Skip to main content Accessibility help
×
Home
Hostname: page-component-59df476f6b-x84v4 Total loading time: 0.264 Render date: 2021-05-18T18:41:37.901Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

Viscoplastic boundary layers

Published online by Cambridge University Press:  26 January 2017

N. J. Balmforth
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
R. V. Craster
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
D. R. Hewitt
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
S. Hormozi
Affiliation:
Department of Mechanical Engineering, Ohio University, Athens, OH 45701-2979, USA
A. Maleki
Affiliation:
Department of Mechanical Engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada
Corresponding
E-mail address:

Abstract

In the limit of a large yield stress, or equivalently at the initiation of motion, viscoplastic flows can develop narrow boundary layers that provide either surfaces of failure between rigid plugs, the lubrication between a plugged flow and a wall or buffers for regions of predominantly plastic deformation. Oldroyd (Proc. Camb. Phil. Soc., vol. 43, 1947, pp. 383–395) presented the first theoretical discussion of these viscoplastic boundary layers, offering an asymptotic reduction of the governing equations and a discussion of some model flow problems. However, the complicated nonlinear form of Oldroyd’s boundary-layer equations has evidently precluded further discussion of them. In the current paper, we revisit Oldroyd’s viscoplastic boundary-layer analysis and his canonical examples of a jet-like intrusion and flow past a thin plate. We also consider flow down channels with either sudden expansions or wavy walls. In all these examples, we verify that viscoplastic boundary layers form as envisioned by Oldroyd. For each example, we extract the dependence of the boundary-layer thickness and flow profiles on the dimensionless yield-stress parameter (Bingham number). We find that, while Oldroyd’s boundary-layer theory applies to free viscoplastic shear layers, it does not apply when the boundary layer is adjacent to a wall, as has been observed previously for two-dimensional flow around circular obstructions. Instead, the boundary-layer thickness scales in a different fashion with the Bingham number, as suggested by classical solutions for plane-parallel flows, lubrication theory and, for flow around a plate, by Piau (J. Non-Newtonian Fluid Mech., vol. 102, 2002, pp. 193–218); we rationalize this second scaling and provide an alternative boundary-layer theory.

Type
Papers
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below.

References

Ancey, C. 2007 Plasticity and geophysical flows: a review. J. Non-Newtonian Fluid Mech. 142, 435.CrossRefGoogle Scholar
Balmforth, N. J. 2017 Viscoplastic asymptotics and other techniques. In Viscoplastic Fluids: From Theory to Application, CISM. Springer.Google Scholar
Balmforth, N. J., Frigaard, I. A. & Ovarlez, G. 2014 Yielding to stress: recent developments in viscoplastic fluid mechanics. Annu. Rev. Fluid Mech. 46, 121146.CrossRefGoogle Scholar
Balmforth, N. J. & Hewitt, I. J. 2013 Viscoplastic sheets and threads. J. Non-Newtonian Fluid Mech. 193, 2842.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bird, R. B., Dai, G. C. & Yarusso, B. J. 1983 The rheology and flow of viscoplastic materials. Rev. Chem. Engng 1 (1), 170.CrossRefGoogle Scholar
Boujlel, J., Maillard, M., Lindner, A., Ovarlez, G., Chateau, X. & Coussot, P. 2012 Boundary layer in pastes: displacement of a long object through a yield stress fluid. J. Rheol. 56, 10831108.CrossRefGoogle Scholar
Chevalier, T., Rodts, S., Chateau, X., Boujlel, J., Maillard, M. & Coussot, P. 2013 Boundary layer (shear-band) in frustrated viscoplastic flows. Europhys. Lett. 102, 48002.CrossRefGoogle Scholar
Dean, E. J., Glowinski, R. & Guidoboni, G. 2007 On the numerical simulation of Bingham visco-plastic flow: old and new results. J. Non-Newtonian Fluid Mech. 142, 3662.CrossRefGoogle Scholar
Green, A. P. 1955 On unsymmetrical extrusion in plane strain. J. Mech. Phys. Solids 3, 189192.CrossRefGoogle Scholar
Hill, R. 1950 The Mathematical Theory of Plasticity. Oxford University Press.Google Scholar
Johnson, W. 1956 Extrusion through square dies of large reduction. J. Mech. Phys. Solids 4, 191198.CrossRefGoogle Scholar
Johnson, W., Sowerby, R. & Venter, R. D. 1982 Plane-strain Slip-line Fields for Metal-deformation Processes. Pergamon.Google Scholar
Mitsoulis, E. 2007 Flows of viscoplastic materials: models and computations. Rheol. Rev. 2007, 135178.Google Scholar
Oldroyd, J. G. 1947 Two-dimensional plastic flow of a Bingham solid: a plastic boundary-layer theory for slow motion. Proc. Camb. Phil. Soc. 43, 383395.CrossRefGoogle Scholar
Piau, J.-M. 2002 Viscoplastic boundary layer. J. Non-Newtonian Fluid Mech. 102, 193218.CrossRefGoogle Scholar
Prager, W. & Hodge, P. G. 1951 Theory of Perfectly Plastic Solids. Wiley.Google Scholar
Roustaei, A., Gosselin, A. & Frigaard, I. A. 2014 Residual drilling mud during conditioning of uneven boreholes in primary cementing. Part 1. Rheology and geometry effects in non-inertial flows. J. Non-Newtonian Fluid Mech. 220, 8798.CrossRefGoogle Scholar
Saramito, P. 2015 Efficient C++ Finite Element Computing with Rheolef. CNRS-CCSD. http://cel.archives-ouvertes.fr/cel-00573970.Google Scholar
Tokpavi, D. L., Magnin, A. & Jay, P. 2008 Very slow flow of Bingham viscoplastic fluid around a circular cylinder. J. Non-Newtonian Fluid Mech. 154, 6576.CrossRefGoogle Scholar
Vinay, G., Wachs, A. & Agassant, J. 2005 Numerical simulation of non-isothermal viscoplastic waxy crude oil flows. J. Non-Newtonian Fluid Mech. 128, 144162.CrossRefGoogle Scholar

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Viscoplastic boundary layers
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Viscoplastic boundary layers
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Viscoplastic boundary layers
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *