Viscoplastic boundary layers
File(s)
Accepted version
Author(s)
Balmforth, NJ
Craster, RV
Hewitt, DR
Hormozi, S
Maleki, A
Type
Journal Article
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 plugged flow and a wall, or buffers for regions of
predominantly plastic deformation. (Oldroyd 1947,
Proc. Camb. Phil. Soc.
43, 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 2002,
J. Non-Newtonian
Fluid Mech.
102, 193 - 218); we rationalize this second scaling and provide an alternative
boundary-layer theory.
flows can develop narrow boundary layers that provide either surfaces of failure between
rigid plugs, the lubrication between plugged flow and a wall, or buffers for regions of
predominantly plastic deformation. (Oldroyd 1947,
Proc. Camb. Phil. Soc.
43, 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 2002,
J. Non-Newtonian
Fluid Mech.
102, 193 - 218); we rationalize this second scaling and provide an alternative
boundary-layer theory.
Date Issued
2017-01-26
Date Acceptance
2017-01-05
Citation
Journal of Fluid Mechanics, 2017, 813, pp.929-954
ISSN
1469-7645
Publisher
Cambridge University Press (CUP)
Start Page
929
End Page
954
Journal / Book Title
Journal of Fluid Mechanics
Volume
813
Copyright Statement
© 2017 Cambridge University Press
Subjects
Fluids & Plasmas
01 Mathematical Sciences
09 Engineering
Publication Status
Published