Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T23:19:14.001Z Has data issue: false hasContentIssue false
  • English
  • Français

Creeping flow of a Herschel–Bulkley fluid with pressure-dependent material moduli

Published online by Cambridge University Press:  11 July 2017

L. FUSI  and
F. ROSSO
L. FUSI
Affiliation:
Dipartimento di Matematica e Informatica “U. Dini”, Viale Morgagni, 67/a, 50134 Firenze, Italy emails: lorenzo.fusi@unifi.it, fabio.rosso@unifi.it
F. ROSSO
Affiliation:
Dipartimento di Matematica e Informatica “U. Dini”, Viale Morgagni, 67/a, 50134 Firenze, Italy emails: lorenzo.fusi@unifi.it, fabio.rosso@unifi.it
Get access Rights & Permissions [Opens in a new window]

Abstract

We model the axisymmetric unidirectional flow of a Herschel–Bulkley fluid with rheological parameters that depend linearly on pressure. Adopting an appropriate scaling, we formulate the mathematical problem in cylindrical geometry exploiting an integral formulation for the momentum equation in the unyielded part. We prove that, under suitable assumptions on the data of the problem, explicit solutions can be determined. In particular, we determine the position of the yield surface together with the pressure and velocity profiles. With the aid of some plots, we finally discuss the dependence of the solution on the physical parameters of the problem.

Keywords

Non-Newtonian fluidsHerschel–BulkleyPoiseuille flowBessel functionsnon-linear constitutive equations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 2 , April 2018 , pp. 352 - 368
Copyright
Copyright © Cambridge University Press 2017 

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

[1] Andrade, E. N. (1930) The viscosity of liquids. Nature 125, 309–310.CrossRefGoogle Scholar
[2] Barus, C. (1893) Isothermals, isopiestics and isometrics relative to viscosity. American J. Sci. 266, 87–96.CrossRefGoogle Scholar
[3] Bridgman, P. W. (1949) The Physics of High Pressure, Bell and Sons, Ltd., London.Google Scholar
[4] Casalini, R. & Bair, S. (2008) The inflection point in the pressure dependence of viscosity under high pressure: A comprehensive study of the temperature and pressure dependence of the viscosity of propylene carbonate. J. Chem. Phys. 128, 084511.CrossRefGoogle ScholarPubMed
[5] Fusi, L., Farina, A. & Rosso, F. (2014) Bingham flows with pressure-dependent rheological parameters. Int. J. Non-Linear Mech. 64, 33–38.CrossRefGoogle Scholar
[6] Fusi, L., Farina, A. & Rosso, F. (2014) On the mathematical paradoxes for the flow of a viscoplastic film down an inclined surface. Int. J. Non-Linear Mech. 58, 139–150.CrossRefGoogle Scholar
[7] Fusi, L., Farina, A., Rosso F. & Roscani, S. (2015) Pressure driven lubrication flow of a Bingham fluid in a channel: A novel approach. J. Non-Newtonian Fluid Mech. 221, 66–75.CrossRefGoogle Scholar
[8] Fusi, L., Farina, A. & Rosso, F. (2015) Ill posedness of Bingham-type models for the downhill flow of a thin film on an inclined plane. Q. Appl. Math. 73, 615–627.CrossRefGoogle Scholar
[9] Fusi, L., Farina, A. & Rosso, F. (2015) Planar squeeze flow of a bingham fluid. J. Non-Newtonian Fluid Mech. 225, 1–9.CrossRefGoogle Scholar
[10] Fusi, L. (2017) Non-isothermal flow of a Bingham fluid with pressure and temperature dependent viscosity. Meccanica, DOI: 10.1007/s11012-017-0655-8.CrossRefGoogle Scholar
[11] Fusi, L. (2017) Unsteady non-isothermal flow of a Bingham fluid with non constant material moduli at low Reynolds number. submitted to Acta Mechanica.CrossRefGoogle Scholar
[12] Harris, K. R. & Bair, S. (2007) Temperature and pressure dependence of the viscosity of diisodecyl phthalate at temperatures between (0 and 100) C and at pressures to 1 GPa. J. Chem. Eng. Data 52 (1), 272–278.CrossRefGoogle Scholar
[13] Hermoso, J., Martínez-Boza, F. & Gallegos, C. (2014) Combined effect of pressure and temperature on the viscous behaviour of all-oil drilling fluids. Oil Gas Sci. Technol.–Revue d'IFP Energies Nouvelles 69 (7), 1283–1296.CrossRefGoogle Scholar
[14] Paluch, M., Dendzik, Z. & Rzoska, S. J. (1999) Scaling of high-pressure viscosity data in low-molecular-weight glass-forming liquids. Phys. Rev. B 60 (5), 2979.CrossRefGoogle Scholar
[15] Stokes, G. G. (1845) On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids.. Trans. Camb. Phil. Soc. 8, 287–341.Google Scholar
[16] Rajagopal, K. R. & Saccomandi, G. (2006) Unsteady exact solution for flows of fluids with pressure-dependent viscosities. Math. Proc. R. Ir. Acad. 106A (2), 115–130.CrossRefGoogle Scholar
[17] Rajagopal, K. R., Saccomandi, G. & Vergori, L. (2009) On the Oberbeck–Boussinesq approximation for fluids with pressure dependent viscosities. Nonlinear Anal.: Real World Appl. 10 (2), 1139–1150.CrossRefGoogle Scholar
[18] Rajagopal, K. R., Saccomandi, G. & Vergori, L. (2012) Flow of fluids with pressure-and shear-dependent viscosity down an inclined plane. J. Fluid Mech. 706, 173–189.CrossRefGoogle Scholar
[19] Rajagopal, K. R. (2015) Remarks on the notion of “pressure”. Int. J. Non-Linear Mech. 71, 165–172.CrossRefGoogle Scholar
[20] Renardy, M. (2003) Parallel shear flows of fluids with a pressure-dependent viscosity. J. Non-Newtonian Fluid Mech. 114, 229–236.CrossRefGoogle Scholar
[21] Srinivasan, S. & Rajagopal, K. R. (2009) Study of a variant of Stokes' first and second problems for fluids with pressure dependent viscosities. Int. J. Eng. Sci. 47 (11–12), 1357–1366.CrossRefGoogle Scholar
[22] Vasudevaiah, M. & Rajagopal, K. R. (2005) On fully developed flows of fluids with a pressure dependent viscosity in a pipe. Appl. Math. 50 (4), 341–353.CrossRefGoogle Scholar