudo-plug since it presents small but non-vanishing relative deformations and does not move as a rigid core. The energy dissipation function for the Bingham fluid flow is written in terms of natural coordinates. Its distribution depends only on the normal shear stress at any point with Bingham number as a parameter. © 2017, Springer Science+Business Media B.V.

AB - Flow of Bingham plastics through straight, long tubes with non-circular cross-sections is studied by means of an analytical method that allows to model a wide spectrum of tube geometries. Shear stress normal to equal velocity lines, velocity field and plug zones are explored, in particular in a tube with equilateral triangular cross-section for small values of the Bingham number Bi, and they are compared with corresponding numerical solutions. We show that a circular plug is present at the center of the triangular tube cross-section, consistent with numerical simulations as well as with previous results in the literature, if calculations up to and including first order in the shape perturbation parameter ϵ are taken into account. However with the inclusion of the second order terms in the algorithm this structure is no longer present and no plug zone is predicted for the same pressure drop. We find that in that case normal shear stresses are always greater than the yield stress of the fluid. As a result, the central region becomes a pseudo-plug since it presents small but non-vanishing relative deformations and does not move as a rigid core. The energy dissipation function for the Bingham fluid flow is written in terms of natural coordinates. Its distribution depends only on the normal shear stress at any point with Bingham number as a parameter. © 2017, Springer Science+Business Media B.V.

KW - Bingham

KW - Non-circular

KW - Tube flow

KW - Viscoplastic

KW - Bins

KW - Deformation

KW - Energy dissipation

KW - Fluid dynamics

KW - Perturbation techniques

KW - Pipe flow

KW - Shear flow

KW - Shear stress

KW - Tubes (components)

KW - Velocity

KW - Yield stress

KW - Energy dissipation function

KW - Non-circular cross-section

KW - Perturbation parameters

KW - Relative deformation

KW - Triangular cross-sections

KW - Flow of fluids

U2 - 10.1007/s11012-017-0716-z

DO - 10.1007/s11012-017-0716-z

M3 - Article

VL - 53

SP - 161

EP - 173

JO - Meccanica

T2 - Meccanica

JF - Meccanica

SN - 0025-6455

IS - 1-2

ER -