Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-22T12:32:22.262Z Has data issue: false hasContentIssue false
  • English
  • Français

On the stability of Poiseuille flow of a Bingham fluid

Published online by Cambridge University Press:  26 April 2006

I. A. Frigaard ,
S. D. Howison  and
I. J. Sobey
I. A. Frigaard
Affiliation:
Industriemathematik, Institut für Mathematik, Johannes Kepler Universität, A-4040, Linz, Austria
S. D. Howison
Affiliation:
Mathematical Institute, University of Oxford, 24–29, St. Giles, Oxford, OX1 3LB, UK
I. J. Sobey
Affiliation:
Oxford University Computing Laboratory, 8–11, Keble Road, Oxford, OX1 3QD, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability to linearized two-dimensional disturbances of plane Poiseuille flow of a Bingham fluid is considered. Bingham fluids exhibit a yield stress in addition to a plastic viscosity and this description is typically applied to drilling muds. A non-zero yield stress results in an additional parameter, a Bingham number, and it is found that the minimum Reynolds number for linear instability increases almost linearly with increasing Bingham number.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 263 , 25 March 1994 , pp. 133 - 150
Copyright
© 1994 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

Beris, A. N., Tsamopoulos, J. A., Armstrong, R. C. & Brown, R. A. 1985 Creeping motion of a sphere through a Bingham plastic. J. Fluid Mech. 158, 219–244.Google Scholar
Comparini, E. 1992 A one-dimensional Bingham flow. J. Math. Anal. Appl. 169, 127–139.Google Scholar
Davey, A. 1977 On the numerical solution of difficult eigenvalue problems. J. Comput. Phys. 24, 331–338.Google Scholar
Davies, S. J. & White, C. M. 1928 An experimental study of the flow of water pipes of rectangular section. Proc. R. Soc. Lond. A 119, 92–107.Google Scholar
Dowell, Schlumberger 1985 Rheology and flow calculations. In Cementing technology, chap. 4. Dowell Schlumberger.
Frigaard, I. A. 1990 The stability of parallel flow of a generalized Bingham fluid. MSc thesis, Oxford University.
Hanks, R. W. 1963 The laminar-turbulent transition for fluids with a yield stress. AIChE J. 9, 306–309.Google Scholar
Hanks, R. W. & Dadia, B. H. 1971 Theoretical analysis of the turbulent flow of non-Newtonian slurries in pipes. AIChE J. 17, 554–557.Google Scholar
Hanks, R. W. & Pratt, D. R. 1967 On the flow of Bingham plastic slurries in pipes and between parallel plates. Soc. Petrol. Engrs J. 7, 342–346.Google Scholar
Herbert, T. 1976 Periodic secondary motions in a plane channel. In Proc. 5th Intl Conf. Numerical Methods in Fluid Dynamics (ed. A. I. Van de Vooren & P. J. Zandbergen). Lecture Notes in Physics, vol. 59, pp. 235–240. Springer.
Oldroyd, J. G. 1947 Two-dimensional plastic flow of a Bingham solid. Proc. Camb. Phil. Soc. 43, 383–395.Google Scholar
Orszag, S. A. 1971 Accurate solution of the Orr—Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.Google Scholar
Orszag, S. A. & Kells, L. C. 1980 Transition to turbulence in plane Poiseuille and plane Couette flow. J. Fluid Mech. 96, 159–205.Google Scholar
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall bounded shear flows. J. Fluid Mech. 128, 347–385.Google Scholar
Patel, V. C. & Head, M. R. 1969 Some observations in skin friction and velocity profiles in fully developed pipe and channel flows. J. Fluid Mech. 38, 181–201.Google Scholar
Scott, M. R. 1973 An initial value method for the eigenvalue problem for systems of ODE's. J. Comput. Phys. 12, 334–347.Google Scholar
Sloan, D. M. 1977 Eigenfunctions of systems of linear ODE's with separated boundary conditions using Riccati transformations. J. Comput. Phys. 24, 320–330.Google Scholar
Sloan, D. M. & Wilks, G. 1976 Riccati transformations for eigenvalues of systems of linear ODE's with separated boundary conditions. J. Inst. Maths Applics. 18, 117–127.Google Scholar
Squire, H. B. 1933 On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls. Proc. R. Soc. Lond. A 142, 621–628.Google Scholar
Thomas, D. G. 1960 Heat and momentum transport characteristics of non-Newtonian aqueous thorium oxide suspensions. AIChE J. 6, 631–639.Google Scholar
Walton, I. C. & Bittleston, S. H. 1991 The axial flow of a Bingham plastic in a narrow annulus. J. Fluid Mech. 222, 39–60.Google Scholar
Wilks, G. & Sloan, D. M. 1976 Invariant imbedding, Riccati transformation and eigenvalue problems. J. Inst. Maths Applics. 18, 99–116.Google Scholar
Zahn, J. P., Toomre, J. Spiegel, E. A. & Gough, D. O. 1974 Nonlinear cellular motions in Poiseuille channel flow. J. Fluid Mech. 64, 319–345.Google Scholar