J. Fluid Mech.
(2022),
v
ol
.
942, A30, doi:10.1017/jfm.2022.384
A layer of yield-stress material on a flat plate
that
moves
sud
denly
Edward M. Hinton
1
, Jesse F. Collis
1
,
2
and John E. Sader
1
,
2
,
†
1
School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
2
ARC Centre of Excellence in Exciton Science, The University of Melbourne, Victoria 3010, Australia
(Received 21 October 2021; revised 28 February 2022; accepted 14 April 2022)
The flow of an unbounded Newtonian fluid above a flat solid plate, set into motion
suddenly with uniform velocity, is a canonical problem investigated by Stokes (
On
the
effect
of
the
inter
nal
fric
tion
of
flu
ids
on
the
motion
of
pen
du
lums,
Trans.
Camb.
Phil.
Soc.
,
vol. 9, 1851) and Rayleigh (
Lond. Edinb. Dubl. Philos. Mag. J. Sci.
, vol. 21, issue 126, 1911,
pp. 697–711). We tackle the same problem but with the fluid replaced by a yield-stress
material of finite height with a free surface; the latter ensures both yielded (fluid-like) and
rigid behaviour. The interface between the yielded and rigid regions – the so-called ‘yield
surface’ – always emerges from the plate at a rate faster than that for momentum diffusion.
The rigid region, adjacent to the free surface, is accelerated by the constant yield stress
acting at this yield surface. As the velocity of this region increases following
start
-
up, its
acceleration climaxes and subsequently diminishes. In this latter period, the thickness of
the rigid region increases owing to relaxation of the velocity gradients and the stress. The
yield surface eventually collides with the plate in finite time, after which the entire material
moves in concert with the plate as a rigid body. Analytical and numerical results are
presented that can form the basis for practical applications. This includes the development
of a ‘rheological microscope’ to directly detect and measure the yield stress. The analysis
focuses on exploring the flow physics of a Bingham material. This is shown to be similar
to that of a general Herschel–Bulkley material with shear-thinning or shear-thickening
rheology.
Key words:
plastic materials, rheology
1. Introduction
In
the
Rayleigh–Stokes problem, an initially stationary and unbounded Newtonian fluid
lying above a horizontal plate is set into motion by the
sud
den in-plane motion of the
†
Email address for correspondence:
jsader@unimelb.edu.au
© The Author(s), 2022. Published by Cambridge University Press. This is an Open Access article,
distributed under the terms of the Creative Commons Attribution licence (
https://creativecommons.
org/licenses/by/4.0/
), which permits unrestricted re-use, distribution, and reproduction in any medium,
provided the original work is properly cited.
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E.M. Hinton, J.F. Collis and J.E. Sader
Newtonian
Newtonian
τ
= 0
τ
= 0
Bingham
u
(
y
,
t
)
t
y
x
U
0
U
0
U
0
u
u
H
Plug
Yielded
(
a
)(
b
)(
c
)
Figure 1.
The
Rayleigh–Stokes problem for a Newtonian fluid and a Bingham material. The solid plate under
the material moves suddenly from rest, and in its plane, with constant
veloc
ity
U
0
ˆ
x
. Curves show the evolution
of the velocity field parallel to the plate, as a function of
time
t
.(
a
) A semi-infinite region of Newtonian fluid.
(
b
) A finite region of Newtonian fluid with a free surface. (
c
) A finite region of Bingham material with a free
surface.
plate. This problem was first proposed and solved by Stokes (
1851
) and later expounded
by Rayleigh (
1911
); it is variably called ‘Stokes’ first problem’, the ‘Rayleigh problem’ or
the ‘
Rayleigh–Stokes problem’ in the literature; we use the latter. For plate motion with
constant velocity in its
plane
U
0
ˆ
x
, where
ˆ
x
is the basis vector in the Cartesian
x
-direction,
the resulting velocity field of the fluid,
u
(
y
,
t
)
ˆ
x
, satisfies the momentum
equation
ρ
∂
u
∂
t
=
∂τ
∂
y
,
(1.1)
where
y
is the Cartesian coordinate normal to the plate,
t
is time,
τ
is the
xy
(shear)
component of the Cauchy stress tensor
,and
ρ
is the density. For the semi-infinite
(unbounded) region of a Newtonian fluid, the velocity disturbance propagates into the
fluid in a self-similar fashion, moving away from the plate with a
√
t
time dependence
(
figure 1
a
):
u
≡
u
N
,
∞
=
U
0
erfc
y
2
√
ν
t
,
(1.2)
where the first subscript,
N
, denotes the flow of a Newtonian fluid
, while the second
subscript specifies the spatial extent of the fluid normal to the plate. Analytical solution
for a finite (bounded) region of Newtonian fluid with a free surface is also
obtained
read
ily
(Liu
2008
); the free surface is initially stationary and
accel
er
ates
grad
u
ally until its velocity
converges with that of the plate (
figure 1
b
).
While these simple solutions provide significant insight into momentum diffusion
through viscous boundary layers, many materials of environmental, industrial and
biological importance do not obey a Newtonian constitutive law. An important class,
known as yield-stress (or viscoplastic) materials, are characterised by constitutive laws
where a sufficiently large stress must be applied to the material before it can flow. A
simple model introduced by Bingham (
1916
) states that the material is rigid below a
critical state of stress, above which the material behaves as a Newtonian fluid. For a
general unidirectional plane flow of velocity,
u
(
y
,
t
)
ˆ
x
, these ‘Bingham’ materials obey
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Yield-stress material on a plate that moves suddenly
the constitutive equation (Bingham
1922
)
∂
u
∂
y
=
0for
|
τ
|
<τ
Y
,
(1.3
a
)
τ
=
μ
∂
u
∂
y
+
τ
Y
sgn
∂
u
∂
y
for
|
τ
|≥
τ
Y
,
(1.3
b
)
where
τ
Y
is the shear yield stress and
μ
is the shear viscosity. This seemingly simple
constitutive model gives rise to rich flow behaviour and applications not possible with
Newtonian fluids.
A key feature of a viscoplastic flow is the presence of a ‘yield surface’, i.e. a surface
that separates a yielded (fluid-like)
region and a rigid region of the material (Balmforth,
Frigaard & Ovarlez
2014
); any particular flow may contain a number of yield surfaces.
These surfaces are often time-dependent and can be created or destroyed depending on the
flow in question (Huilgol
2004
).
In many industrial applications, viscoplastic boundary layers play an important role. For
example, Peixinho
et al.
(
2005
) studied the flow of both Newtonian fluids and yield-stress
materials through pipes. They found that the presence of a yield stress stabilises the
flow,
thus higher Reynolds numbers can be achieved before the onset of turbulence.
Moreover, it has been reported that Poiseuille flow of a viscoplastic material comes to rest
in finite time when the forcing is removed (Huilgol, Mena & Piau
2002
; Chatzimina
et al.
2005
). Transient viscoplastic and bi-viscous flows in pipes, with a time-dependent pressure
gradient or wall velocity, have also been well studied, with detailed mathematical solutions
developed for the evolution of the yield surface (Safronchik
1959
a
,
1960
; Comparini
1992
;
Burgess & Wilson
1997
). The flow exterior to a cylinder with general applied angular
velocity has also been considered (Safronchik
1959
b
). These flows do not have a free
surface, which, as we show here, exerts a leading-order effect on the flow physics.
Applications also arise in natural processes, where a Bingham material has been
employed to model the effect of shallow water waves on a muddy sea-bed (Mei & Liu
1987
). The key finding from these works is that the viscoplastic sea-bed acts as a damper
to the waves (Ng
2000
; Chan & Liu
2009
; Roustaei, Gosselin & Frigaard
2015
). This
problem forms a complementary view to
Rayleigh–Stokes type problems because the free
surface of the material, rather than its solid base, is disturbed.
For the semi-infinite (unbounded)
Rayleigh–Stokes problem, replacing the Newtonian
fluid with a Bingham material does not alter the velocity field (Pascal
1989
), which is
identical to (
1.2
). The key difference is that the shear stress in a Bingham material is offset
from its Newtonian counterpart by the yield stress, throughout the flow domain, i.e. the
magnitude of the shear stress is held above the yield stress,
τ
=
μ
∂
u
N
,
∞
∂
y
−
τ
Y
,
(1.4)
for which
∂
u
N
,
∞
/∂
y
<
0; see (
1.2
). This means that the shear stress at the solid plate
approaches a constant
value
−
τ
Y
as
t
→∞
, unlike Newtonian flow where the shear
stress approaches zero. That is, a greater shear stress must be applied to the plate to
move an unbounded region of Bingham material relative to its Newtonian counterpart.
Equation (
1.4
) shows that the Bingham material is always yielded.
Hence it never contains
rigid regions
or a yield surface
– a key characteristic that, as discussed above, leads to much
of the interesting behaviour associated with yield-stress materials. The closely related
‘shear-driven’
Rayleigh–Stokes problem for an unbounded region of Bingham material,
i.e. constant shear stress applied by the plate, also admits an analytical solution for the
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E.M. Hinton, J.F. Collis and J.E. Sader
velocity field identical to its Newtonian counterpart (Duffy, Pritchard & Wilson
2014
)
,
pro
vided
that the applied stress at the plate exceeds the yield stress (see also Sekimoto
1991
). No motion occurs if the applied stress is less than the material’s yield stress.
Non-Newtonian versions of Stokes’ second problem, where the plate oscillates
periodically in time (rather than moving abruptly with constant velocity), have drawn more
attention than the first problem. Balmforth, Forterre & Pouliquen (
2009
) considered the
role of yield stress in the second problem for a Bingham material of finite height, bounded
by a free surface. Their numerical results showed that isolated regions of yielded material,
separated by rigid regions,
move
peri
od
i
cally from the plate to the free surface. The
second problem, with power-law and thixotropic materials in semi-infinite and unbounded
domains, has also been explored (Pritchard, McArdle & Wilson
2011
; McArdle, Pritchard
& Wilson
2012
). For shear-thickening materials, these studies show that the velocity
vanishes beyond a finite distance from the plate
, in contrast to the Newtonian case. Stokes’
second problem differs from the first problem because the spatial extent of momentum
diffusion in the former is controlled by the plate oscillation
time
scale. While the Laplace
transform links these two problems for a Newtonian fluid (Stokes
1851
), flow of a Bingham
material is an intrinsically nonlinear process that alters the flow physics (and analyses) of
the first and second problems.
We consider the
Rayleigh–Stokes problem for a Bingham material of finite height with
a free surface, whose behaviour differs considerably
from that of a semi-infinite region
or a layer bounded by no-slip boundaries (discussed above). The velocity gradient is
sin
gu
lar
ini
tially at the solid plate due to its sudden motion, ensuring that the yield surface
propagates from the plate into the material immediately following
start
-
up. The shear stress
at the free surface vanishes, ensuring that the material is rigid in its vicinity. Because the
material is always rigid near the free surface and
yields
ini
tially near the plate, a yield
surface exists
that can evolve in time. Importantly, the
Rayleigh–Stokes problem for a
Bingham material of finite height is yet to be reported and its fluid physics explored. This is
the focus of the present study, which includes both numerical and analytical solutions. We
also show how the results generalise to other related materials, in particular for a general
Herschel–Bulkley constitutive law that exhibits either shear-thinning or shear-thickening
post-yield behaviour.
The paper is structured as follows: The theoretical foundation for the
Rayleigh–Stokes
problem involving a Bingham material of finite height with a free surface is formulated
in §
2
,
and illustrated in
figure 1
(
c
). We non-dimensionalise the problem, which identifies
the Bingham
num
ber
B
, defining the magnitude of the yield stress relative to the viscous
shear stress generated by the plate. Numerical results as a function of Bingham number
arereportedin§
2.1
. The physical mechanisms driving the flow, reported in §
2.1
,are
explored in §
3
. This begins by identifying the general properties of the rigid and yield
regions of the material in §§
3.1
and
3.2
, respectively. An upper bound for the ‘collision
time’, at which the entire material becomes rigid, is derived in §
3.3
.In§
3.4
, we show that
at early time, the yield surface does not travel far from the plate and the material behaves
as if it were unbounded; the Newtonian solution for the velocity field applies. In §
3.5
,
the Newtonian solution is deployed to analyse the problem for small
B
. This describes the
evolution of the yield surface and the collision time. Immediately prior to the collision
time, the material is almost entirely rigid. In §
3.6
, we use this property to prove that the
velocity field is self-similar in this regime. The dynamics at large Bingham number
is
explored in §
3.7
, for which a rescaling reduces the motion to a universal system defining
an unusual type of Stefan problem. A comparison of the dynamics for small and large
B
is reported in §
3.8
, and approximate formulas are given in §
3.9
that may be useful
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Yield-stress material on a plate that moves suddenly
in practice. In §
4
, we propose a rheometer
that uses the dynamics of the free surface to
study the existence of the yield stress and measure its value. Each result in the paper can
be extended to more complicated viscoplastic constitutive laws. Throughout, we highlight
where the analysis for a Bingham material applies either to any viscoplastic material or to a
general Herschel–Bulkley material. The mathematical details are relegated to
Appendix A
,
where we itemize the generalisation of each section of the main text. Importantly, we find
that the flow physics in all cases is dominated by the yield stress. Concluding remarks are
provided in §
5
, with details of the numerical methods used in this study given
in
Appendix
B
.
2. Theoretical formulation
We consider a Bingham material that is initially at rest, bounded by a solid plate at
y
=
0
and a free surface at
y
=
H
;see
figure 1
(
c
). The solid plate is suddenly set into motion
at
time
t
=
0, with constant
veloc
ity
U
0
ˆ
x
. This motion generates a unidirectional and
incompressible flow of constant density and velocity
u
(
y
,
t
)
ˆ
x
. The governing equations
for the resulting flow are provided by (
1.1
)and(
1.3
), with initial and boundary
con
di
tions
u
(
y
,
0
)
=
0
,
u
(
0
,
t
)
=
U
0
,τ(
H
,
t
)
=
0
,
(2.1
a
–
c
)
corresponding to no initial flow, no-slip at the solid plate
, and no shear stress at the free
surface, respectively.
The
y
-coordinate is scaled by the height of the material,
H
;
time
t
is non-dimensionalised
by the
time
scale for momentum diffusion,
H
2
/ν
; the shear
stress
τ
is scaled by
μ
U
0
/
H
for consistency with the chosen
time
scale; and the material
veloc
ity
u
is scaled by that of
the plate,
U
0
. In their corresponding dimensionless forms, (
1.1
)and(
1.3
) become
∂
u
∂
t
=
∂τ
∂
y
(2.2)
and
∂
u
∂
y
=
0for
|
τ
|
<
B
,
(2.3
a
)
τ
=
∂
u
∂
y
+
B
sgn
∂
u
∂
y
for
|
τ
|≥
B
,
(2.3
b
)
where the Bingham number is
B
≡
τ
Y
H
μ
U
0
,
(2.4)
defining the ratio of the yield to viscous stresses. The initial and boundary conditions in
(
2.1
a
–
c
)
become
u
(
y
,
0
)
=
0
,
u
(
0
,
t
)
=
1
,τ(
1
,
t
)
=
0
.
(2.5
a
–
c
)
All variables
will henceforth refer to their dimensionless quantities, unless
indi
cated
oth
er
wise.
While our analysis focuses on a Bingham material, it is straightforward to extend each
of the results to a general Herschel–Bulkley material and in some cases to any viscoplastic
material; the details of this are given in
Appendix A
. The flows of all such materials are
similar because they share the same key ingredient: a yield stress.
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E.M. Hinton, J.F. Collis and J.E. Sader
1.0
0.5
0
1.0
0.5
0
1.0
0.5
0
1.0
0.5
0
1.0
0.5
0
1.0
0.5
0
0.2
0.4
0.6
0.2
0.4
0.6
0.5
1.0
1.5
0.5
1.0
1.5
0.02
0.04
0.02
0.04
0.8
–20
–40
–60
–80
–2
–4
–6
–8
–2
–4
–6
0.6
0.4
0.2
0
0.8
0.6
0.4
0.2
0
0.8
0.6
0.4
0.2
0
y
y
y
tt
u
τ
u
τ
u
τ
B
=20
B
=20
B
=1
B
=1
B
=1/20
B
=1/20
(
a
)(
b
)
(
c
)(
d
)
(
e
)(
f
)
Figure 2. Numerical solutions of (
2.2
)–(
2.5
a
–
c
)forthe
Rayleigh–Stokes problem involving a Bingham
material, for
B
=
20
,
1
,
1
/
20. Colour maps show the velocity field (
a
,
c
,
e
) and shear stress (
b
,
d
,
f
). The yield
sur
face
y
=
Y
(
t
)
is denoted by a red solid curve.
2.1.
Numerical solutions
We commence our discussion of the
Rayleigh–Stokes problem for a layer of Bingham
material (
figure 1
c
) by presenting numerical solutions of the governing equations spanning
the small, intermediate and large Bingham number regimes. This is achieved by
solv
ing
numer
i
cally (
2.2
)–(
2.5
a
–
c
) directly for the shear
stress
τ
through
out the
domain 0
≤
y
≤
1, using a semi-implicit method. Acceleration of the material, and hence its velocity
field
u
, then follow from (
2.3
). The details of the numerical method, including its extension to
a general Herschel–Bulkley material, are given in §
B.1
of
Appendix B
.
Results for the
veloc
ity
u
and shear
stress
τ
of the material for
B
=
20
,
1
,
1
/
20 are
shown in
figure 2
, in the form of space–time colour maps. We denote the position of the
yield surface, where
|
τ
|=
B
,by
y
=
Y
(
t
)
; the material is yielded within 0
≤
y
≤
Y
(
t
)
and
rigid for
Y
(
t
)<
y
≤
1. The yield surfaces in
figure 2
are indicated by the red curves.
In all cases, we observe that the entire layer of Bingham material comes to move rigidly
with the solid plate in finite time, denoted
t
=
t
f
(>
0
)
– a general feature for
B
>
0.
Numerical results for this ‘collision time’ are given in
figure 3
(
a
), which corresponds to
collision of the yield surface with the plate; see
figure 2
. This behaviour differs from
the singular limit of a Newtonian fluid, i.e.
B
=
0(
figure 1
b
), where the fluid never moves
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Yield-stress material on a plate that moves suddenly
0
0.2
0.4
0.6
0.8
1.0
10
–2
10
–1
10
0
Numerical
10
1
t
f
∼
1/
B
t
f
∼
4
π
–2
log(
π
/
B
)
Numerical
Y
max
∼
1 – 0.54
B
Y
max
∼
1.23
B
–1/2
B
10
–2
10
–1
10
0
10
1
B
10
–1
t
f
Y
max
10
0
(
b
)
(
a
)
Figure 3. (
a
) Collision
time
t
=
t
f
for a Bingham material to move entirely as a rigid body with the plate
veloc
ity
u
=
1
.(
b
)
Max
i
mum height of the yield surface,
Y
max
, as a function of the Bingham
num
ber
B
.
Numerical results as per §
2.1
.
rigidly, but rather asymptotically approaches rigid body motion as time evolves. Moreover,
the finite collision time required to reach rigid body motion throughout the Bingham
material decreases with increasing
B
;see
figure 3
(
a
). The spatial extent of the yield surface
also decreases as
B
increases, i.e. less of the material yields before complete rigid body
motion; cf.
figures 2
(
a
)and
2
(
e
),
quan
ti
fied in
figure 3
(
b
).
These observations of the material can be understood by noting that an increase in
B
can be achieved by increasing the yield stress while holding all other parameters constant.
For very high yield stress, virtually none of the material yields before moving rigidly and
almost instantly with the plate, i.e. the material behaves as a rigid solid block. In contrast,
for very small yield stress, nearly all the material yields almost instantly after
start
-
up, i.e.
it
behaves
ini
tially as a Newtonian fluid. As the material accelerates, the shear stress then
decreases
, with the material eventually recovering its solid-like behaviour prior to
start
-
up.
It is clear from
figure 2
that the dynamics of the yield surface
con
trols the material
motion, and
the
dynam
ics
dif
fers for small and large
B
. The physical mechanisms and
details of this dependency
– features of the yield curve and its effect on the overall material
dynamics
– are investigated in the following sections.
3. Physical mechanisms
In this section, we explore the physical mechanisms driving the dynamics of a layer of
Bingham material whose lower surface is suddenly set into motion. This includes using
an alternative analysis to that above, which enables direct calculation of both the velocity
field and yield surface; in §
2.1
, the stress field was calculated directly. This alternative
analysis facilitates the deployment of both analytical and numerical methods.
Due to sudden motion of the solid plate at
t
=
0, i.e. infinite acceleration, the shear
stress at the plate initially exceeds the yield stress. Because material away from the solid
plate cannot move abruptly, it follows that the yield surface must originate at the plate.
Following
start
-
up, the yield surface moves away from the plate before returning to the
plate in finite time, after which the entire material is rigid; see
figure 2
(cf. Ancey & Bates
2017
).
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E.M. Hinton, J.F. Collis and J.E. Sader
3.1.
Rigid region
In the rigid region above the yield surface, i.e.
Y
(
t
)<
y
≤
1, the velocity field of the
material is constant:
u
≡
u
R
(
t
),
(3.1)
where the
sub
script
R
denotes the rigid material and is used throughout for clarity. At the
position of the yield surface, the shear stress
is
τ
=−
B
,
hence the material is on the verge
of flowing; see (
2.3
). Note that the shear
stress
τ
is always non-positive owing to the sign
of the velocity gradient and the chosen Cartesian coordinate system; see
figure 1
.Inthis
rigid region, we differentiate the momentum
equation (
2.2
)
with respect to
y
to
obtain
∂
2
τ
R
∂
y
2
=
0
.
(3.2)
This shows that the shear stress in the rigid region increases linearly from
τ
R
=−
B
at
y
=
Y
(
t
)
,to
τ
R
=
0 at the free
sur
face
y
=
1. This result applies to any viscoplastic material
because the rigid region arises from the existence of the yield stress, and is independent of
the constitutive law when the material yields. For further details, see
Appendix A
.
3.2.
Yielded region
In the yielded region, i.e. 0
≤
y
≤
Y
(
t
)
, the magnitude of the shear
stress
τ
exceeds the
yield stress, and the momentum
equation (
2.2
) reduces to its Newtonian form,
∂
u
∂
t
=
∂
2
u
∂
y
2
,
(3.3)
which is supplemented by three boundary conditions.
First,
after
start
-
up (
t
>
0)
,the
material velocity matches that of the plate,
u
(
0
,
t
)
=
1
,
(3.4)
while
sec
ond, at the yield surface,
∂
u
∂
y
y
=
Y
(
t
)
=
0
,
(3.5)
because the shear stress must be continuous at the yield surface.
The third boundary condition (Balmforth
et al.
2009
) is obtained by first integrating
(
2.2
)overtherigid
region
Y
(
t
)<
y
≤
1, and using the properties that
τ
R
=−
B
at
y
=
Y
(
t
)
,and
τ
R
=
0at
y
=
1. Noting that the velocity is continuous at the yield surface, i.e.
u
(
Y
(
t
),
t
)
=
u
R
(
t
)
, then gives the required
result
(see also Sekimoto
1993
)
:
(
1
−
Y
)
∂
u
∂
t
y
=
Y
(
t
)
=
(
1
−
Y
)
∂
u
R
∂
t
=
B
.
(3.6)
Equation (
3.6
) is simply a restatement of Newton’s
sec
ond law, that the shear
stress
τ
=
τ
R
=−
B
at the yield surface accelerates the rigid region above it of
thick
ness 1
−
Y
.
Since
B
>
0, (
3.6
) establishes that acceleration of the rigid region is strictly non-negative.
The three boundary conditions (
3.4
), (
3.5
)and(
3.6
) apply to the yielded region for any
viscoplastic material, although the governing equation (
3.3
) in the yielded region will
generally be nonlinear (for further details, see
Appendix A
).
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Yield-stress material on a plate that moves suddenly
1
2
3
∂
u
R
/∂
t
B
= 0
B
= 1
B
= 5
2
4
6
8
0
0.1
0.2
0.3
0.4
0
0.2
0.4
tt
t
Y
(
t
)
0
0.2
0.4
0
0.1
0.2
0
0.2
0.4
0.6
0.8
4
1
2
3
(
b
)
(
a
)(
c
)
Figure 4.
Accel
er
a
tion
∂
u
R
/∂
t
of the rigid region adjacent to the free surface (black curves, left-hand axes),
and the location of the yield
sur
face
Y
(
t
)
(red dashed curves, right-hand axes), for (
a
)
B
=
0 (Newtonian fluid),
for which there is no yield surface,
(
b
) a Bingham material
with
B
=
1
,and(
c
)
a
Bing
ham
mate
rial
with
B
=
5.
Numerical results as per §
2.1
. The dotted horizontal specifies the minimum acceleration, i.e.
∂
u
R
/∂
t
=
B
,as
per (
3.8
). Numerical results for the acceleration exclude the zero acceleration conditions specified at the end
points
t
=
0and
t
f
;
see
§
B.2
of
Appendix
B
.
Rewriting (
3.6
)as
Y
=
1
−
B
∂
u
R
/∂
t
(3.7)
shows that
Y
(
t
)
is a monotonically increasing function of
∂
u
R
/∂
t
. That is, maximum
Y
corresponds to maximum acceleration of the rigid region. This is because the shear stress
– and hence driving force – is constant at the yield
sur
face
y
=
Y
(
t
)
, and given by
τ
=−
B
.
Plots showing the time evolution of
Y
(
t
)
and
∂
u
R
/∂
t
,for
B
=
1 and 5, are given in
figure 4
,
obtained using the numerical results in §
2.1
.
To explain this observed transient evolution of the location of the yield surface, we
first explore how the rigid material near the free surface accelerates. Immediately after
start
-
up, i.e. at ‘early time’, the free surface accelerates at a growing rate as momentum
diffuses out from the plate where the material has yielded. Later, the material moves at
speeds comparable to the plate
, which reduces the velocity gradients. The shear stress at
the plate, and hence the acceleration of the material, subsequently decreases with time.
This behaviour is shown by the black solid curves in
figure 4
, which give the acceleration
of the rigid region, and hence the free surface. The location of the yield surface for
B
=
1
and
B
=
5(reddashedcurvesin
figure 4
) moves monotonically with the acceleration of
the free surface; the yield curve has a turning point owing to the change in the rate of
acceleration of the free surface.
Equation (
3.7
) also demonstrates that for small
B
, the yield surface is close to the free
surface (i.e.
Y
≈
1) when the acceleration of the rigid region is not small, i.e. much larger
than
B
. This occurs very shortly after initiation of the motion; see
figures 2
(
e
,
f
). However,
the acceleration is subsequently much slower to decay than for a material with larger yield
stress; see
figure 4
(
a
).
Hence for small
B
,the
time
scale for relaxation to the fully rigid
motion is longer. These effects are discussed in more detail in §§
3.4
and
3.5
.
3.3.
Collision time
Equation (
3.6
) is derived assuming that a yield curve exists where
τ
=−
B
, i.e. a yielded
region occurs below the yield surface. From (
3.6
), it then follows that acceleration of the
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E.M. Hinton, J.F. Collis and J.E. Sader
rigid region near the free surface must
sat
isfy
∂
u
R
∂
t
≥
B
(3.8)
when a yielded region exists, which is evident in
fig
ures
4
(
b
,
c
). This behaviour contrasts
with flow of a Newtonian fluid for which the shear stress at the plate (and hence the
material acceleration) approaches but never reaches zero as time increases, for
t
1.
Evidently
, for a Bingham material, a time must be reached where (
3.8
) is violated
,
otherwise the material would accelerate
ad infinitum
. This corresponds to the collision
time
t
=
t
f
(defined above), where the yield surface collides with the solid plate and
extinguishes the yield region. At and after this time, shear stress at the plate, and
throughout the material, drops to zero
, and the entire material moves as a rigid body in
concert with the plate.
An upper bound for this collision time can be derived. Integrating the inequality in (
3.8
)
with respect to
time
t
, while noting that the rigid material is initially at rest, i.e.
u
R
|
t
=
0
=
0,
and cannot exceed the plate velocity, i.e.
u
R
≤
1, gives the required result
:
t
f
≤
1
B
.
(3.9)
Equation (
3.9
) is shown as a dashed blue curve in
figure 3
(
a
), where it indeed provides an
upper bound to the full numerical solution for all
B
.
For large
B
, the yield curve stays close to the solid plate at all times, i.e.
Y
1, and
almost the entire material region remains rigid. That is, the inequalities in (
3.8
)and(
3.9
)
become equalities as
B
→∞
, which is evident in the comparison provided in
figure 3
(
a
).
The results for the bound on the rigid acceleration, the bound on the collision time, and
the collision time for large
B
, apply generally to any viscoplastic material.
3.4.
Early time
At early time, immediately following
start
-
up, the material dynamics possess two key
properties: (i) the velocity field is non-negligible only in the immediate neighbourhood
of the plate; and (ii) the yield surface remains close to the plate (Burgess & Wilson
1997
).
These properties imply that the free surface cannot yet influence the material dynamics.
That is, the velocity field for
start
-
up of a semi-infinite (unbounded) region – identical
to that of Newtonian flow, (
1.2
) – applies in the yielded region close to the plate. In
dimensionless form, (
1.2
) becomes
u
N
,
∞
=
erfc
y
2
√
t
.
(3.10)
Also, provided that the yield
curve
Y
(
t
)
grows faster than
√
t
, which we show
a posteriori
,
∂
u
N
,
∞
/∂
y
∼
0at
y
=
Y
(
t
)
for early time. In other words, the yield surface moves more
quickly than transport due to momentum diffusion, which is evident in the numerical
results of
figure 2
.Since
Y
1, the boundary condition at the yield surface in (
3.6
)
becomes
∂
u
∂
t
=
B
for
y
=
Y
(
t
),
(3.11)
which can be integrated by noting that the yield surface originates from the plate, i.e.
Y
(
0
)
=
0, to give
u
(
Y
(
t
),
t
)
=
Bt
.
(3.12)
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Yield-stress material on a plate that moves suddenly
0
0.01
4 × 10
–4
2 × 10
–4
5 × 10
–5
10
–4
t
Bt
Y
0.02
0.2
0.4
0.6
0.8
1.0
0.10
0.08
0.06
0.04
B
= 25
B
= 1/25
B
= 1/25
B
1/2
Y
B
= 25
(
b
)
(
a
)
Figure 5. Location of the yield
sur
face
Y
at early times, for
B
=
1
/
25
,
1
/
5
,
1
,
5
,
25. Accurate numerical
results for arbitrary
B
,asper§
2.1
(solid black curves). (
a
) Data rescaled for small time as per (
3.14
a
–
c
).
Asymptotic solution is (
3.15
) (red dashed curve). (
b
) Data not rescaled. Improved asymptotic solution is (
3.23
)
(red dot-dashed curves). The early time
con
di
tion (
3.22
)
for
B
=
25 is violated over the time duration shown.
Substituting (
3.10
)into(
3.12
) gives the evolution equation for the yield surface (for early
time)
:
erfc
Y
2
√
t
=
Bt
,
(3.13)
which shows that the appropriate rescalings for early time are
t
∼
1
/
B
and
Y
∼
1
/
√
B
.
Defin
ing
̄
t
=
Bt
,
̄
y
=
√
By
,
̄
Y
=
√
BY
,
(3.14
a
–
c
)
then transforms the evolution equation (
3.13
)
into one that is independent of
B
:
̄
Y
=
2
√
̄
t
erfc
−
1
̄
t
.
(3.15)
This prediction is compared to the numerical results for the yield surface evolution in
figure 5
(
a
) for a range of values of
B
.
For small
̄
t
,(
3.15
) yields the asymptotic
result
̄
Y
∼
2
√
̄
t
1
2
log
⎛
⎜
⎜
⎝
2
π
̄
t
2
log
2
π
̄
t
2
⎞
⎟
⎟
⎠
,
(3.16)
while the velocity field (
3.10
)
dif
fuses away from the plate with the time dependence
:
̄
y
v
el
∼
2
√
̄
t
,
(3.17)
with the subscript ‘vel’ referring to the velocity field. Equations (
3.16
)and(
3.17
)thengive
lim
t
→
0
+
̄
y
v
el
̄
Y
=
lim
t
→
0
+
̄
y
v
el
̄
Y
=
0
,
(3.18)
via L’Hopital’s rule, where
refers to the time derivative. Consistency of (
3.18
)withthe
imposed assumption that the yield surface moves away from the plate at a much faster rate
than the velocity field, just after
start
-
up, confirms the validity of this assumption.
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We also calculate the shear stress applied by the plate,
τ
plate
, to the material at early
time, to ensure
that the plate moves at constant
veloc
ity
U
0
ˆ
x
. Substituting (
3.10
)into(
1.4
)
(in its dimensionless form) gives the required result:
τ
plate
=−
1
√
π
t
+
B
.
(3.19)
3.4.1.
Duration of the early time period
We next calculate the duration of the early time period. Equation (
3.13
) relies on two
conditions. The first is that the velocity field in the rigid region (away from the plate),
just after
start
-
up, is close to zero, i.e.
u
R
1. Because
u
R
=
Bt
immediately following
start
-
up, as established by (
3.12
), this small velocity condition in the rigid region is
equivalent to
t
1
B
.
(3.20)
Second, the yield surface remains close to the plate just after
start
-
up, i.e.
Y
1.
Equation (
3.16
) implies that
Y
∼
√
t
, which then imposes a second constraint that
t
1
.
(3.21)
The conditions (
3.20
)and(
3.21
) are different when
B
/
=
1. This indicates that the
early-time regime is shorter for larger values of
B
, but its extent is independent of
B
for
small
B
. The intersection of (
3.20
)and(
3.21
) gives the required duration of the early time
period:
t
min
1
,
1
B
,
‘early time period’
.
(3.22)
This highlights the different physical mechanisms at play in the small and large
yield-stress regimes, i.e. small and large
B
, respectively. Equation (
3.22
)gives
t
1
for small
B
, consistent with the velocity field and yield surface both being controlled by
momentum diffusion; see the chosen (dimensional) time scale of
H
2
/ν
in §
2
. In contrast
,
for large
B
, the time scale is reduced by a factor of
B
,
which
cor
re
sponds
dimen
sion
ally
to
ρ
U
0
H
/τ
Y
. This time scale arises from acceleration of the rigid region above the yield
surface, driven by its (constant) stress,
τ
Y
, with momentum diffusion playing a negligible
role.
3.4.2.
Dynamics of the yield surface just after the early time period
Here, we calculate the yield surface
posi
tion
Y
when it is no longer very close to the solid
plate, but the velocity of the rigid region remains small;
u
R
1. This situation can arise
because the yield surface moves away from the plate at a rate faster than that of the velocity
disturbance (see §
3.4
); it is irrelevant for large
B
where the yield surface always stays close
to the plate.
Since
Y
1 no longer holds, we retain the 1
−
Y
term in (
3.6
)togivetherefined
evolution equation for the yield surface:
(
1
−
Y
)
Y
2
√
π
t
3
/
2
exp
−
Y
2
4
t
=
B
,
(3.23)
whose solution for
Y
is compared to the accurate numerical results for arbitrary
B
(as per
§
2.1
)in
figure 5
(
b
).
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Yield-stress material on a plate that moves suddenly
For a general Herschel–Bulkley material, the results are qualitatively similar but with
different scalings for the thickness of the yielded region. The time scale for the early-time
behaviour is the same as for a Bingham material. For further details, see
Appendix A
.
3.5.
Small Bingham number
For zero yield stress,
B
=
0, the velocity field is given by that for Newtonian flow, i.e.
u
N
(
y
,
t
)
. Because flow occurs everywhere in a Newtonian fluid, the yield surface of a
Bingham material with
B
=
0 will reside at its free surface, i.e.
Y
=
1. We use these
Newtonian results in a regular perturbation expansion for small
B
:
u
(
y
,
t
)
=
u
N
(
y
,
t
)
+
o
(
1
),
Y
(
t
)
=
1
−
BY
1
(
t
)
+
o
(
B
),
(3.24
a
,
b
)
where
Y
1
is to be determined, while the velocity field for a Newtonian fluid of finite extent
with a free surface, is
u
N
(
y
,
t
)
=
1
−
∞
n
=
0
4
(
2
n
+
1
)
π
sin
n
+
1
2
π
y
exp
−
n
+
1
2
2
π
2
t
.
(3.25)
While (
3.10
) is valid only at early times for a finite layer of Newtonian fluid, (
3.25
)is
valid at all times because it accounts for the free surface. This latter solution is needed
for the present analysis because the yield surface comes close to the free surface, and the
rigid region has non-negligible velocity. This is in contrast to early times, for which the
yield surface remains near the plate with the rigid material approximately stationary, so
that (
3.10
) is applicable.
Equa
tions (
3.24
a
,
b
)
cor
re
spond to the velocity field being approximately Newtonian,
except near the free surface (1
−
BY
1
<
y
≤
1)
, where the yield stress becomes important
and rigid motion occurs. Substituting (
3.24
a
,
b
)into(
3.6
), and equating terms of equal
order in
B
, produces
Y
1
(
t
)
=
∂
u
N
∂
t
y
=
1
−
1
,
(3.26)
and using (
3.25
) gives the required result
:
Y
1
(
t
)
=
∞
n
=
0
(
−
1
)
n
(
2
n
+
1
)
π
exp
−
n
+
1
2
2
π
2
t
−
1
.
(3.27)
Equations (
3.24
a
,
b
)and(
3.27
) evidently do not capture the true behaviour of
Y
(
t
)
at
t
=
0. Namely, the yield surface must begin at the solid plate, i.e.
Y
(
0
)
=
0, for
non
-
zero
B
.
Equa
tions (
3.24
a
,
b
)
apply only when the yield surface is close to the free surface. There
is an early time start-up solution with a shorter
time
scale,
which
is evident in
figure 2
(
e
)
andreportedin§
3.4
.
Importantly, (
3.27
) contains no adjustable parameters and can be compared to full
numerical solutions for small but arbitrary
B
, which is provided in
figure 6
. Good
agreement is observed for the smallest values of
B
=
1
/
25 and
B
=
1
/
125, with deviations
growing as
B
is increased, as expected.
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E.M. Hinton, J.F. Collis and J.E. Sader
0
0.2
0.4
0.6
t
0.8
1.0
B
= 1/5
Asymptotic
B
= 1/25
B
= 1/125
1
2
(1 –
Y
)/
B
3
4
Figure 6. Location of the yield surface in rescaled coordinates using accurate numerical solutions for arbitrary
B
(as per §
2.1
), and the small-
B
asymptotic solution in (
3.27
) (dashed line). Results
are shown for
B
=
1
/
5
,
1
/
25
,
1
/
125.
3.5.1.
Maximum height of the yield surface
Equation (
3.27
) is observed to provide a good approximation for the closest approach
position,
Y
=
Y
max
, that the yield surface makes to the free surface at
time
t
=
t
max
,
corresponding to maximum height of the yield surface. Including only two terms (
n
=
0
and 1) in the infinite series in (
3.27
)
gives the analytical
approx
i
ma
tion
Y
max
≈
1
−
3
3
/
8
9
8
π
B
+
o
(
B
)
≈
1
−
0
.
541
B
+
o
(
B
),
(3.28)
which occurs at
t
max
≈
log 17
2
π
2
+
o
(
1
)
≈
0
.
167
+
o
(
1
).
(3.29)
Equation (
3.28
) is compared to the accurate numerical solution in
figure 3
(
b
), where it
is seen to capture the behaviour for small
B
. The time at which this maximum in the yield
surface occurs, given in (
3.29
), is consistent with the end of the early time period; see
(
3.22
).
3.5.2.
Estimation of the collision time
For long times, i.e.
t
≥
O
(
1
)
, the expression for
Y
1
in (
3.27
) is dominated by its first term
(
n
=
0). Substituting this approximate (
n
=
0) formula for
Y
1
into (
3.24
a
,
b
)gives
Y
(
t
)
≈
1
−
1
π
exp
π
2
t
4
B
.
(3.30)
An approximation for the collision
time
t
=
t
f
, where
Y
(
t
f
)
=
0, can then be obtained from
therootof(
3.30
), giving
t
f
≈
4
π
2
log
π
B
.
(3.31)
Strictly,
this represents an extrapolation because (
3.30
) is based on an asymptotic
expansion for small
B
, in which
Y
must remain close to 1. Nonetheless, comparison to
the numerical results in
figure 3
(
a
)showsthat(
3.31
) is accurate for small
B
.
Equation (
3.31
) shows that the
time
scale to accelerate the entire Bingham material, from
rest to the plate velocity, scales as log
(
1
/
B
)
for small
B
. This is
slower
asymp
tot
i
cally
than
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Yield-stress material on a plate that moves suddenly
the
time
scale for the early time period; see (
3.22
). The enhanced
time
scale in (
3.31
) arises
from the diminishing acceleration of the rigid region, whose thickness strongly increases –
from 1
−
Y
≈
0 (just after the early time period) to 1 (at the collision time) – while being
driven by the constant yield stress at the yield surface.
If the height of the rigid region were not to change, then the time scale for collision
would be 1
/
B
,asper(
3.6
); this latter condition is relevant to the following
sub
sec
tions.
Results for a general Herschel–Bulkley material at small
B
are qualitatively similar, with
the yield surface moving close to the free surface, i.e.
(
1
−
Y
)
∼
B
. However, its analysis
is complicated by the lack of an exact solution for
B
=
0; see
Appendix A
.
3.6.
Behaviour immediately prior to collision
Just before the entire material becomes rigid at the collision
time
t
=
t
f
, its velocity is
close to that of the plate, i.e.
u
≈
1, and the position of the yield surface is near the plate,
i.e.
Y
≈
0. These properties and the governing equations for the yielded region, (
3.3
), (
3.5
)
and (
3.6
), indicate a similarity solution of the
form
u
(
y
,
t
)
=
1
−
B
(
t
f
−
t
)
f
(η),
(3.32)
where
η
=
y
√
t
f
−
t
,
(3.33)
which at the yield
sur
face
y
=
Y
is
η
|
y
=
Y
≡
η
yield
=
Y
√
t
f
−
t
.
(3.34)
Importantly, (
3.32
)–(
3.34
) hold for arbitrary collision
time
t
f
.
The shape
func
tion
f
(η)
in (
3.32
) must satisfy the momentum equation (
3.3
), which
becomes
f
−
1
2
η
d
f
d
η
=−
d
2
f
d
η
2
,
(3.35)
whose boundary conditions follow from (
3.4
)–(
3.6
)andthe
prop
erty
Y
→
0as
t
→
t
f
:
f
(
0
)
=
0
,
f
(η
yield
)
=
1
,
d
f
d
η
η
=
η
yield
=
0
,
(3.36
a
–
c
)
where
η
yield
is to be determined. Solving (
3.35
)and(
3.36
a
–
c
) gives the required result:
f
(η)
=
2
η
exp
η
2
4
+
√
π
2
−
η
2
erfi
η
2
2
η
yield
exp
η
2
yield
4
+
√
π
2
−
η
2
yield
erfi
η
yield
2
,
(3.37)
with
√
π
2
η
yield
exp
−
η
2
yield
4
erfi
η
yield
2
=
1
,
(3.38)
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E.M. Hinton, J.F. Collis and J.E. Sader
10
–2
10
0
B
= 20
B
= 1
B
= 1/20
Equation (3.34)
t
f
–
t
Y
(
t
)
10
–2
10
–1
10
0
Figure 7. Position of the yield surface just before collision. Comparison of the similarity solution (dashed
curve) (
3.34
)
to accurate numerical results (solid curves) that hold for arbitrary time; results shown for
B
=
1
/
20
,
1
,
20.
whose only positive real solution is
η
yield
=
1
.
848
.
(3.39)
The similarity solution for the yield surface (
3.34
)
is compared to the accurate numerical
solution in
figure 7
, where good agreement is observed. For the similarity solution to
apply,
u
must be close to 1 across the layer. From (
3.32
), this condition corresponds to
(
t
f
−
t
)
B
−
1
. Hence the similarity solution is accurate over a shorter period for larger
B
;see
figure 7
.
The shear stress applied by the plate,
τ
plate
, just before collision, follows from (
2.3
b
),
(
3.32
), (
3.37
)
and (
3.39
):
τ
plate
=−
B
1
+
f
(
0
)
t
f
−
t
,
(3.40)
where
f
(
0
)
=
0
.
7868.
These results are easily extended to a general Herschel–Bulkley material; see
Appendix A
.
3.6.1.
Abrupt change in acceleration at the collision time
Because
Y
=
0 at the collision time – where the material moves in concert with the
plate – it follows from (
3.6
) that the
free
sur
face acceleration jumps discontinuously from
∂
u
R
/∂
t
=
B
to 0 at the collision time; see
figure 4
. This phenomenon occurs for all
B
and is a signature of
non
-
zero yield stress, which should be observable experimentally.
Indeed, the result is not specific to a Bingham constitutive law, with the abrupt change in
acceleration occurring for any viscoplastic material.
3.7.
Large Bingham number
As discussed in §
2.1
, the yield surface always stays close to the solid plate in the large
yield-stress regime,
B
1. Imposing this constraint on the boundary condition at the
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