Journal of Non-Newtonian Fluid Mechanics 167–168 (2012) 59–74
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Journal of Non-Newtonian Fluid Mechanics
journal homepage: http://www.elsevier.com/locate/jnnfm
Incomplete fluid–fluid displacement of yield stress fluids in near-horizontal pipes:
Experiments and theory
S.M. Taghavi a, K. Alba b, M. Moyers-Gonzalez c, I.A. Frigaard d,⇑
a
Department of Chemical and Biological Engineering, University of British Columbia, 2360 East Mall, Vancouver, BC, Canada, V6T 1Z3
Department of Mechanical Engineering, University of British Columbia, 2054-6250 Applied Science Lane, Vancouver, BC, Canada V6T 1Z4
c
Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand
d
Department of Mathematics and Department of Mechanical Engineering, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada V6T 1Z2
b
a r t i c l e
i n f o
Article history:
Received 10 August 2011
Received in revised form 11 October 2011
Accepted 11 October 2011
Available online 21 October 2011
Keywords:
Yield stress fluids
Static residual layers
UDV
Displacement flow
Buoyancy
Carbopol
a b s t r a c t
We present results of a primarily experimental study of buoyant miscible displacement flows of a yield
stress fluid by a higher density Newtonian fluid along a long pipe, inclined at angles close to horizontal.
We focus on the industrially interesting case where the yield stress is significantly larger than a typical
viscous stress in the displacing fluid, but where buoyancy forces may be significant. We identify two distinct flow regimes: a central-type displacement regime and a slump-type regime for higher density
ratios. In the central-type displacement flows, we find non-uniform static residual layers all around
the pipe wall with long-wave variation along the pipe. In the slump-type displacement we generally
detect two propagating displacement fronts. A fast front propagates in a thin layer near the bottom of
the pipe. A much slower second front follows, displacing a thicker layer of the pipe but sometimes stopping altogether when buoyancy effects are reduced by spreading of the front. In the thin lower layer the
flow rate is focused which results in large effective Reynolds numbers, moving into transitional regimes.
These flows are frequently unsteady and the displacing fluid can channel through the yield stress fluid in
an erratic fashion. We show that the two regimes are delineated by the value of the Archimedes numbers
(equivalently, the Reynolds number divided by the densimetric Froude number), a parameter which is
independent of the imposed flow rate. We present the phenomenology of the two flow regimes. In simplified configurations, we compare computational and analytical predictions of the flow behaviour (e.g.
static layer thickness, axial velocity) with our experimental observations.
Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction
There are many industrial processes in which it is necessary to
remove a gelled material or soft-solid from a duct. Examples include bio-medical applications (mucus [27,34], biofilms [7,52]),
cleaning of equipment and food processing [6,8], oil well cementing and waxy crude oil pipeline restarts. A wide range of material
models are used to describe residual deposits in these situations.
Some of these flows are turbulent, but equally often process limitations dictate that the flows be laminar. It is this case that we
study here. Our industrial motivation comes from the oil industry,
and we consider that the fluid to be removed is either a drilling
mud or a pipeline full of waxy crude oil, and that these fluids have
a yield stress. We study downward displacement flows along pipes
that are inclined at angles close to horizontal (but not horizontal),
as for Newtonian fluid flows we can see significant differences
⇑ Corresponding author. Tel.: +1 604 822 3043; fax: +1 604 822 6074.
E-mail address: [email protected] (I.A. Frigaard).
0377-0257/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.jnnfm.2011.10.004
when fully horizontal. We have recently studied in detail such displacement flows, in the Newtonian fluid setting, in [44–47].
The main feature of a yield stress fluid is that the fluid does not
deform until a critical shear stress is exceeded locally. Therefore,
when these fluids fill ducts and are displaced by other fluids, there
is a tendency for the yield stress fluid to remain stuck to the duct
walls and in particular in parts of the duct where there are constrictions or corners. This type of feature was first recognised in
the context of oil well cementing by McLean et al. [33], who identified potential bridging of a static plug of mud on the narrow side
of an eccentric annulus. Avoidance of this feature has since been an
ingredient of industrial design rules for oilfield cementing
[10,29,40], and latterly also simulation based design models
[3,41]. Further features of oilfield cementing are discussed in
[36], but here our geometry is simpler.
In waxy crude oil pipeline restarts (see [5,12,42,49]) a large pressure is applied at one end of the pipe, to break the gel of the waxy oil.
The waxy state has formed due to a drop in temperature below the
wax appearance temperature, often related to stopping the pipeline
for maintenance or other issues. Temperature is not particularly
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S.M. Taghavi et al. / Journal of Non-Newtonian Fluid Mechanics 167–168 (2012) 59–74
important in the restart process itself [51]. It is common to displace
the in situ oil with an oil of different physical properties (often this is
the same oil at higher temperature and thus with a Newtonian viscosity). In the displacement it is possible for static residual layers
to form on the walls of the pipeline; see [17,50].
The phenomenon of a static wall layer in a plane channel was
first studied by Allouche et al. [2] who studied symmetric displacement flows of two visco-plastic fluids. As is intuitive, a necessary
condition for the existence of a static wall layer is that the yield
stress of the displaced fluid exceeds that of the displacing fluid
(if there is one). In [2] a constant displacement flow rate was imposed. A number of two-dimensional (2D) simulations were performed with no density difference, and the static layer thickness
was measured from the results. Additionally, expressions were derived for the maximal static layer thickness (for vertical plane
channels in the presence of a density difference), and for a prediction of the actual layer thickness (no density difference). Frigaard
et al. [20] extended this approach, showing that in a steady displacement flow with a uniform static wall layer the thickness of
the layer and the shape of the interface are non-unique for the
steady displacement problem and consequently must result from
transient aspects of the flow. The concept of maximal static wall
layers was further explored in [18]. More recently in [53], an
extensive computational study of static layer thickness in iso-density fluid displacements (Newtonian fluid displacing Bingham
fluid) was performed, including the effect of flow rate oscillations.
This has shed further light on the effects of the main three dimensionless parameters (Reynolds number, Bingham number and viscosity ratio), in the absence of density differences.
In contrast to the amount of computational work, there are relatively few experimental studies of displacement of yield stress
fluids by other fluids. Gabard [21], and Gabard and Hulin [22]
investigated iso-density miscible displacements in which a more
viscous fluid is displaced by a Newtonian fluid. In their experimental investigation the geometry used was a vertical tube. They observed the effect of rheology of the displaced fluid and the flow
velocity on the transient residual film thickness during the displacement process. They showed that in displacements of shearthinning fluids the residual wall layer thickness decreases compared to Newtonian fluid displacements. The shear thinning fluid
displacements were characterised by slowly evolving interfacial
instabilities of (inverse) bamboo type, which further reduced the
initially symmetric residual wall layer. For yield stress fluids static
residual wall layers were observed of uniform thickness. Axisymmetric computations were also carried out in [21] with results
which were qualitatively similar to the experimental results.
Other experimental studies involving two fluid flows of yield
stress fluids in the pipe geometry include [11,32] who have studied
the exchange flow problem (i.e. buoyancy driven flow in a closed
ended pipe). The focus of these studies is stopping the motion
using the yield stress of one of the fluids. Huen et al. [28] and Hormozi et al. [26] have studied core–annular flows, using a yield
stress fluid for the outer lubricating layer and a range of different
Newtonian and non-Newtonian fluids for the core. The start-up
phase of these experiments is displacement-like, although the final
steady state is a multi-layer flow.
In the Hele-Shaw geometry, Lindner et al. [30,31] studied the
Saffman–Taylor (viscous fingering) instability while displacing
yield stress fluids. They observed a yield stress dominated regime
at low velocity and a viscous dominated regime when the velocity
was higher. The former regime shows branched patterns because
in simple words each finger does not really feel the presence of
walls or other fingers due to the fluid’s yield stress. In the viscous
dominated regime, yield stress does not play an important role.
Other investigations of viscous fingering (with stability analyses)
include [9] and the earlier Darcy-flow analogues of [37–39].
Finally, a number of authors have considered the displacement
of yield stress fluids by a gas. De Souza Mendes et al. [13] investigated the displacement of viscoplastic flows in capillary tubes
experimentally through gas injection. They showed that below a
certain critical flow rate, the visco-plastic liquid is completely displaced by the displacing fluid. However above this critical flow rate
small lumps of unyielded liquid will remain on the walls. For increased values of imposed flow rate a smooth liquid layer of uniform thickness forms. They reported that the thickness of this
layer increases with the dimensionless flow rate. There have also
been extensive computational studies of these flows; see
[14,15,43,48]. Finally, there is a limited amount of analytical work
concerning bubble propagation/displacement in Hele-Shaw geometries; see [1].
The aim of our study is to deepen our understanding of yield
stress fluid displacements in pipes in a regime that has not been
previously studied. Namely we study displacement of fluids with
large yield stress and where buoyancy is significant. Near horizontal pipelines and wells lead to situations in which buoyancy forces
promote slumping and asymmetry for Newtonian displacement
flows. When a yield stress fluid is involved the displaced fluid rheology may counter the tendency to stratify, but these flows are
poorly understood. The situations of primary industrial interest
are those in which the yield stress is very large, so that static residual layers may persist.
The outline of our paper is as follows. In Section 2 we outline
the scope of our study and the experimental methods used. Results
are presented in Section 3. We first describe the main finding of the
paper, namely the observation of two principal types of flow delineated by the ratio of Reynolds number to densimetric Froude number (equivalent to the square root of the Archimedes number). We
then describe in more detail the features of the central (Section
3.2) and slump (Section 3.4) type displacements. The paper ends
with a brief summary.
2. Scope of the study and methodology
As explained in the introduction, the aim of our study is to
better understand displacement flows of visco-plastic fluids in
near-horizontal pipes. The choice of a near-horizontal pipe follows from our previous work on Newtonian–Newtonian fluid displacements, where this range of pipe inclinations is found to
exhibit interesting transitions between inertia-buoyancy and viscous-buoyancy dominated regimes; see [44]. In moving from an
iso-viscous buoyant Newtonian–Newtonian fluid displacement
to a displacement flow of a typical shear-thinning visco-plastic
fluid by a Newtonian fluid, we have at least three more dimensionless parameters, e.g. for a Herschel–Bulkley model. Although
we consider pipe inclinations b 90°, a comprehensive experimental study of flow variations with the remaining six dimensionless parameters is infeasible. Therefore, we focus on
displacing visco-plastic fluids with large yield stress, with the
motivation that these are the flows that are most problematic
from an industrial perspective. However, the idea of large yield
stress needs qualifying.
^Y )
Suppose we displace a yield stress fluid (with yield stress s
^ , by imposing a flow rate
with a Newtonian fluid of viscosity l
b ¼ pV
b i.e. V
b 0 is the
b0D
b 2 =4, through a long pipe of diameter D,
Q
mean velocity. Inevitably the fluid will finger through some part
of the pipe cross-section, potentially leaving behind residual layers
as the displacement front propagates. Apart from close to the tip of
the finger we might suppose that the Newtonian flow in the bulk of
the finger becomes near-parallel and generates viscous stresses of
b 0 = D.
b If we wish to study flows in which it is possible
^v ¼ l
^V
order s
for the visco-plastic fluid to be left behind as the displacement
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S.M. Taghavi et al. / Journal of Non-Newtonian Fluid Mechanics 167–168 (2012) 59–74
front propagates along the pipe, it is clear that a first requirement
is:
l^ Vb 0 = Db ¼ s^v s^Y :
Tank
^
ð1Þ
β
V0
UDV probe
Dyed light fluid
We may reformulate this as
BN 1;
s^Y Db
BN ¼
:
l^ Vb 0
Drain
ð2Þ
Here BN is a form of Bingham number, but with the viscous stress
scale coming from the Newtonian fluid, as would be appropriate
in this type of flow.
Secondly, since the flow close to the displacement front will be
three-dimensional we must consider inertial stresses as well as
b 2 . If we were to consider
^t ¼ q
^V
viscous. The inertial stress scale is s
0
^t s
^v , meaning Re O(1), then (2) would imply
flows for which s
^t s
^Y . It is then unlikely that we would see much variation
that s
b 0 is varied. Consequently, we have targeted our
in our results as V
study at the range Re > 1, where inertial effects are dominant close
the displacement front. Further, we wanted to see how changes in
flow rate might affect the type of displacement observed and so
have selected flow parameters such that:
q^ Vb 20 ¼ s^t / s^Y :
ð3Þ
^t > s
^Y , then we enter a regime in which inertial stresNote that if s
ses alone might be sufficient to yield and fully displace the viscoplastic fluid; hence the inequality above, which is equivalent to
Re/BN / 1.
A third consideration for our study is that we wish to observe
buoyancy effects. The appropriate scale for buoyant stresses transb sin b, where Dq
^b ¼ Dq
^ g^ D
^ is the absolute denverse to the pipe is s
sity difference between fluids (axial buoyancy stresses are much
smaller than this since b 90°). If we hope to observe significant
effects of buoyancy on the type of displacement flow, we would expect that the buoyancy stress contributes to yielding at the front.
Thus, we have selected fluid parameters such that:
b sin b ¼ s
^Y :
^b s
^ g^ D
Dq
ð4Þ
^b s
^Y it is likely that there would be no effect of buoyIf instead s
ancy. The three conditions (2)–(4) frame the parameter space of our
experiments.
2.1. Experimental description
Our experimental study was performed in a 4 m long, 19.05 mm
diameter, transparent pipe with a gate valve located 80 cm from
one end; see Fig. 1. The pipe was mounted on a frame which could
be tilted to any angle. Initially, the lower part of the pipe is filled
with a less dense fluid (fluid 2) coloured1 with a small amount of
ink. The upper part of the pipe, above the gate valve, is filled by
the denser fluid 1. To avoid pump disturbances, the displacing
upper fluid was fed by gravity from a large elevated tank. The flow
rate was controlled by a valve and measured by both a rotameter
and a magnetic flowmeter, located downstream of the pipe. At
the start of the experiment the gate valve is opened. Images of
the displacing fluid are recorded using two cameras, and subsequently analyzed to characterize different aspects of the flow.
Velocity is also measured through the central plane of the pipe at
a position downstream of the gate valve, using an Ultrasonic Doppler Velocimeter (UDV). These methods are described below in Section 2.1.2.
1
For interpretation of colour in Figs. 2–4 and 10–19, the reader is referred to the
web version of this article.
Transparent
heavy fluid
Gate valve (open)
Fig. 1. Schematic view of experimental set-up (the shape of the interface is
illustrative only).
Our experiments were conducted at two inclinations: b = 83°
and 85°. The upper displacing fluid was denser than the lower
fluid. The density difference is characterised dimensionlessly via
the Atwood number, At:
At ¼
q^ 1 q^ 2
:
q^ 1 þ q^ 2
ð5Þ
Our experiments were conducted over the ranges:
1:2 103 6 At 6 1:6 102 ;
b 0 6 72 mm:s1 ;
06V
which is comparable to that used previously in our Newtonian fluid
displacement flows. Two other dimensionless numbers characterise
the flow: the Reynolds number Re and the densimetric Froude number, Fr:
Re ¼
b
b0
V
q^ 1 Vb 0 D
; Fr ¼
b 1=2
l^ 1
ðAtg^ DÞ
ð6Þ
Our experiments cover the ranges: Re 2 [200, 1400] and
Fr 2 [0.2, 4.71]. In our previous Newtonian experiments qualitative
changes in flow behaviour occur around Fr 1.
2.1.1. Fluid preparation and rheology
The displacing fluid 1 was always a Newtonian salt-water solution. Fluid 2 was always a yield stress fluid, namely a solution of
CarbopolÒ EZ-2 polymer (Noveon Inc.). Carbopol is widely used
as thickener, stabilizer and suspending agent. The rheology of Carbopol is largely controlled by the concentration and pH of the solution. Once mixed with water, Carbopol makes an acidic solution
with no yield stress. The yield stress is developed at intermediate
pH on neutralising with a base agent (in our case NaOH). The neutralised solution is fairly transparent and has the same density as
water (for low concentrations).
In preparing fluid 2 we gradually added the Carbopol powder to
water in a mixing tank, while the mixing blade was rotating. Mixing proceeded slowly for 24 h. However, since the Carbopol concentration in our experiments was not very high, the rheometry
was not found to be sensitive to the blade shape, its rotating speed
(which in our case was usually set between 100 and 400 rpm), nor
the time taken to mix (within this range). When adding NaOH to
the Carbopol–water solution we were very cautious about mixing
and lowered the rotation rate to minimise introduction of air bubbles into the gel-like solution. Carbopol solutions can be mildly
thixotropic and to avoid any artifacts we always sampled the Carbopol directly before each experiment and conducted our rheometric measurements a consistent time after the experiment.
Four different Carbopol solutions were used for the experiments. Their composition is shown in Table 1. The rheological measurements for Carbopol solutions A–C were performed using a
Bohlin CVOR (Malvern) digital controlled stress-shear rate rheometer. A cone-and-plate geometry was used, with 40 mm cone diameter, 60 mm plate diameter, 4° cone angle and 150 lm gap at the
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S.M. Taghavi et al. / Journal of Non-Newtonian Fluid Mechanics 167–168 (2012) 59–74
Table 1
Composition and properties of the displaced fluid used in our experiments.
Carbopol solution
Carbopol %
(wt/wt)
NaOH %
(wt/wt)
s^Y (Pa)
n
j^ ðPa sn Þ
A
B
C
D
0.1125
0.12
0.15
0.14
0.0322
0.0343
0.0429
0.040
1.17
3.05
6.51
3.12
0.45
0.60
0.39
0.26
4.49
8.24
4.63
20.44
cone tip. The cone and plate were roughened with a thin layer of
sand paper (400 grit roughness), to avoid slip. Identical loading
procedures were followed in all tests. Temperature was controlled
to match that in the experimental room. Supplementary experiments were carried out with Carbopol solution D, nearly
18 months after the original experiments with A–C. Rheological
measurements were carried out using the same procedure but
now with a parallel plate geometry and a different rheometer
(HR Nano from Malvern). A rheological model that fits well the
shear behaviour of Carbopol behavior is the Herschel–Bulkley
model:
s^ ¼ s^Y þ j^ c^_n :
ð7Þ
This includes the simpler Bingham, power law and Newtonian mod^, a
els and is defined by three parameters: a fluid consistency index j
^Y , and a power law index n. From our rheometer data,
yield stress s
we determined the yield stress value through the shear stress value
at the global maximum of the viscosity. Afterwards, we subtracted
the yield stress value from the remaining shear stress data and then
we found the best fit to a power law curve. The error in determining
the yield stress value of the Carbopol solution in this way is esti^Þ
mated to be in the range 5–27%. The errors in the consistency ðj
and the power law index (n) were below 7% and 12% respectively.
In fact the Carbopol rheology plays little apparent role in our
experiments, as we target the range BN 1 where much of the displaced fluid will be unyielded, i.e. it is important to have a large
yield stress, but the rheology after yielding is probably irrelevant
for our particular experiments. Determined values of the rheological constants for each of the four Carbopol solutions that we have
used are shown in Table 1. An example flowcurve from the rheometer compared with the fitted Herschel–Bulkley model data is
shown in Fig. 2.
2.1.2. Image processing and local velocity measurement
In order to help the visualization of the two fluids, the pipe was
illuminated from behind by a light box containing fluorescent light
tubes filtered through a diffusive paper to give a homogeneous
light. Light absorption calibration was carried out for both cameras.
During the experiment (after opening the gate valve), images were
Fig. 2. Example flowcurve for Carbopol B. The solid line shows the curve fit with
parameters shown in Table 1.
obtained at regular time intervals, which enabled us to create spatiotemporal diagrams of the averaged concentration profiles along
the length of the pipe. The fronts were marked on these diagrams
by a sharp boundary between the different relative concentrations
of the fluids. The front velocities were obtained from the slope of
this boundary.
We also measured the velocity profile at 80 cm below the gate
valve, using an Ultrasonic Doppler Velocimeter DOP2000 (model
2125, Signal Processing SA) with 8 MHz, 5 mm (TR0805LS) transducers (with a duration of 0.5 l s). This velocimetry technique
suits our experimental needs well since it does not require transparent fluids and is completely non-intrusive. The measuring volume has a cylindrical shape and its the axial resolution in our
fluids is around 0.375 mm and the lateral resolution is equal to
the transducer diameter (5 mm) slightly varying with depth. The
slightly diverging ultrasonic beam enters the fluids by passing
through a 3.175 mm-thick plexiglass pipe wall. This technique is
based on the pulse-echo technique and allows measurement of
the flow velocity projection on the ultrasound beam, in real time
[25]. For the tracer, we used polyamid seeding particles with a
mean particle diameter of 50 lm, with volumetric concentration
of 0.2 g l1 in both fluids. Following [4], there is a trade off between
a good signal to noise ratio and small ultrasonic signal reflections,
achieved by mounting the probe at an angle in the range 68–72 °
relative to the axis of the pipe.
3. Results
Two qualitatively distinct flows were observed in our experiments. In some flows the displacing fluid propagated approximately centrally along pipe, leaving behind residual layers on all
walls. We call this a central type displacement and describe its
characteristics below in Section 3.2. In other displacements the
heavier fluid appeared to slump to the lower part of the pipe and
propagate along the lower wall. As far as could be observed, the
interface was approximately horizontal as measured in a transverse plane and the flow stratified progressively in the length-wise
direction. We call this a slump type displacement and describe its
characteristics below in Section 3.4.
3.1. The transition between central and slump displacements
It was not surprising that the slump displacements appeared to
occur for larger density differences. However, we sought a more
quantitative description for their occurrence. All our experiments
were purposefully designed to satisfy (2)–(4), meaning a large yield
stress. This suggested that the yield stress itself would not play a
significant role in determining flow type. Equally, since all flows
had significant residual layers it appeared that the Carbopol must
be yielded only close to the front and far ahead of the displacement
front in thin wall layers (i.e. Poiseuille flow). Thus, it also seemed
unlikely that the sheared rheology of the Carbopol would be particularly relevant to our experiments.
With the above considerations, we were led to consider only
those dimensionless parameters relevant to the Newtonian displacing fluid: the Atwood number, At, the Reynolds number Re
and the densimetric Froude number Fr. In buoyant displacement
flows, At independently influences only the inertial terms in the
momentum balance. For density differences of less than 10% this
effect can be largely ignored between the fluids, and our At falls
in this range. Neglecting At, buoyancy still has a significant influence through the densimetric Froude number Fr. On analyzing
our data we discovered that the transition between regimes was
governed by the ratio Re/Fr and was largely independent of all
other dimensionless groups we considered. A selection of plots,
S.M. Taghavi et al. / Journal of Non-Newtonian Fluid Mechanics 167–168 (2012) 59–74
showing this dependency on Re/Fr, and independency with respect
to other parameters, is shown in Fig. 3.
The parameter Re/Fr is interesting in that it is independent of
b 0 . For small At, Re/Fr is equivalent to the square
the mean velocity V
root of the Archimedes number, Ar:
b3
^1 q
^ 2 ½q
^1 þ q
^ 2 g^ D
½q
Ar ¼
;
2
^1
2l
rffiffiffiffiffi
Ar
½1 þ OðAtÞ
¼
2
b 3 Þ1=2
^1 q
^ 2 g^ D
^ 1 ð½q
Re q
¼
1=2
Fr
^1
^1 þ q
^2 l
½q
ð8Þ
One could also include a sin b term with the gravitational constant
above, but for simplicity this is neglected. The Archimedes number
occurs commonly in flows where both forced and natural convective forces are involved: large Ar indicates dominance of the buoyancy forces, as indicated here by the stratified slumping.
A dependency on Re/Fr was also evident in our studies of Newtonian fluid displacements; see [44]. In those experiments, different
flow regimes are delineated in the plane of Fr versus Re cos b/Fr. Here
we have not explored variation with b, using only two pipe inclinations. However, it is worth pointing out that the effects are anyway
markedly different to the Newtonian displacements. Firstly, there is
little apparent effect of the velocity (captured here in Fr). Secondly,
in the Newtonian fluid studies, stratified viscous regimes were
Fig. 3. Classification of our experiments:
63
associated with smaller values of Re/Fr, whereas here the reverse
is observed: the central regime is found for smaller Re/Fr.
3.2. Central-type displacements
Examples of central-type displacements are shown in Figs. 4
and 6, at different imposed flow rates, inclinations and for different
Carbopol solutions. Figs. 4a and 6a show sequences of images as
the displacing fluid advances steadily through the Carbopol. The
front shape is skewed towards the top of the pipe, which suggests
inertial dominance at the tip/front. Purely viscous effects would
lead to slumping. The bottom image on each figure shows the scale
for the images, which can be interpreted as a mean concentration
at each position. After the displacement front has passed we see
darker regions at the top and bottom of the pipe, but also at
mid-height the images indicate that there is residual displaced
fluid. Darkness at the top and bottom is simply because we are
viewing from the side of the pipe, essentially looking through the
residual layer rather than perpendicular to the layer. This pattern
is consistent with the presence of a residual wall layer all around
the pipe and the images suggest that the layer is not uniform.
Fig. 4b shows the spatiotemporal plot corresponding to the
displacement of Fig. 4a. The boundary separating dark and light regions indicates the position of the propagating front. It is relatively
easy to calculate the front velocity from such images. Behind the
– slump type displacement;
– central type displacement.
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S.M. Taghavi et al. / Journal of Non-Newtonian Fluid Mechanics 167–168 (2012) 59–74
b 0 ¼ 32 mm s1 with Carbopol solution A: (a) images of the displacement at ^t ¼ 4; 6; . . . ; 20; 22 s after opening the
Fig. 4. Central displacement for b ¼ 83 ; At ¼ 3 103 ; V
gate valve; (b) spatiotemporal image of the same displacement. The plot shows a 833 mm long section of the pipe a few centimeters below the gate valve. The rectangular
region marked by the broken line is explained in the text and is used in the following figures.
front, we observe vertical streaks in the spatiotemporal plot, indicating that the residual layers are static. Computation of the mean
concentration at different positions in the region behind the displacement front suggests that 30–40% of the Carbopol is not displaced. A similar range is found for the displacement of Fig. 6a.
Concentration-based estimates of the residual layer thickness
can be compared with estimates of the mean layer thickness, made
from the front velocity measurements. For example, in Fig. 6a the
b 0 ¼ 44 mm s1 whereas the measured
imposed flow velocity is V
b f ¼ 64 mm s1 . Assuming the residual layers
front velocity is V
are static, this suggests a mean layer thickness
vffiffiffiffiffiffi3
2
u
b
ub
D
^ ¼ 41 t V 0 5 1:63 mm;
h
bf
2
V
with 31% of the Carbopol remaining. In general we have found
that such estimates are self-consistent, but as we can see in both
Figs. 4a and 6a there is variation in layer thickness along the pipe.
To quantify this variation in layer thickness we have analysed
data from the rectangular time-space region indicated by the
broken line in Fig. 4b. First of all we have averaged with respect
to the axial distance ^
x and with respect to time, to give a mean
^Þ at each depth in the pipe; see Fig. 5a. Looking
concentration Cðy
^Þ is skewed about
in the mid range of depths, we can see that Cðy
the centre-line, meaning that the centrally propagating fronts are
not axisymmetric (presumably a buoyancy effect). Close to the
top and bottom of the pipe the data in Fig. 5a is hard to interpret
quantitatively, since curvature effects also come into play.
In Fig. 5b we look at the axial variation of the concentration.
Here we average with respect to depth y and also with respect to
time. This plot and the various insets show that the main larger
amplitude fluctuations are occurring on the order of 50–100 mm.
The small high frequency oscillations correspond approximately
to the pixel scale, 1 pixel 1 mm, and are likely to be image-related. Thus, the variations evident in figures such as Fig. 4 are
long-wavelength.
Fig. 6a presents a second example of a central displacement,
and has been discussed above. A representative example of the
velocity profiles obtained from the UDV measurement is shown
in Fig. 6b, for the same experiment as Fig. 6a. The UDV probe is
fixed at 80 cm below the gate valve angled at 68° to the surface
of the pipe. The velocity readings are taken through the pipe centreline in a vertical section. The vertical axis shows depth measured
from the top of the pipe. The velocity contours are averaged timewise over 25 velocity profiles (3 s). We can see that the main flow
^Þ and (b) Cð^
Fig. 5. Variation of (a) Cðy
xÞ in the rectangular region (illustrated by a broken line) in Fig. 4b. The insets in (b) show that the interfacial modes of the mean static
layer apparently have long wavelength variations, of the order of 50–100 mm.
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S.M. Taghavi et al. / Journal of Non-Newtonian Fluid Mechanics 167–168 (2012) 59–74
b 0 ¼ 44 mm s1 , with Carbopol solution C: (a) images of the displacement at ^t ¼ 1; 2; . . . ; 16; 17 s after opening the
Fig. 6. Central displacement for b = 85°, At = 4 103 and V
gate valve. The length of pipe shown in (a) is a 990 mm long section of the pipe, starting a few centimeters below the gate valve. (b) contours of velocity profiles obtained from
the Ultrasonic Doppler Velocimeter at 80 cm below the gate valve. Assuming a symmetric displacement, velocity values from a simple Poiseuille profile surrounded by static
layers are superimposed onto this plot. The broken lines show the position of the symmetric static layer, estimated from the mean concentration.
Fig. 7. An example of central displacement: sequence of images of increasing the imposed flow showing propagation of interface along a 1386 mm long section of the pipe,
1711 mm below the gate valve. Other parameters are b = 83° and At = 1.2 103 with Carbopol solution D: from top to bottom the images are taken at
b 0 ¼ 24; 44; 55; 71 mm s1 .
V
is more or less in the center of the pipe. Also looking carefully we
can see very thin static residual layers close to the walls. Although
the resolution of the readings falls off close to the walls, we shall
see later that identical readings are obtained for slump-like displacements where the upper layer is very thick. Thus, we consider
these layers to be static.
The dark broken curves superimposed on the UDV contours
indicate where a perfectly axisymmetric concentric flow would
have its static layer, (based on the same mean concentration as
that measured). The numbers within the velocity field indicate
the speed (mm/s) that this same axisymmetric concentric velocity
profile would have.
Although we have not observed much sensitivity of the centralb 0 , there are other qualitative
slump transition to changes in V
b
changes in the flow as V 0 is increased. An example sequence of
b 0 . The flow
central displacements is shown in Fig. 7, for increasing V
becomes progressively unsteady during this sequence. The residual
layer thickness generally decreases.
3.2.1. Maximal static residual layers
Some insight into the presence of residual static layers is
gleaned from a simple one-dimensional model. Suppose that the
flow is steady, laminar, axisymmetric and that the outer layer of
fluid is static. If the interface between the Newtonian inner fluid
and outer static layer is at radius ^ri then it is straightforward to
show that the magnitude of the shear stress at the outer wall of
the pipe is given by:
b0 1 1
^V
8l
b g^ cos b;
^1 q
^ 2ÞD
s^w ¼
ð1 ki Þðq
b
D ki 4
ð9Þ
b < 1. Provided that s
^w 6 s
^Y then the fluid layer
where ki ¼ 2^r i = D
may remain static.2 We readily see that (9) can be interpreted in
^Y an outer layer may remain
alternate way. For a given yield stress s
static only for ki P ki,min, for which:
s^Y ¼
b0 1
^V
8l
1
b g^ cos b:
^1 q
^ 2ÞD
ð1 ki;min Þðq
b
k
4
i;min
D
We see that ki,min is given by:
ki;min
2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3
2
/B 4
8/
¼
1þ
1 1 B 2 5 ;
/B
4
B 1 þ /B
N
ð10Þ
4
where the two dimensionless groups are:
/B ¼
b g^ cos b
^1 q
^ 2ÞD
ðq
;
^
sY
BN ¼
s^Y Db
:
l^ Vb 0
ð11Þ
The first of these is the ratio of axial buoyant stress to the yield
stress of the fluid. Since we have designed our experiments so that
(4) is satisfied, we typically have /B cot b 1. The second parameter BN appears in (2), which states BN 1 for our experimental
design.
Fig. 8 shows contours of 1 ki,min in the /B–BN plane. In this figure the shaded area marks the limit where no static wall layers are
possible. We can interpret 1 ki,min as a dimensionless maximal
2
Strictly speaking, some additional assumptions are needed to ensure that the
second term on the right-hand side above is not too large, or we may have a buoyancy
driven flow backwards against the mean flow. These assumptions are anyway met by
our experimental conditions.
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S.M. Taghavi et al. / Journal of Non-Newtonian Fluid Mechanics 167–168 (2012) 59–74
the Newtonian fluid. At the pipe wall, w(z, y) = 0. Both the shear
stress and velocity are continuous at the interface. Finally, since
the flow rate is fixed the following flow rate constraint is satisfied:
Z
X1
Fig. 8. Contours of the maximal static layer thickness (1 ki,min), in the BN–/B
plane. The shaded area marks the limit where no static wall layers are possible.
static layer thickness (scaled with the pipe radius). For our range of
experiments (/B 1 and BN 1), we have the approximation:
ki;min 2
/
24/B
1 B
þ BN
4
BN
ð12Þ
Thus, the maximal layers in our experimental range are always predicted to be close to 1, i.e. a very thin radial channel down the centre of the pipe. This is very far from what we observe: the actual
layer thicknesses (residual volume fractions) are far less than the
maximal possible. This is not unexpected and has been observed
in other geometries, e.g. [2,21,53]. The cause of the over-prediction
of the residual layer is the flow at the displacement front is threedimensional, so that the above analysis is only valid behind the
front. In this three-dimensional frontal region inertial stresses are
significant.
3.3. Axial flow computations
To gain insight into the observed phenomena, and in particular
to give evidence at the computational level that residual layers are
indeed static, we compute the flow in a simplified configuration,
assuming the flow to be steady and uniaxial along the pipe. The
cross-section of the pipe is assumed to be divided into two domains: X1 (for the displacing fluid) and X2 (for the displaced fluid).
b 0,
We scale the axial velocities with the mean imposed velocity V
b and adopt a stress-scale l
b based on the visb 0 = D,
^V
lengths with D
cous shear stress in the Newtonian fluid. The scaled velocity
w(z, y) satisfies the following problem:
@2w @2w
þ 2 ; ðz; yÞ 2 X1 ;
@z2
@y
@
@
b f ¼ sxz þ sxy ; ðz; yÞ 2 X2 ;
@z
@y
BN
jrwj; () jðsxz ; sxy Þj > BN ;
ðsxz ; sxy Þ ¼ jjrwjn1 þ
jrwj
f ¼
ðz; yÞ 2 X2 ;
jrwj ¼ 0; () jðsxz ; sxy Þj 6 BN ;
ð13Þ
ð16Þ
b n1 =ðl
b n1 Þ is a dimensionless consistency, f is the
^V
^D
where j ¼ j
0
dimensionless modified pressure gradient and b > 0:
b ¼ /B BN ¼
wðz; yÞdzdy ¼
X2
p
4
:
ð17Þ
This constraint is used to find f iteratively.
Problems of this nature are considered theoretically in [19] and
computationally in [35]. The computations shown below and later
in the paper have been computed using a finite element method,
as described in detail in [35]. The numerical code solves the flow
problem for a general uniaxial flows of two Herschel–Bulkley fluids
(although in our case one fluid is always Newtonian). The algorithm
used is the augmented Lagrangian algorithm (ALG2) of Glowinski
and co-workers (see [16,23,24]), with slight modifications.
We present two example computations, for the same experimental parameters as in Fig. 4. Using the mean concentration from
the experiment we have defined the area fraction of the Newtonian
displacing fluid and then computed the velocity and stress field,
assuming that the interface is circular. In Fig. 9a and b, the interface is concentric and the velocity solution can be validated against
the analytical solution (easily obtained). In Fig. 9c and d we have
eccentered the interface towards the top of the pipe. In this figure,
the eccentricity e is defined as the distance between the centre of
the circular Newtonian fluid region and the centre of the circular
pipe, divided by the difference in radii of these two circles.
The advantage of using the augmented Lagrangian method for
this type of problem, with a yield stress fluid, is that below the
yield stress the strain rate is exactly zero. We see that the residual
layers in Fig. 4a and c are unyielded and have zero velocity. The
shear stresses increase radially outwards from the centre of the
Newtonian fluid domain, even when eccentric, and the maximal
shear stresses are found at the wall. The effects of (axial) buoyancy
are minimal here. The jump in shear stress gradient is barely perceptible across the interface. In the eccentric case we have slightly
larger stresses at the wall, but in any case these maximal stresses
are far below the yield stress (expressed dimensionlessly by
BN = 550.7). When the yield stress is so much larger than the wall
shear stresses generated, the axial flow of the inner fluid 1 is completely decoupled from that of fluid 2, i.e. fluid 2 is simply a solid.
3.4. Slump-type displacements
At the beginning of Section 3 we presented the main finding of
our study, namely that central and slump displacement regimes
are sharply delineated by the value of Re/Fr. The interesting feature
of this parameter is that it is independent of the imposed flow rate
b 0 ). Although this is the case, we do in fact see significant
(i.e. V
b 0 is increased. We describe
qualitative changes in the flow as V
these changes below, via examination of the results of two experb 0.
imental sequences of increasing V
ð14Þ
ð15Þ
ðz; yÞ 2 X2 :
S
b 2 g^ cos b 2At D
b 2 g^ cos b
^1 q
^ 2ÞD
ðq
¼
;
l^ Vb 0
m^ Vb 0
^ is defined using the mean
which is a buoyancy parameter. Here m
^ ¼ ðq
^1 þ q
^ 2 Þ=2 of the two fluids and the viscosity l
^ of
density q
3.4.1. Sequence 1: b = 85°, At = 102, Carbopol C
In essentially all of our slump-type displacements the displacing fluid was observed to propagate along the bottom of the pipe
in a fairly thin but fast moving layer. A significantly slower second
front followed, typically with a much thicker layer of displacing
fluid and an orientation that was (at least initially) approximately
perpendicular to the pipe axis. Fig. 10 shows the velocity of the two
b 0 within this sequence. In this sefronts for different imposed V
quence, neither front stops during the time of the experiment.
At lower flow rates the second displacement front moves very
slowly. Fig. 11a illustrates such a displacement. The initially perpendicular second front becomes progressively sloped as the displacement continues, but still moves very slowly. Presumably
S.M. Taghavi et al. / Journal of Non-Newtonian Fluid Mechanics 167–168 (2012) 59–74
67
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Fig. 9. 2D computational results with the parameters of the experiment shown in Fig. 4 (BN = 550.7, /B = 0.1286): (a) w(z, y); (b) s2xz þ s2xy ; (c) w(z, y); (d) s2xz þ s2xy . The
interface has a circular shape and is assumed concentric in a and b, eccentric in c and d (e 0.41 towards the upper wall). The eccentricity e is defined as the distance between
center of the Newtonian fluid flow and the center of the pipe, divided by the difference in radii. Broken white lines indicate the interface.
b f with V
b 0 for a sequence of
Fig. 10. Variation of measured front velocity V
experiments with Carbopol solution C at b = 85° and At = 102; – leading front,
– second front.
inertial forces are relatively weak here and the buoyancy gradients
are weakened as the front elongates, i.e. there is some form of slow
relaxation.
We see that the interface between fluids appears to slowly
move upwards, i.e. the residual layer is thinned over time. In
Fig. 11b we show the velocity contours from the UDV system for
the same experiment, superimposing also the interface height
(estimated from the measured mean concentration). We observe
that the thick layer towards the top of the pipe is static. This is
not surprising: the flow and interface are pseudo parallel and since
BN 1 we expect the viscous stresses (transmitted across the
interface) to be insufficient to yield the upper layer.
Curious however is that the interface still moves and the upper
layer thins. We can only speculate that this might be due to erosion
from the interface over time. We can see that the initial velocity in
the lower layer is quite high and only relaxes later as the residual
layer thins. The effective Reynolds number in the lower layer is of
order 103 so that we might expect some unsteadiness in the flow,
as is suggested by the UDV. Unsteady interfacial stress fluctuations
could weaken the gelled upper layer allowing mixing and dilution
of the Carbopol solution.
b 0 in Fig. 12. We
Two further examples are shown at increased V
show again snapshots of the displacement and also the UDV contours. The most notable difference is that here the second interface moves steadily along at a significant speed. Although there
is evidence of some residual fluid near the top of the pipe, behind
the second front, the displacement is quite effective. In the early
part of the experiment, when only the fast moving lower front
has passed the UDV position, we observe that the upper residual
layer is apparently moving in a plug like fashion with decay in
velocity close to the upper wall. Since all the flow does not pass
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S.M. Taghavi et al. / Journal of Non-Newtonian Fluid Mechanics 167–168 (2012) 59–74
b 0 ¼ 26 mm s1 : (a) a sequence of snapshots showing a 990 (mm) long section of the pipe a few centimeters
Fig. 11. Displacement of Carbopol C for b = 85°, At = 102 at V
below the gate valve at ^t ¼ 30; 60; . . . ; 570; s600 after opening the gate valve; (b) contours of velocity from the UDV at 80 cm below the gate valve: readings taken through the
pipe centreline in a vertical section. The vertical axis shows depth measured from the top of the pipe. Velocity data is averaged time-wise over 25 velocity profiles (3 s), but no
spatial averaging/filtering is applied. The broken line illustrates the depth of the interface, as inferred from the normalized concentration across the pipe at the UDV position.
b 0 ¼ 42 mm s1 ; (c and d) show data for V
b 0 ¼ 57 mm s1 . (a) Snapshots of the
Fig. 12. Displacement of Carbopol solution C for b = 85°, At = 102: (a and b) show data for V
displacement at ^t ¼ 1; 3; . . . ; 31; 33 s in a 990 mm long section of the pipe a few centimeters below the gate valve. (b) UDV contours and superimposed interface position
estimated from the mean concentration. (c) Snapshots of the displacement at ^t ¼ 1; 2; . . . ; 16; 17 s in a 990 mm long section of the pipe a few centimeters below the gate valve.
(d) UDV contours and superimposed interface position estimated from the mean concentration.
in the lower layer with the first front, it is necessary that the
upper layer moves, in order to conserve mass. As the second front
passes the upper residual layer is thinned considerably, but now
becomes static. The entire imposed flow rate is now passing below the interface. At the larger flow rate the final residual layer
is thinned.
S.M. Taghavi et al. / Journal of Non-Newtonian Fluid Mechanics 167–168 (2012) 59–74
3.4.2. Sequence 2: b = 85°, At = 1.6 102, Carbopol B
The second sequence we examine is for Carbopol B, and with a
slightly larger density difference. The main interesting feature of
this sequence, by comparison to the previous one is that although
there are two fronts, the second front stops at long times. It
appears that there is some form of relaxation of the stresses close
to the second front, as the displacement progresses. Possibly the
buoyant stresses diminish as the interface slope changes. A second
possibility is that the initial front channels through making a sufficiently wide channel to accommodate the entire flow rate without
yielding the upper layer. This then would have the effect of reducing the inertial stresses close to the second front. We have seen in
the central displacements that at large BN the maximal static layer
thickness is very large. Below we shall perform a similar calculation for slump-like configurations.
An example of a slump-like displacement where the second
front stops is shown below in Fig. 13. The deceleration of the second front is evident in the images of Fig. 13a. For this displacement
the second front does not reach the UDV position before stopping.
In Fig. 13b the velocity contours therefore show the mobile lower
layer only, which is increasing gradually in thickness over this time
frame. We observe as the layer expands the maximum velocity decreases slightly. Spatiotemporal diagrams shown in Fig. 13c and d
indicate that the second front, although static, has displaced all the
Carbopol. We can see that further down the pipe there are light
69
spots in the spatiotemporal diagram. These indicate unsteadiness
in the flow, which we discuss later. However, we also observe that
the concentration lines in spatiotemporal diagram becomes progressively vertical as the displacement progresses, suggesting that
the residual layer is static.
Qualitatively similar displacements are found at other flow
b 0 Þ in this sequence. As the velocity is increased we observe
rates ð V
that the initial front speed also increases and that the depth of static layer (say dstatic, measured from the top of the pipe), also decreases. These effects are shown in Fig. 14. We also find that the
distance that the second front travels before stopping increases
b 0 . These effects are perhaps intuitive in that larger displacewith V
ment velocities generally give rise to larger stresses.
In Fig. 15 we show the axial flow solutions for the final static
layer depths of two of the experiments in the sequence of
Fig. 14. We have assumed a perfectly horizontal interface of the
same height as that in the experiments and computed the velocity
and stress profiles. We observe firstly that in both cases the computed solutions are indeed static. However, compared to our earlier computations with the circular interface, the stresses induced
in the static layer appear significantly larger. Part of this is an effect
of restricted flow area, but part is due to the interface shape. Note
in particular the high stresses in the upper layer close to the interface at each side wall. This suggests that, for equal flow areas, the
stratified configuration will yield before the central configuration.
b 0 ¼ 30mm.s1, b = 85°, At = 1.6 102, Carbopol solution B). (a) Images at
Fig. 13. An example of slump-like displacement for which the second front stops; ( V
^t ¼ 2; 6; . . . ; 62; 66 s after opening the gate valve for a 990 (mm) long section of the pipe a few centimeters below the gate valve. (b) Velocity contours from the UDV
measurement, at 80 (cm) below the gate valve. The normalized concentration across the pipe is interpreted as an interface height (shown by the broken line). (c and d)
Spatiotemporal diagrams, both close to the gate valve and further downstream.
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S.M. Taghavi et al. / Journal of Non-Newtonian Fluid Mechanics 167–168 (2012) 59–74
Fig. 14. (a) Normalized static layer depth dstatic; (b) Leading front velocity: (b = 85°, At = 1.6 102, Carbopol B). The layer depth is averaged over a 37 (mm) section at 2407
(mm) along the pipe, late in the experiment.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Fig. 15. 2Dqcomputational
solution with a horizontal interface from the sequence of Fig. 14 (b = 85°, At = 1.6 102 with Carbopol solution B): (a) w(z,y); (b) s2xz þ s2xy ; (c)
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
1
b 0 ¼ 23 mm s ; BN ¼ 2497:2; / ¼ 0:1751; dstatic ¼ 0:76.
b 0 ¼ 61mm.s , BN = 960.3, /B = 0.1751, dstatic = 0.57; for figures c and d: V
w(z,y); (d) s2xz þ s2xy . For figures a and b: V
B
We should note that in the case that the upper fluid is static, the
stress field computed is simply an admissible candidate stress
field.
By iterating the 2D computational solution we may compute
the maximal depth of the static layer dmax (measured down from
the top of the pipe, scaled with the pipe diameter). We have com-
puted this for three values of /B that span the range of our experiments; see Fig. 16a. To interpret this figure, at fixed layer depth d
the layer is static for any BN above the curve, or alternatively for
any fixed BN the layer is static for any depth d 6 dmax. We observe
a slight increase in dmax with /B. Note that /B represents a buoyancy force upwards along the pipe, opposing the mean flow and
S.M. Taghavi et al. / Journal of Non-Newtonian Fluid Mechanics 167–168 (2012) 59–74
71
Fig. 16. (a) Maximal static layer depth dmax measured from the top of the pipe, computed iteratively from the 2D computational solution: /B = 0.01 h; /B = 0.1 e;
/B = 0.5 . (b) Fraction of total flow rate flowing in the lower layer as a function of layer depth d and BN, all computed at fixed buoyancy parameter, b = BN/B = 250.
b 0 ¼ 36 mm s1 with Carbopol solution C. (a) From top to bottom we show images for
Fig. 17. Unsteady slump-like displacement for b ¼ 85 ; At ¼ 102 ; V
^t ¼ 1; 2:5; 4; . . . ; 11:5; 13; 14:5 s after opening the gate vale. The figure shows a 990 mm long section of the pipe a few centimeters below the gate valve. The last image at
the bottom is the colourbar of the concentration values. (b) Spatio-temporal diagram for the same experiment. (c) Velocity contours as measured by the UDV, situated at
80 cm below the gate valve.
thus reduces the shear stress in the upper layer, increasing the
layer thickness. However, for larger values of /B this buoyancy
force would eventually yield the upper layer, moving it backwards
against the mean flow. This also happens in the concentric interface case considered earlier. In Fig. 16b we have numerically computed the flow rate through the displacing fluid layer (expressed as
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S.M. Taghavi et al. / Journal of Non-Newtonian Fluid Mechanics 167–168 (2012) 59–74
a fraction of the total flow rate), for large BN and for a range of layer
depths d. For these computations we have fixed the buoyancy
parameter, b = BN/B = 250 (roughly an upper bound for our experimental range). We can see that even when the displaced fluid
layer does yield, the fraction of flow rate in the upper layer is relatively small (0–25%).
3.4.3. Unsteady displacements
Within the slump-like displacements there were a number of
experiments that gave rise to distinctly unsteady flows. Typically
this unsteady behaviour was found in the layer close to the base
of the pipe, where the first front is displacing. This type of flow is
characterised in less extreme cases by the occurrence of irregularly
shaped regions of displaced Carbopol above the narrow lower
layer. We have already observed this type of effect (e.g. see
Fig. 12a and c (quite faint), and Fig. 13a).
In more extreme cases the unsteady flow can leave the lower
part of the pipe and channel through in a disorderly fashion. An
example of this is shown in Fig. 17a, with spatiotemporal plot in
Fig. 17b and UDV data in Fig. 17c. In this flow the front channels
through the Carbopol initially higher up in the pipe, before settling
down again to the lower part of the pipe. We can observe from the
spatiotemporal plot that after the front has passed by any fixed position in the pipe the spatiotemporal pattern becomes largely stationary in time, indicating that the residual layers and shape of
yielded regions are static even though quite irregular. The UDV signal indicates temporal fluctuations in the velocity field.
What is interesting about this type of flow is that we have observed some of the most unsteady flows at intermediate flow rates.
We are uncertain of the causes of this type of flow, but make the
following comments. Firstly, the apparent bias towards the lower
part of the channel suggests that buoyancy is perhaps important
in these flows, exerting a stabilizing influence on the orientation
of the channel. Secondly, we suspect that the yield stress fluid is
largely passive in determining the direction of the propagating
front. We are in the regime BN 1 for which it is possible for a
Newtonian fluid to channel through a pipe in any variety of shaped
channels, leaving the outer fluid static. To illustrate this we have
taken the experimental values from Fig. 17 and computed from
the mean concentration a representative area fraction corresponding to the channeling Newtonian fluid. In Fig. 18 we show the computed 2D axial velocity for a selection of different channel crosssection shapes. Although the stress field is different in these different cases, in no case are we close to yielding the outer fluid.
Thirdly, although our experiments are conducted at significant
Re, these Re values are below transition for a single fluid pipe flow.
However, this led us to speculate that the effective Reynolds number could be much larger within the narrow channel formed at the
base of the pipe as the first front propagates. To explore this we
have calculated a relative hydraulic Reynolds number Reh for three
sets of experiments in slump type displacement regime at b = 85°.
Fig. 18. Velocity profiles, w(z,y), obtained though 2 D computation with parameters of the experiment shown in Fig. 17, (BN = 3405.4, /B = 0.04906). (a) an eccentric circular
interface with e 0.48; (b) a centred square interface; (c) a plane interface. In all plots the area fractions are equal, corresponding approximately to a channel area fraction
based on the mean concentration in Fig. 17. Broken white lines indicate the interface.
S.M. Taghavi et al. / Journal of Non-Newtonian Fluid Mechanics 167–168 (2012) 59–74
Fig. 19. Variation of Reh versus Re for three sets of experiments in slump type
displacement at b = 85°. The data correspond to experiments with At = 1.6 102
and Carbopol solution B ( ), At = 1 102 and Carbopol solution C (j) and
At = 1 102 and Carbopol solution B ( ).
This Reynolds number is based on the displacing fluid properties
and on the mean velocity and hydraulic diameter of the lower
layer. The term relative reflects the fact that not all the displacing
fluid is flowing along this initial channel.
For slumping displacement flows there usually exist two fronts
b f 1 and V
b f 2 , where we take V
bf1 > V
b f 2 . We denote by
with speeds V
^1 and h
^ 2 the corresponding averaged heights at which these fronts
h
^2 Þ for the slow front is
propagate. The averaged interface height ðh
relatively thick and can be computed with higher accuracy than
^1 Þ, which is typically very thin. Therefore, we comthe fast front ðh
^ 1 though a control volume method, using h
^2 ; V
b f 1, V
b f 2 and
pute h
^2 ; V
b f 2 and V
b 0 can be obtained with
b 0 . Although the values of h
V
b f 1 from the spatiotemporal conacceptable accuracy, estimation of V
^1
tains is less accurate, so that significant errors in our estimate of h
^1 and assuming a horizontal intermust be acknowledged. Using h
face, the area and perimeter of the fast moving channel is estimated
and Reh is computed. The results are shown in Fig. 19.
Fig. 19 indicates that Reh > 1000 is quite common in the propagating channel, in each of these three experimental sequences. As
is evident from the spatiotemporal plots that we have seen earlier,
we have significant long-wave axial variation in the geometry of
the displacing fluid channel. At these high Reh and with a varying
geometry we speculate that the flow is quite possibly turbulent
or transitional. Potentially this is the cause of the erratic direction
taken by the advancing front. We have seen earlier that the displacing fluid seems to intermittently rupture the Carbopol, opening
up into a larger irregular channel. This larger channel will evidently slow the flow, perhaps re-laminarising locally. Although
speculative, we feel this is a possible stabilising influence, countering the transitional/turbulent flow regions in the narrower channels, and that overall this is a plausible propagation mechanism
for unsteady displacements.
On the other hand we can note some anomalies. Considering for
example the sequence at At = 1 102 and Carbopol solution C,
shown in Fig. 19, we can see that Reh increases with Re. On the
other hand, the lower flow rates (hence Re) appear more unstable
than the higher flow rates. Compare Figs. 12 and 17. At the higher
flow rates the secondary fronts in Fig. 12 appear to propagate fairly
rapidly and stably. Although there is an initial very fast precursor
front at the bottom of the pipe, it may well be that this front exits
our experimental apparatus before any instability is observed.
4. Summary
We have explored displacement flows of a yield stress fluid by a
Newtonian fluid, with a laminar imposed flow along a long pipe in-
73
clined close to horizontal. The study has been focused at the regime where the yield stress is far larger than the characteristic
viscous stress and, although laminar, the Reynolds numbers are
significant. We have observed two distinct flow regimes: a central-type regime where the displacing fluid propagates in a finger
along the centre of the pipe, and a slumping configuration, where
the displacing fluid moves along the bottom of the pipe. The transition between these two characteristic flow types appears to occur
at a critical ratio of the Reynolds number to the densimetric Froude
number, and has been found largely independent of other dimensionless groups.
In both regimes, due to the large yield stress, we find residual
layers of displaced fluid present at long times. Both our UDV measurements and computations from a simplified axial flow model,
suggest that these residual layers are fully static. In each case we
see slow axial variation in layer thickness along the pipe axis and
it appears that the residual layers are significantly thinner than
the computed maximal static layers.
At larger displacement flow rates we have generally seen a decrease in the residual layer thickness. For the central displacements the flows became progressively unstable as the flow rate
increases. For the slump like displacements we have observed a
number of different evolutions. Firstly, there are typically two
fronts: a rapidly propagating lower front along the bottom of the
pipe, followed by a slower front that displaces a larger fraction of
the Carbopol. The second front may in some cases stop completely.
The first front has been observed to destabilise and propagate
erratically along the pipe.
Acknowledgements
This research has been carried out at the University of British
Columbia, supported financially by NSERC and Schlumberger
through CRD Project 354716-07.
References
[1] A.N. Alexandrou, V. Entov, On the steady-state advancement of fingers and
bubbles in a Hele-Shaw cell filled by a non-Newtonian fluid, Euro. J. Appl.
Math. 8 (1997) 73–87.
[2] M. Allouche, I.A. Frigaard, G. Sona, Static wall layers in the displacement of two
visco-plastic fluids in a plane channel, J. Fluid Mech. 424 (2000) 243–277.
[3] S.H. Bittleston, J. Ferguson, I.A. Frigaard, Mud removal and cement placement
during primary cementing of an oil well; laminar non-Newtonian
displacements in an eccentric Hele-Shaw cell, J. Eng. Math. 43 (2002) 229–253.
[4] B. Brunone, A. Berni, Wall shear stress in transient turbulent pipe flow by local
velocity measurement, J. Hydraul. Eng. – ASCE 136 (10) (2010) 716–726.
[5] C. Chang, Q.D. Nguyen, H.P. Ronningsen, Isothermal start-up of a pipeline
transporting waxy crude oil, J. Non-Newton. Fluid Mech. 87 (1999) 127–154.
[6] G.K. Christian, P.J. Fryer, The effect of pulsing cleaning chemicals on the
cleaning of whey protein deposits, Trans. IChemE C 84 (2006) 320–328.
[7] N.G. Cogan, J.P. Keener, Channel formation in gels, SIAM J. Appl. Math. 65 (6)
(2005) 1839–1854.
[8] P.A. Cole, K. Asteriadou, P.T. Robbins, E.G. Owen, G.A. Montague, P.J. Fryer,
Comparison of cleaning of toothpaste from surfaces and pilot scale pipework,
Food Bioprod. Process. 88 (2010) 392–400.
[9] P. Coussot, Saffman–Taylor instability in yield-stress fluids, J. Fluid Mech. 380
(1999) 363–376.
[10] Guillot D.J. Hendriks H. Couturier, M. and F. Callet, Design rules and associated
spacer properties for optimal mud removal in eccentric annuli, Soc. Pet. Eng.
Paper Number SPE 21594, 1990.
[11] J.P. Crawshaw and I.A. Frigaard, Cement plugs: Stability and failure by
buoyancy-driven mechanism, Soc. Pet. Eng. Paper Number SPE 56959, 1999.
[12] M.R. Davidson, Q.D. Nguyen, C. Chang, H.P. Ronningsen, A model for restart of a
pipeline with compressible gelled waxy crude oil, J. Non-Newton. Fluid Mech.
123 (2004) 269–280.
[13] P.R. de Souza Mendes, E.S.S. Dutra, J.R.R. Siffert, M.F. Naccache, Gas
displacement of viscoplastic liquids in capillary tubes, J. Non-Newton. Fluid
Mech. 145 (1) (2007) 30–40.
[14] Y. Dimakopoulos, J. Tsamopoulos, Transient displacement of a viscoplastic
material by air in straight and suddenly constricted tubes, J. Non-Newton.
Fluid Mech. 112 (1) (2003) 43–75.
[15] Y. Dimakopoulos, J. Tsamopoulos, Transient displacement of Newtonian and
viscoplastic liquids by air in complex tubes, J. Non-Newton. Fluid Mech. 142
(1–3) (2007) 162–182.
74
S.M. Taghavi et al. / Journal of Non-Newtonian Fluid Mechanics 167–168 (2012) 59–74
[16] M. Fortin, R. Glowinski, Augmented Lagrangian Methods, North-Holland, 1983.
[17] I.A. Frigaard, G. Vinay, A. Wachs, Compressible displacement of waxy crude
oils in long pipeline startup flows, J. Non-Newton. Fluid Mech. 147 (1–2)
(2007) 45–64.
[18] I.A. Frigaard, S. Leimgruber, O. Scherzer, Variational methods and maximal
residual wall layers, J. Fluid Mech. 483 (2003) 37–65.
[19] I.A. Frigaard, O. Scherzer, Uniaxial exchange flows of two bingham fluids in a
cylindrical duct, IMA J. Appl. Math. 61 (1998) 237–266.
[20] I.A. Frigaard, O. Scherzer, G. Sona, Uniqueness and non-uniqueness in the
steady displacement of two viscoplastic fluids, ZAMM 81 (2) (2001) 99–118.
[21] C. Gabard, Etude de la stabilité de films liquides sur les parois d’une conduite
verticale lors de l’ecoulement de fluides miscibles non-newtoniens. PhD thesis,
These de l’Universite Pierre et Marie Curie, Orsay, France, 2001.
[22] C. Gabard, J.-P. Hulin, Miscible displacements of non-Newtonian fluids in a
vertical tube, Eur. Phys. J. E 11 (2003) 231–241.
[23] R. Glowinski, Numerical Methods for Nonlinear Variational Problems,
Springer-Verlag, 1983.
[24] R. Glowinski, J.L. Lions, R. Tremolieres, Numerical Analysis of Variational
Inequalities, North-Holland, 1981.
[25] I. Grants, C. Zhang, S. Eckert, G. Gerbeth, Experimental observation of swirl
accumulation in a magnetically driven flow, J. Fluid Mech. 616 (2008)
135–152.
[26] S. Hormozi, K. Wielage-Burchard, I.A. Frigaard, Entry, start up and stability
effects in visco-plastically lubricated pipe flows, J. Fluid Mech. 673 (2011)
432–467.
[27] P.D. Howell, S.L. Waters, J.B. Grotberg, The propagation of a liquid bolus along a
liquid-lined flexible tube, J. Fluid Mech. 406 (2000) 309–335.
[28] C. K Huen, I. A Frigaard, D. M Martinez, Experimental studies of multi-layer
flows using a visco-plastic lubricant, J. Non-Newton. Fluid Mech. 142 (2007)
150–161.
[29] A. Jamôt, Déplacement de la boue par le latier de ciment dans l’espace
annulaire tubage-paroi d’un puits, Revue Assoc. Franc. Techn. Petr. 224 (1974)
27–37.
[30] A. Lindner, L’instabilité de Saffman–Taylor dans les fluides complexes: relation
entre les propriétés rhéologiques et la formations de motifs. PhD thesis, These
de l’Universite Paris VI, Paris, France, 2000.
[31] A. Lindner, P. Coussot, D. Bonn, Viscous fingering in a yield stress fluid, Phys.
Rev. Lett. 85 (2000) 314–317.
[32] S. Malekmohammadi, M.F. Naccache, I.A. Frigaard, D.M. Martinez, Buoyancy
driven slump flows of non-Newtonian fluids in pipes, J. Petr. Sci. Eng. 72 (2010)
236–243.
[33] R.H. McLean, C.W. Manry, W.W. Whitaker, Displacement mechanics in primary
cementing, Soc. Pet. Eng. Paper Number SPE 1488, 1966.
[34] J.A. Moriarty, J.B. Grotberg, Flow-induced instabilities of a mucus-serous
bilayer, J. Fluid Mech. 397 (1999) 1–22.
[35] M.A. Moyers-González, I.A. Frigaard, Numerical solution of duct flows of
multiple visco-plastic fluids, J. Non-Newton. Fluid Mech. 122 (2004) 227–241.
[36] E.B. Nelson, D. Guillot, Well Cementing, second ed., Schlumberger Educational
Services, 2006.
[37] H. Pascal, Dynamics of moving interface in porous media for power law fluids
with a yield stress, Int. J. Eng. Sci. 22 (5) (1984) 577–590.
[38] H. Pascal, Rheological behaviour effect of non-Newtonian fluids on dynamic of
moving interface in porous media, Int. J. Eng. Sci. 22 (3) (1984) 227–241.
[39] H. Pascal, A theoretical analysis of stability of a moving interface in a porous
medium for Bingham displacing fluids and its application in oil displacement
mechanism, Can. J. Chem. Eng. 64 (1986) 375–379.
[40] S. Pelipenko, I.A. Frigaard, Two-dimensional computational simulation of
eccentric annular cementing displacements, J. Fluid Mech. 520 (2004) 343–
377.
[41] S. Pelipenko, I.A. Frigaard, Two-dimensional computational simulation of
eccentric annular cementing displacements, J. Eng. Math. 64 (6) (2004) 557–
583.
[42] J. Sestak, M.E. Charles, M.G. Cawkwell, M. Houska, Start-up of gelled crude oil
pipelines, J. Pipelines 6 (1987) 1524–1532.
[43] D.A. Sousa, E.J. Soares, R.S. Queiroz, R.L. Thompson, Numerical investigation on
gas-displacement of a shear-thinning liquid and a visco-plastic material in
capillary tubes, J. Non-Newton. Fluid Mech. 144 (2007) 149–159.
[44] S.M. Taghavi, K. Alba, T. Seon, K. Wielage-Burchard, D.M. Martinez, I.A.
Frigaard, Miscible displacement flows in near-horizontal ducts at low Atwood
number, J. Fluid Mech., submitted for publication
[45] S.M. Taghavi, T. Seon, D.M. Martinez, I.A. Frigaard, Buoyancy-dominated
displacement flows in near-horizontal channels: the viscous limit, J. Fluid.
Mech. 639 (2009) 1–35.
[46] S.M. Taghavi, T. Seon, D.M. Martinez, I.A. Frigaard, Influence of an imposed
flow on the stability of a gravity current in a near horizontal duct, Phys. Fluids
22 (2010) 031702.
[47] S.M. Taghavi, T. Seon, K. Wielage-Burchard, D.M. Martinez, I.A. Frigaard,
Stationary residual layers in buoyant Newtonian displacement flows, Phys.
Fluids 23 (2011) 044105.
[48] R.L. Thompson, E.J. Soares, R.D.A. Bacchi, Further remarks on numerical
investigation on gas-displacement of a shear-thinning liquid and a viscoplastic material in capillary tubes, J. Non-Newton. Fluid Mech. 165 (2010)
448–452.
[49] G. Vinay, Modélisation du redemarrage des écoulements de bruts parafiniques
dans les conduites pétrolieres. PhD thesis, These de l’Ecole des Mines de Paris,
Paris, France, 2005.
[50] G. Vinay, A. Wachs, J.-F. Agassant, Numerical simulation of non-isothermal
viscoplastic waxy crude oil flows, J. Non-Newton. Fluid Mech. 128 (2–3) (2005)
144–162.
[51] G. Vinay, A. Wachs, J.-F. Agassant, Numerical simulation of weakly
compressible Bingham flows: the restart of pipeline flows of waxy crude
oils, J. Non-Newton. Fluid Mech. 136 (2–3) (2006) 93–105.
[52] O. Wanner, H.J. Eberl, E. Morgenroth, D. Noguera, C. Picioreanu, B.E. Rittmann,
M.C.M. Van Loosdrecht, Mathematical modeling of biofilms, IWA Scientific and
Technical Report Series, vol. 18, IWA Publishing, 2006, pp. 1–199.
ISBN:1843390876.
[53] K. Wielage-Burchard, I.A. Frigaard, Static wall layers in plane channel
displacement flows, J. Non-Newton. Fluid Mech. 166 (2011) 245–261.