An experimental investigation on the Newtonian–Newtonian and viscoplastic–Newtonian displacement in a capillary tube

Accepted Manuscript

An experimental investigation on the Newtonian–Newtonian and viscoplastic–Newtonian displacement in a capillary tube Hilton M. Caliman, Edson J. Soares, Roney L. Thompson PII: DOI: Reference:

S0377-0257(17)30215-X 10.1016/j.jnnfm.2017.08.001 JNNFM 3914

To appear in:

Journal of Non-Newtonian Fluid Mechanics

Received date: Revised date: Accepted date:

15 May 2017 31 July 2017 1 August 2017

Please cite this article as: Hilton M. Caliman, Edson J. Soares, Roney L. Thompson, An experimental investigation on the Newtonian–Newtonian and viscoplastic–Newtonian displacement in a capillary tube, Journal of Non-Newtonian Fluid Mechanics (2017), doi: 10.1016/j.jnnfm.2017.08.001

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Highlights • We experimentally analyse a displacement of a Newtonian liquid by a

CR IP T

second Newtonian liquid or a viscoplastic material; • We use a capillary tube;

• The dimensionless numbers that govern the problem are the capillary number, the viscosity ratio and the plastic number;

AN US

• We found that the displacement efficiency increases with the plastic

AC

CE

PT

ED

M

number.

1

ACCEPTED MANUSCRIPT

Hilton M. Caliman

CR IP T

An experimental investigation on the Newtonian–Newtonian and viscoplastic–Newtonian displacement in a capillary tube LabReo, Department of Mechanical Engineering, Universidade Federal do Esp´ırito Santo, Avenida Fernando Ferrari, 514, Goiabeiras, 29075-910, Vit´ oria, ES, Brazil.

Edson J. Soares

AN US

LabReo, Department of Mechanical Engineering, Universidade Federal do Esp´ırito Santo, Avenida Fernando Ferrari, 514, Goiabeiras, 29075-910, Vit´ oria, ES, Brazil.

Roney L. Thompson

M

COPPE, Department of Mechanical Engineering, Universidade Federal do Rio de Janeiro, Centro de Tecnologia, Ilha do Fundo, 21945-970, Rio de Janeiro, RJ, Brazil.

ED

Abstract

The immiscible displacement of a viscous Newtonian liquid by a second Newtonian liquid or a viscoplastic material in a capillary tube is experimen-

PT

tally analysed in the low inertia regime and very similar mass densities. The dimensionless numbers that govern the problem are the capillary number

CE

(Ca), the viscosity ratio of the displaced to the displacing fluids (Nη ), and the plastic number, P l, a dimensionless normalized yield stress. The tube is

AC

initially filled with a fluid when a second fluid is injected with a flow rate control, displacing the first one. The displacement efficiency of the Newtonian–

Email addresses: [email protected] (Hilton M. Caliman), [email protected] (Edson J. Soares), [email protected] (Roney L. Thompson)

Preprint submitted to Elsevier

August 3, 2017

ACCEPTED MANUSCRIPT

Newtonian case is found to be in good agreement with the numerical data of the literature. Pictures of the flow reveal that for Nη < 2 the radius of the

CR IP T

displacing fluid varies along its length, being thinner near the front of the interface. In addition, within this range, buoyant effects can arise even for low discrepancies in mass density. We found that the displacement efficiency and geometric mass fraction decay with the increase of plastic effects if we define

the viscosity ratio taking into account the yield stress in the viscoplastic ma-

AN US

terial. We also found that plastic effects associated to the injected material

can be responsible for the appearance of irregular (non-parallel) interface with respect to the wall. 1. Introduction

M

Fluid–fluid displacement is an important subject of Fluid Mechanics, which can be found in many applications, such as molding, injection of liq-

ED

uid medicine in blood vessels, oil recovery in porous media, wells cementing process, and many others. Even the laminar version of this problem is pro-

PT

hibitively complex: it can be miscible or immiscible, isothermal or not, with or without buoyancy effects, with or without a dynamic contact line at the

CE

wall, etc.

This problem has its origins in the experimental investigation conducted

AC

by Taylor (1961), where the efficiency of the gas–liquid displacement was plotted against the capillary number. This work was generalized by considering the Newtonian–Newtonian liquid–liquid displacement Goldsmith and Mason (1963); Hodges et al. (2004); Soares and Thompson (2009); Lac and Sherwood (2009); Freitas et al. (2011a); Soares et al. (2015); Hasnain and 3

ACCEPTED MANUSCRIPT

Alba (2017), and the gas displacement of non-Newtonian fluids (Alexandrou and Entov, 1997; Lee et al., 2002; Dimakopoulos and Tsamopoulos, 2003;

CR IP T

Kamisli, 2006; de Souza Mendes et al., 2007; Sousa et al., 2007) or when long bubbles are considered (Zamankhan et al., 2011; Jalaal and Balmforth, 2016). The study of the Saffman-Taylor instability when a gas displaces a viscoplastic material was explored in a number of papers (Coussot, 1999; Lindner et al., 2000; Maleki-Jirsaraei et al., 2005; Eslami and Taghavi, 2017).

AN US

The viscoplastic displacement by a Newtonian fluid is more commonly inves-

tigated in the literature (Alba and Frigaard, 2016; Zare et al., 2016; Mois´es et al., 2016). Another generalization step was achieved by investigating the liquid–liquid displacement when the displacing fluid is non-Newtonian (Freitas et al., 2011b; Thompson and Soares, 2012), or when both (Allouche

M

et al., 2000b; Wielage-Burchard and Frigaard, 2011; Freitas et al., 2013) are non-Newtonian. Buoyancy effects were analysed Taghavi et al. (2009); Al-

ED

louche et al. (2000a); Malekmohammadi et al. (2010) with one viscoplastic fluid and investigated by Taghavi et al. (2012); Alba et al. (2013) when both

PT

fluids were viscoplastic.

In the case of the oil displacement by an injected fluid, one way to inves-

CE

tigate how this process can be optimized is to analyse how the rheological properties of the injected fluid affect the displacement efficiency. Reservoirs which are wetted by the oil have a natural extra-resistance for removing

AC

the oil from the pores, since part of the oil remains attached to the wall as the displacing fluid is injected. In this connection, in the present investigation we focus on the analysis of immiscible fluids where interfacial forces are important, in the laminar regime, where buoyancy and inertial effects are

4

ACCEPTED MANUSCRIPT

negligible. In addition, we are particularly interested in the case where the displaced fluid wets the wall duct and, therefore, a layer of this fluid still

CR IP T

remains attached to the wall. As discussed by Soares et al. (2015), there are two quantities of interest

that can be used in order to grasp the amount of liquid of the displaced fluid that is ‘left behind’: The so-called mass displacement efficiency and the so-

called geometrical residual mass fraction, which coincide in the gas–liquid

AN US

displacement problem, but are different measures in the liquid–liquid one.

The numerical scheme adopted by Soares et al. (2005) and Soares and Thompson (2009) assumed that the flow is not intrinsically transient, i.e. there is an observer that evaluates that this problem can be described in a steady state regime. This observer is the one attached to the tip of the inter-

M

face. It is important to note that Soares and Thompson (2009) reported some difficulties in obtaining convergent results for a viscosity ratio in the range

ED

Nµ < 2, i.e. when the viscosity of the displaced fluid is less than two times the viscosity of the displacing one. Lac and Sherwood (2009) and Soares and

PT

Thompson (2009) have independently shown that the Newtonian–Newtonian displacement problem has two branches of flow patterns, where Nµ = 2 is

CE

the border of the two cases. In particular, it was shown that is impossible to have a by-pass regime when Nµ ≤ 2. The experimental work conducted by Soares et al. (2015) reported the same tendency obtained numerically by

AC

Soares and Thompson (2009) with a quantitative disagreement although they have made an effort to reproduce the conditions of the numerical study and concluded that the reasons for the discrepancy needed further investigation. Another important aspect of the present analysis concerns the dimension-

5

ACCEPTED MANUSCRIPT

less numbers that govern the problem, especially when a viscoplastic fluid is involved. Thompson and Soares (2016) discussed the maintenance in a

CR IP T

viscoplastic flow of the original meaning of a certain dimensionless number conceived in the Newtonian paradigm. The main conclusion is that the yield stress must be taken into account in every dimensionless number that considers viscous effects. This conclusion was drawn by the evidence that the

characteristic viscosity should be representative of the diffusion of momen-

AN US

tum, and the yield stress contributes to this term. This form of construction

of dimensionless viscoplastic numbers was shown to be more effective at collapsing some data into master curves and obtaining a fairer comparison with the Newtonian case, a limit of the non-Newtonian behavior and, therefore, a reference. This methodology requires a more laborious procedure since any

account for viscous effects.

M

change in the flow rate or the fluid redefines the dimensionless numbers that

ED

In connection to what was presented above, the objectives of the present experimental study are: 1) to address the discrepancies between the experi-

PT

mental and numerical results obtained in Soares et al. (2015) and Soares and Thompson (2009) in the Newtonian–Newtonian displacement; 2) to explore

CE

the Nµ < 2 range in the Newtonian–Newtonian displacement problem; 3) to investigate the plastic effects (in the sense described by Thompson and Soares (2016)) of the displacement of a Newtonian fluid by a viscoplastic

AC

material.

Next, we set forth an outline of the sections to come. In Sec. 2, Theoretical

analysis, a dimensionless analysis of the problem is presented and the significance of the output variables are highlighted. Section 3, Experiments, de-

6

ACCEPTED MANUSCRIPT

scribes the experimental setup, the Newtonian fluids employed, the rheological characterization of the viscoplastic fluid, and the experimental procedure.

CR IP T

In the subsequent section, Results, we show Newtonian–Newtonian results of the fraction of mass that remains attached to the wall and some pictures of flow that illustrate the configuration of the interface in a diversity of situ-

ations. A similar set of results is shown for the viscoplastic–Newtonian displacement and a comparison with the corresponding Newtonian–Newtonian

AN US

displacement is provided. Finally, the last section, Conclusions, is dedicated to the conclusions of the present work. 2. Theoretical Analysis

2.1. Dimensionless governing equations

M

Choosing the tube radius, R0 , as the characteristic length and the drop velocity U as the characteristic velocity, together with the properties of Fluid 2

ED

(displaced), we can follow a traditional procedure to deduce the dimensionless forms of the governing equations, i.e. continuity, momentum balance, interface impermeability, and interface balance.

PT

The governing equations can be non-dimensionalized using appropriate characteristic velocity and length scales, the front velocity U , and the radius

CE

of the capillary tube R0 , respectively, together with a characteristic stress based on the displaced (Newtonian) fluid, µ2 U/R0 . The conservation of mass

AC

equation assuming incompressibility is simply given by ∇∗ · u∗k = 0.

(1)

As will be shown, the SMD constitutive equation (de Souza Mendes and Dutra, 2004) was able to fit quite well the flow curve of the viscoplastic 7

ACCEPTED MANUSCRIPT

(2)

CR IP T

material considered in the present study. This equation is given by    ηo γ˙ τ = 1 − exp − (τy + K γ˙ n ) , τy

where τy is an apparent yield stress, ηo is the high viscosity in the limit where γ˙ → 0, K is the consistency index and n is the exponent index. Following the

guidelines adopted by Thompson and Soares (2016), where the yield stress

is present in the characteristic viscosity of any viscoplastic material, we can

Nη =

AN US

define the viscosity ratio, Nη , by

τy RU0

and the plastic number by

τy  n . τy + K RU0

(3)

(4)

M

Pl =

µ2  n−1 , + K RU0

ED

We define the following dimensionless quantities labeled with a superscript ∗

PT

∇∗ ≡ R0 ∇; p∗k ≡

pk R0 ; µ2 U

γ˙ 1∗ ≡

γ˙ 1 R0 ; U

D∗k ≡

2R0 Dk . U

(5)

CE

where pk is the mechanical pressure in fluid k, Dk is the symmetric part of the p velocity gradient corresponding to fluid k, and γ˙ 1 ≡ 0.5tr(2D1 )2 . There-

fore, the dimensionless forms of the stresses in the displaced and displacing

AC

fluids are given by (using γ˙ c ≡ U/R0 for convenience) 2µ2 D2 τ ∗2 = = D∗2 τc µ2 U/R0 h  i
2 1 − exp − η0τyγ˙ 1 τy /γ˙ 1 + K γ˙ 1n−1 D1 τ ∗2 =

τ ∗1 =

τ ∗1 = τc

µ2 U/R0

8

(6)

ACCEPTED MANUSCRIPT



    η0 γ˙ 1 τy K γ˙ 1n−1 = 1 − exp − + D∗1 τy µ2 γ˙ 1 µ2

(7)

τy τy + K γ˙ cn Pl τy = = ∗ n µ2 γ˙ 1 τy + K γ˙ c µ2 γ˙ 1 γ˙ c Nη γ˙ 1∗

(8)

CR IP T

Since

and

K γ˙ cn τy + K γ˙ cn ∗n−1 1 − P l ∗n−1 K γ˙ 1n−1 K γ˙ cn ∗n−1 = = , = γ˙ 1 γ˙ 1 γ˙ µ2 µ2 γ˙ c τy + K γ˙ cn µ2 γ˙ c Nη 1

AN US

the dimensionless stress tensor in fluid 1 is given by   ∗    1 η0 Nη γ˙ 1∗ Pl ∗ ∗n−1 τ1 = 1 − exp − + (1 − P l) γ˙ 1 D∗1 ∗ Nη Pl γ˙ 1

(9)

(10)

Therefore, the momentum equations in dimensionless form, assuming an

M

inertialess flow and the same mass densities, are given by   ∗     Pl η0 Nη γ˙ 1∗ 1 ∗ ∗n−1 ∗ ∗ + (1 − P l) γ˙ 1 ∇ · 1 − exp − D∗1 = 0 −∇ p1 + Nη Pl γ˙ 1∗ (11)

ED

and

−∇∗ p∗2 + ∇∗ · D∗2 = 0.

(12)

PT

In Eq. (11) there also appear the viscosity ratio, Nη , and the plastic number P l. When the yield stress vanishes, and so when P l = 0, the power-

CE

law-Newtonian displacement explored by Thompson and Soares (2012) is recovered. If in addition n = 1, the Newtonian–Newtonian liquid displace-

AC

ment, investigated by Soares and Thompson (2009), is recovered. The continuity at the interface states that there is no mass crossing this

surface and that the tangential velocity is continuous through it. Therefore, it is given by v2∗ = v1∗ = vt∗ , 9

(13)

ACCEPTED MANUSCRIPT

and the force balance at the interface states that there is a normal stress jump due to capillary effects and that tangential stresses are continuous. In

n · (T2 − T1 ) =

σ Rm

CR IP T

dimension form, this condition is given by (14)

where Tk = −pk + 2ηk Dk is the total stress at fluid k. Dividing this equa-

tion by the characteristic stress, we find the dimensionless form of the force

n (p∗1

−

AN US

balance at the interface, which is given by

   ∗     1 η0 Nη γ˙ 1∗ Pl 1 1 ∗ ∗n−1 + (1 − P l) γ˙ 1 D2 − 1 − exp − D∗1 n, = ∗ ∗ Nη Pl γ˙ 1 Ca Rm (15)

p∗2 )+n·

∗ = Rm /R0 is the dimensionless mean radius of currespectively, where Rm

M

vature and Ca is the capillary number based on the displaced fluid, defined

ED

by

Ca =

µ2 U . σ

(16)

2.2. Geometric mass fraction and displacement efficiency

PT

The geometric mass fraction, mg , and the displacement efficiency, me , were defined in Soares et al. (2015). Their definitions are repeated here

CE

for convenience. The geometric mass fraction is the fraction of mass that remains attached to the wall in the case of a uniform layer of the displaced

AC

fluid. Hence, mg is given by Rb2 mg = 1 − 2 Ro

(17)

where Rb is the radius of the injected fluid. The displacement efficiency, me , represents the fraction of mass that is not recovered by the injection of the 10

ACCEPTED MANUSCRIPT

displacing fluid, and is given by me = 1 −

U¯2 , U

(18)

CR IP T

where U¯2 is the mean velocity of the displaced fluid. In the gas-displacement problem, these two quantities coincide, i.e. mg = me . 3. Experiments 3.1. Experimental setup

AN US

In what follows, we describe the elements of the setup. At the beginning of the line, there is a Cole–Parmer syringe pump (2) filled with the displacing

fluid. This syringe is controlled by software in a computer (1). The syringe is connected to a capillary tube made of glass (6), 150 cm in length, 6 mm in

M

external diameter, and a 2-mm internal diameter. This tube is sustained in three metallic sticks by means of hooks (3). A plexiglass box (5) of 4x4x10

ED

cm is placed somewhere in the tube domain and a Nikon photograph camera (4) is placed perpendicularly to the box. A ruler (7), 150 cm in length, marks the position of the moving front. A B-TEC 2200 Tecnal balance (8) is used to

PT

weigh the fluid, which is collected in a glass beaker (9). A plexiglass cylinder reservoir (11) stores the displaced fluid and is connected to the flow line by

CE

a valve (10) at the end. Before the start of the experiment, the capillary tube (6) is cleaned and

AC

dried and then horizontally positioned as in the schematic figure. The horizontal configuration is guaranteed by means of a leveler. The plexiglass box (5) is empty at this stage and is inserted 35 cm from the beginning of the tube. After that, a flexible tube is used to connect the reservoir (11) with the other extremity (B) and the valve is opened, allowing the flow of the 11

ACCEPTED MANUSCRIPT

displaced fluid inside the tube until the complete filing of its volume, with a small overflow in extremity (A). At this moment, valve (10) is closed. After

CR IP T

that, the syringe pump (2) is connected to the extremity (A) of the tube and the hose at extremity (B) is removed. The plexiglass box (5) is filled with glycerine, which has a refraction index similar to that of glass, in order

to improve the quality of the captured images by avoiding refraction of the light. At this point, the camera (4) is positioned perpendicularly to the box

AN US

(5) and its focus is adjusted.

After setting the flow rate of the pump (2) in the computer (1), the flow is initiated. In order to avoid edge effects, the first 25 cm are neglected. To this end, the position of the moving front of the displacing fluid is followed with the aid of the ruler (7). When this front achieves the desired position,

M

the beaker (9) is placed at the end of the tube (B) initiating the computation of the quantity of mass which is recovered. The beaker (9) is removed when

ED

the moving front achieves the position 125 cm by the ruler (7), neglecting the last 25 cm also. The beaker filled with the displaced fluid is weighed.

(4).

PT

During this period of time, images are continuously captured by the camera

CE

3.2. Newtonian fluids

In order to achieve a final composition where the desired properties are

AC

obtained, we have prepared two sets of solutions, water based and oil based. To the water based solutions we added ethanol and polyethylene-glycol (PEG, Mv =8,000 g/mol), with the compositions displayed in Table 1. The Newtonian solutions were mixed in a magnetic Nalgon HOTLAB II

mixer to the point where they become homogeneous. Then their properties 12

AN US

CR IP T

ACCEPTED MANUSCRIPT

Figure 1: Scheme of the experimental setup.

% Ethanol % PEG µ [m.Pa.s] ρ [kg/m3 ] 16.6 41.7 103.3 1035.8 31 0 2.4 971.5 51.25 27.25 30 944.8

M

% Water 41.7 69 20.5

ED

Sample Water A Water B Water C

PT

Table 1: Three water-based solutions with their viscosities and mass densities.

AC

CE

Sample % Castor bean oil Castor bean oil 100 Soybean oil 0 Oil X 73.7 Oil Y 24 Oil Z 49.4

% Soybean oil 0 100 26.3 76 50.6

µ [m.Pa.s] ρ [kg/m3 ] 961.5 966.5 61 924.3 410.4 955.7 102.7 933.3 205.6 944.6

Table 2: The oil-based solutions with their viscosities and mass densities.

13

ACCEPTED MANUSCRIPT

CR IP T

System Displacing Fluid Displaced Fluid Nµ Nρ σ[mN/m] S1 Water B Castor bean oil 397 0.995 16.41 S2 Water A Oil X 3.97 0.923 6.75 S3 Water C Soybean oil 2.03 0.978 5.85 S4 Water A Oil Y 0.99 0.901 6.94 S5 Oil Z Water A 0.5 1.09 6.91

Table 3: The employed pair of fluids: viscosity ratio (Nµ ), mass density ratio (Nρ ), and interfacial tension (σ). These quantities are calculated with respect to the displaced fluid.

AN US

were measured. If they did not match the necessary requirements of mass

density and viscosity, the procedure was repeated with different compositions. In order to measure the viscosity of a Newtonian fluid, a Cannon–Fenske viscosimeter made by SCHOTT Instruments was used. The mass density was determined by a pycnometer. The interfacial tension between the fluids

M

was measured by a Radian Series 300 tensiometer made by Thermo Scientific with a Du Nouy ring. The ring is placed in the fluid which is below and

ED

starts moving up towards the fluid which is above. The maximum value of the measured force is used to compute the interfacial tension (dividing this

PT

value by two times the circumferential length). 3.3. Characterization of the viscoplastic material properties

CE

The viscoplastic material employed was a 7.5% by weight water based

solution of a commercial hair gel (Bozzano), which is a Carbopol-based dis-

AC

persion. In order to characterize the material, the Haake Mars III rheometer of Thermo Scientific was employed. The resulting flow curve is presented in Fig. 2. The parallel-plate geom-

etry with cross-hatched surface (roughness ≈ 0.40 mm, see the subfigure at the left up corner of this figure) was selected in order to avoid wall slip, a 14

AN US

CR IP T

ACCEPTED MANUSCRIPT

M

Figure 2: Flow curve for the hair gel used in the present work and a curve fitting using a SMD model.

ED

known problem in viscoplastic materials. The material exhibited an apparent yield stress fluid behavior and, in this connection, a curve fitting obtained from the SMD model (de Souza

PT

Mendes and Dutra, 2004) was chosen. The parameters of the SMD model that correspond to the curve fitting obtained in Fig. 2 are displayed in Table

CE

4.

In order to measure the interfacial tension between the viscoplastic ma-

AC

terial and a second Newtonian fluid, a non-standard procedure is required. The first thing to notice is that the viscoplastic material is placed under the Newtonian fluid. This procedure is employed in order to avoid the instabilities that could arise in the case where the mass density of the Newtonian

15

Property Value % Hair gel 7.5 ρ [kg/m] 1008.9 τy [Pa] 0.95 n 0.48 K [Pa.sn ] 1.1 ηo [Pa.s] 375

CR IP T

ACCEPTED MANUSCRIPT

Table 4: Properties of the viscoplastic material.

AN US

fluid is slightly below the mass density of the viscoplastic material. If the viscoplastic material is under the Newtonian one, then in the case where the mass density of the Newtonian fluid is slightly above the corresponding property of the viscoplastic material, unstable motion will not occur since the yield stress of the viscoplastic material prevents this from happening.

M

With this in mind, we observe that the first stage of the process, where the ring moves in the fluid which is placed below, consists of a constant

ED

velocity motion immersed in the viscoplastic material. Therefore, an extra force is needed in order to overcome the yield stress. Figure 3 illustrates

PT

this point, comparing the time evolution for the force when a viscoplastic material is below a Newtonian fluid and the Newtonian–Newtonian case.

CE

In the Newtonian–Newtonian case, the ring motion in the lower layer fluid occurs with negligible force until the interface is reached, since this motion

AC

has a very low speed. On the other hand, after a short transient, the motion inside the viscoplastic lower layer material, although slow, needs a minimum force to occur, which is the plateau observed before the ring reaches the interface. After this instant, the force grows continually up to the point where the force reaches its maximum, i.e. the point associated to the interfacial 16

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

Figure 3: Typical interfacial tension measurement of a viscoplastic–Newtonian pair (in red) exemplified by the case of the Carbopol based gel and the Castor Bean oil and a Newtonian–Newtonian pair (in blue) exemplified by the water solution Water A (see Table 1) and oil Y (see Table 2).

17

ACCEPTED MANUSCRIPT

tension. Because of what was described above, we can infer that the interfacial

CR IP T

force when one of the materials is viscoplastic, is the force associated to the maximum force obtained from the ring dislocation discounting the extra force associated to the motion of the ring inside the viscoplastic material,

as indicated in Fig. 3. Therefore, σ = ∆F/(2C), where C is the length of

the ring circle. The results for the viscoplastic–Newtonian displacement were

AN US

obtained with the castor beam oil as the Newtonian fluid and the measured interfacial tension between them was 5.07 mN/m.

This kind of carbopol-based material can exhibit elastic effects which are more pronounced below the apparent yield stress. Because of that, we kept the characteristic shear rate of the experiment (by controlling the shear rate)

M

much higher than the critical shear rate corresponding to the yield stress value (in the present case, γ˙ c ≈ 0.008s−1 ).

ED

3.4. Experimental procedure

The viscoplastic–Newtonian displacement problem is more challenging

PT

when it is investigated experimentally, as initially advanced by Thompson and Soares (2016). After matching the mass densities of the two fluids, a

CE

Newtonian and a viscoplastic one, the interfacial tension between the fluids, σ, is measured. The main problem is that changing the fluid is not as easy

AC

as doing so numerically. On the other hand, to change the front velocity, although not difficult, leads to changes in the viscosity ratio and the capillary number at the same time, since the viscosity of the viscoplastic material is dependent on the characteristic deformation rate. A possible solution is to proceed the other way around and start with the viscoplastic–Newtonian prob18

ACCEPTED MANUSCRIPT

lem to generate the initial data and after that do the Newtonian–Newtonian displacement problem corresponding to the viscoplastic one, in order to infer

CR IP T

the effect of the plastic number. Because of the very time-consuming task that would be undertaken to investigate the role of the viscosity ratio, capillary number, and plastic number, we compared a number of cases where the

viscosity ratio and Ca are the same for a Newtonian and viscoplastic case, which restricts the difference between the two results to the plastic number.

AN US

We have computed the ratio mvp /mN (the fraction of mass obtained with the

viscoplastic material divided by the corresponding fraction of mass obtained with the Newtonian fluid for the same viscosity ratio and capillary number) for me and mg .

In addition to the above concerns, we have tried to provide conditions

M

of low inertia and negligible buoyancy effects. The Froude and Reynolds numbers vary between F r = 0.003 and Re = 0.0025 (Nµ = 4.0, Ca = 0.04)

ED

till F r = 0.75 and Re = 3.18 (Nµ = 2.0, Ca = 1.95) for the Newtonian tests. In the case of the viscoplastic ones the range was from F r = 0.0023

PT

and Re = 0.00065 (Ca = 0.05) and F r = 0.05 and Re = 0.015 (Ca = 1.79). 4. Results

CE

4.1. Newtonian–Newtonian results Figures 4 and 5 show the comparisons between the present experimental

AC

results and the ones obtained numerically by Soares and Thompson (2009) for me and mg as functions of the capillary number, for viscosity ratios of Nµ = 2 and Nµ = 4, respectively. There is observed a much better agreement than the one obtained by Soares et al. (2015). The conclusion is that the procedure adopted here is more reliable and the main difference of the 19

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

ED

Figure 4: Comparison of the results obtained by Soares and Thompson (2009) (numerical), Soares et al. (2015) (experimental), and the present work (Experimental) for me and mg as functions of the capillary number for Nµ = 2 and the high viscosity ratio case as a reference.

PT

approaches was the use of controlled flow rate syringe-pump, in the place of the pressure controlled system employed by Soares et al. (2015). The curve

CE

for a very high viscosity ratio (where me and mg tend to the same value) was obtained in order to validate the adopted procedure by means of a compar-

AC

ison with the data obtained by Taylor (1961). This result also shows that the assumptions adopted by Soares and Thompson (2009) are reasonable if one adopts the experimental procedure employed here, since the matching between numerical and experimental results is remarkable. Figure 6 shows the results for me and mg for all the Newtonian–Newtonian 20

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

Figure 5: Comparison among the results obtained by Soares and Thompson (2009) (numerical), Soares et al. (2015) (experimental) and the present work (Experimental) for me and mg as functions of the capillary number for Nµ = 4 and the high viscosity ratio case as a reference.

21

ACCEPTED MANUSCRIPT

Nµ = 4 0.000919 0.225151 0.718025 0.764561

Nµ = 2 0.000385 0.524234 0.962584 0.546483

Nµ = 1 -0.000075 0.344171 0.961682 0.657004

Nµ = 0.5 0.001331 0.31088 1.017175 0.637601

CR IP T

Parameter a b c d

Table 5: Parameters a, b, c, and d for the curve fitting of mg .

cases analysed in the present work, i.e. Nµ = 4; 2; 1; 0.5. As discussed by

AN US

Soares et al. (2015), the curves for mg detach from gas–liquid displacement, showing higher values of this quantity as the viscosity ratio decreases. The opposite tendency takes place in the curves of me . However, for this quantity, the curves seem to reach an asymptotic profile for low viscosity ratios. The fitted curves shown in Fig. 6 were obtained with the following em-

M

pirical expression,

ED

m=

ab + c Cad b + Cad

(19)

where a, b, c, and d are constant parameters for each fixed viscosity ratio. It

PT

is worth noticing that some works Fairbrother and Stubbs (1935); Bretherton (1961); Feng (2009) have proposed equations for the dependency of the mass

CE

fraction with respect to the capillary number for particular cases. However, as shown by Freitas et al. (2011a), these expressions are only valid for small

AC

values of Ca <≈ 0.25, and, therefore, cannot fit the data covered in the present work. The value of the parameter of the curves obtained with the expression given by Eq. (19) are given in the following tables. This relation between the fraction of mass and the capillary number was

used not only for the fitting, but also as a means to evaluate the corresponding 22

ACCEPTED MANUSCRIPT

Nµ = 1 Nµ = 0.5 0.000187 0.00051 0.261036 0.231228 0.511558 0.497807 0.617985 0.603483

CR IP T

Parameter Nµ = 4 Nµ = 2 a 0.0002 0.001369 b 0.1223 0.191175 c 0.5017 0.50672 d 0.7849 0.694774

Table 6: Parameters a, b, c, and d for the curve fitting of me .

AN US

System Displacing Fluid Displaced Fluid Nµ Nρ σ[mN/m] S1 Water B Castor bean oil 397 0.995 16.41 S2 Water A Oil X 3.97 0.923 6.75 S3 Water C Soybean oil 2.03 0.978 5.85 S4 Water A Oil Y 0.99 0.901 6.94 S5 Oil Z Water A 0.5 1.09 6.91

M

Table 7: The employed pair of fluids: viscosity ratio (Nµ ), mass density ratio (Nρ ), and interfacial tension (σ). These quantities are calculated with respect to the displaced fluid.

Newtonian–Newtonian counterpart of the viscoplastic–Newtonian case, as

ED

will be explained later.

The configuration of the front of the injected fluid as a function of the capillary number and the viscosity ratio is shown in Fig. 7. It can be noticed

PT

that the thickness of the mass of displaced liquid is fairly constant at this distance from the front. The effect of Ca on the radius of the injected

CE

fluid is visually detected. The decrease of this radius when the capillary number changes from Ca ≈ 0.15 to Ca ≈ 1.5 is more pronounced for lower

AC

viscosity ratios. Although the columns do not correspond to a fixed capillary number, we have tried to maintain, at least, the same order of magnitude, so that the effect of the viscosity ratio can also be visually evaluated. For approximately the same Ca, lower values of viscosity ratios lead to an increase

23

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

Figure 6: Experimental results for the geometrical mass fraction, mg , and displacement efficiency, me as functions of the capillary number for different values of the viscosity ratio.

24

ACCEPTED MANUSCRIPT

of the thickness of the layer of displaced fluid. In order to understand how the displacing fluid evolves inside the tube,

CR IP T

pictures were taken at different relative positions with respect to the tip of the interface. At the front, i.e. near the tip, we can see that the gap

between the displacing fluid and the wall is more uniform. However, at positions farther from the tip, the center and the end, a clear dislocation

of the injected fluid towards the lower part of the tube occurs, revealing a

AN US

buoyant effect. In addition, we can observe an increase of the thickness of the displacing fluid at positions farther from the tip of the interface. This effect is more pronounced for high values of the capillary number. Since the injection flow rate is constant, this fact can only be justified by a decrease of the mean cross-section velocity of the displacing fluid.

M

Although there are some differences in the behavior of the displacing fluid depending on its position, these effects are slight when compared to

ED

what happens for lower values of Nµ , especially when Nµ < 2. Figures 9, 10, and 11 illustrate this point. We notice that the internal diameter of the tube

PT

is D = 2mm and this length is a reference for the subfigures displayed in each figure. For equal viscosities, the injected fluid starts to exhibit wavy shapes,

CE

as shown in Fig. 9. We can notice an increase of the frequency of these waves as the capillary number increases. We can also observe that these waves are preferentially located at the upper part of the injected fluid.

AC

In the case where the viscosity of the injected fluid is twice the viscosity

of the displaced one, illustrated in Fig. 10, we have observed a phenomenon that is independent of Ca. At a certain position, there is a localized decrease

in the diameter, which does not generate further waves. However, after a

25

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

Figure 7: Configuration of the interface for fixed values of the viscosity ratio displayed in the rows and similar values of the capillary number displayed in columns. The images were captured at 40 cm from the entrance of the tube.

26

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

27 Figure 8: Pictures of the relative position with respect to the displacing fluid: FRONT, CENTER, END, for different values of capillary number and fixed Nµ = 4.

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

Figure 9: Pictures of the wavy shape configurations of the displacing fluid for Nµ = 1 (displacing fluid: Water A, displaced fluid: Oil Y, see tables II and III) and different values of the capillary number.

28

ACCEPTED MANUSCRIPT

certain length, we can find successive ruptures of the displacing fluid, leading to the formation of drops of similar sizes and shapes. Figure 11 illustrates

CR IP T

this point, showing pictures of the rupture process of the injected fluid and the formation of drops as a consequence. All pictures correspond to values

of the dimensionless parameters of Nµ = 0.5 and Ca = 0.5. In the first two pictures, we can see the accumulation of drops at the left side of the

displacing fluid. In the third picture, we can see a line of these drops, which

AN US

are similar in size and shape. In the last two pictures we can observe a neck forming at the left end of the injected fluid. In this case we can see the action of the capillary forces struggling to form the liquid into a new drop.

Recently, Govindarajan and Sahu (2014) made an important review on the subject of instability analysis in stratified flow (see also Joseph et al.

M

(1997) and the references threin). They were emphatic to the fact that stratified flows tend to induce instabilities. In general, when the more viscous

ED

fluid is in the thinner layer, the flow is susceptible to long-wave instabilities. Since the fraction of mass at the wall is an increasing function of the capillary

PT

number, this situation would correspond to where Nµ < 1 and Ca is high or Nµ > 1 and Ca is low. We did not see any kind of instability of this second

CE

type. Unstable results for the viscosity ratio below unity is predicted in the linear stability analysis for mg >≈ 0.5, which means Ca >≈ 0.15, according to our results, which means almost all the cases analyzed for Nµ = 0.5. This

AC

is in accordance to our results. One important point stressed by Govindarajan and Sahu is the fact that, in experiments, since the region of parallel flow is not infinite, very long or short waves rarely exist. In addition, surface tension generally acts to stabilize the flow and the curvature of the pipe also

29

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

Figure 10: Configuration of the displacing fluid for Nµ = 0.5 (displacing fluid: Oil Z, displaced fluid: Water A, see tables II and III) and different values of capillary number.

30

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

Figure 11: The breakage process of the displacing fluid into drops for the case of a Newtonian–Newtonian displacement, with Nµ = 0.5 and Ca = 0.5163 (displacing fluid: Oil Z, displaced fluid: Water A, see tables II and III).

31

ACCEPTED MANUSCRIPT

leads to more stable flows when compared to the ones obtained in a parallelPoiseulle geometry. This explains why we did not obtain some instabilities

CR IP T

theoretically or numerically predicted. For example, the “bamboo instabilities” reported by Kouris and Tsamopoulos (2001) were not visualized in the present work. It seems that the elapsed time needed to develop such instabil-

ity is higher than the elapsed time needed for the injected material to cross the entire tube. However, such instability could rise if we kept injecting the

AN US

displacing fluid for much longer times. 4.2. Viscoplastic–Newtonian results

As a preliminary result, we plotted in Fig. 12 the geometric mass fraction, mg , and the mass fraction efficiency, me as a function of the capillary number for different values of plastic number and viscosity ratio. We can notice that

M

both quantities, mg and me , increase with capillary number for the cases analyzed. It is important to notice that although P l and Nη are changing

ED

from point-to-point, the three variables have a monotonic response, namely that decreasing P l, increasing Nη , and increasing Ca, leads to an increase

PT

in mg and me . From this way of displaying the results we cannot conclude which variable(s) is (are) responsible for the increase of those quantities. The

CE

Newtonian–Newtonian result has shown that the viscosity ratio affects both quantities in opposite ways, if capillary number is fixed.

AC

A different approach lead to the results for mg and me , shown in Fig. 13.

There, we have computed this information as a ratio of these quantities obtained in the viscoplastic–Newtonian displacement to the corresponding quantity obtained in the Newtonian–Newtonian experiment with the same viscosity ratio and capillary number. To this end, we have employed Eq. (19) 32

CR IP T

ACCEPTED MANUSCRIPT

0.8

AN US

0.6

Nm = 0.879 Pl = 0.368 0.4

Nm = 1.098 Pl = 0.331

Nm = 1.988 Pl = 0.248

M

Mass fraction, m

Nm = 2.475 Pl = 0.219

Nm = 1.344 Pl = 0.304

Nm = 0.564 Pl = 0.443

ED

0.2

Displacement efficiency, me

PT

Nm = 0.223 Pl = 0.577

Geometric mass fraction, mg

0

0.4

0.8

1.2

1.6

2

Capillary number, Ca

AC

CE

0

Figure 12: The ratio of mass fraction obtained by the injection of a viscoplastic material with respect to the one obtained with a Newtonian fluid at the same viscosity ratio and capillary number.

33

ACCEPTED MANUSCRIPT

using the parameters obtained for the curve of the same viscosity ratio and setting the capillary number of viscoplastic–Newtonian experiment. The re-

CR IP T

sulting (Newtonian) fraction of mass, mN , was used to construct the dimensionless mass fractions shown in Fig. 13. The quantities m∗g = mg /mN g and m∗e = me /mN e are plotted as functions of the plastic number and this can

be seen as the main result of the present work. These curves show how the plastic effect of the yield stress affects the common outputs of the problem,

AN US

associated to thickness and efficiency. In other words, since the viscosity ratio takes into account the viscous effect of the viscoplastic nature of the material, the plastic effects are isolated in P l. The general trends of these curves are

approximately linear with mg presenting a less pronounced dispersion. Since this fraction is below one, for both quantities, the Newtonian–Newtonian

values of Nη and Ca.

M

fractions of mass are higher than the viscoplastic–Newtonian ones for fixed

ED

From the obtained results, we can conclude that displacing a Newtonian material by the injection of a viscoplastic one is more efficient, since the

PT

mass fraction left behind is higher in the Newtonian case. These results are in qualitative agreement with the numerical ones obtained by Freitas et al.

CE

(2011b). It is worth noticing, as discussed by Thompson and Soares (2016), that the conclusion could be different if the viscosity ratio was not defined as in the present work.

AC

Figures 14, 15, and 16 show a direct comparison of the configuration of

the interface between two pair of fluids, for very similar values of viscosity ratio and capillary number. In other words, what differs from one case to another is, essentially, the plastic number, which in the upper case of each

34

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

Figure 13: The ratio of mass fraction obtained by the injection of a viscoplastic material with respect to the one obtained with a Newtonian fluid at the same viscosity ratio and capillary number.

35

AN US

CR IP T

ACCEPTED MANUSCRIPT

Figure 14: Comparison of the interface configuration between two pair of fluids, of Nµ ≈ 0.5 and Ca ≈ 0.22. A Newtonian–Newtonian and a viscoplastic–Newtonian interface with P l = 0.443 for different relative positions with respect to the injected material: FRONT, CENTER, END.

M

figure is zero, i.e. is a Newtonian–Newtonian interface, and in the lower case is a finite number, which indicates it is a viscoplastic–Newtonian problem.

ED

In the three cases, we can observe that the viscoplastic–Newtonian case contrasts with the Newtonian–Newtonian one, by exhibiting a non-flat shape

PT

far from the tip of the interface. This kind of configuration is clearly different from the ones observed in the wavy shapes in the Newtonian–Newtonian displacement. When the viscoplastic material is injected in the conditions

CE

of Figs. 14, 15, and 16, it seems that the material is dented, in a plastic-like deformation.

AC

Comparing Figs. 15, and 14 where the plastic number was very similar,

it seems that when the viscosity ratio is higher, the frequency and amplitude of those indentations do increase. Figure 17 corroborates this point. In this figure different interface configurations for the viscoplastic–Newtonian case

36

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

Figure 15: Comparison of the interface configuration between two pair of fluids, of Nµ ≈ 1 and Ca ≈ 0.6. A Newtonian–Newtonian and a viscoplastic–Newtonian interface with P l = 0.331 for different relative positions with respect to the injected material: FRONT, CENTER, END.

Figure 16: Comparison of the interface configuration between two pair of fluids, of Nµ ≈ 2 and Ca ≈ 1.4. A Newtonian–Newtonian and a viscoplastic–Newtonian interface with P l = 0.248 for different relative positions with respect to the injected material: FRONT, CENTER, END.

37

ACCEPTED MANUSCRIPT

are shown. Unlike the wavy shapes found in the Newtonian–Newtonian case, we can notice that the lower part of the interface also exhibits a non-flat

CR IP T

configuration, although in the upper part this effect is more pronounced. It is difficult to trace the reason why the wavy shapes are more frequent and with higher amplitude from top to bottom in this figure, since P l, Nη , and

Ca are changing monotonically in these figures. It seems that is not an effect of the plastic number, since the wavy effects are more pronounced when P l

AN US

is lower.

The traveling velocity of the wavy shape interface obtained in the present work is the same as the front velocity, i.e. once the wave of the viscoplastic material is formed, it travels with the same shape. This propagation is similar to what was obtained by Hormozi et al. (2014), although in our case

M

the viscoplastic material is in the core while in their case the viscoplastic material is in the annular space. One thing that remains to be addressed is

PT

5. Conclusions

ED

how the initial wave is formed, since no oscillations are imposed.

In the present work we have investigated some quantities defined in Soares

CE

et al. (2015), such as the geometric mass fraction and displacement efficiency, to understand the displacement of one fluid by another in a capillary tube from the experimental point of view. A new setup was developed to replace

AC

the one used in Soares et al. (2015) in order to address the quantitative mismatch between the results obtained in that work and the numerical data obtained in Soares and Thompson (2009). With a syringe attached to a pump where the flow rate can be accurately set, the results obtained here

38

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

Figure 17: Configuration of the interface in the viscoplastic–Newtonian displacement for different relative positions with respect to the injected material: FRONT, CENTER, END; for different values of viscosity ratio and capillary number. The plastic numbers from top to bottom are P l = 0.47; 0.39; 0.29; 0.23, respectively.

39

ACCEPTED MANUSCRIPT

were in much better agreement with Soares and Thompson (2009) than setting the pressure drop, where only qualitative agreement was found. In this

CR IP T

sense, we can conclude that when the displaced fluid is at least two times more viscous than the displacing one, the assumptions made by Soares and

Thompson (2009) in their numerical study were reproduced, i.e. negligible

inertial effects, a uniform layer of the displaced fluid where the stratified flow develops, and constant interfacial tension along the interface.

AN US

A different scenario was found for the Newtonian–Newtonian displacement when the displacing fluid is as viscous as the displaced one, or when the injected fluid is more viscous. We found significant changes in the thickness of the fluid layers as the flow evolved in time. Therefore, the assump-

tions made by Soares and Thompson (2009) seem to be no longer valid in

M

this range. The problem is intrinsically unsteady for Nµ < 2 and different patterns of interface are formed. In particular, for Nµ = 0.5, we could see

ED

that a long injected fluid cannot be sustained and capillary forces break this fluid, forming drops of the same size and shape that start flowing inside the

PT

capillary tube. Therefore, the quantity mg has no meaning in this case. The quantity me cannot be computed from Eq. 18, however it can be calculated

CE

from an integral computation where the injected and discharged quantities are measured.

The viscoplastic displacement of a Newtonian fluid was also investigated.

AC

A hair gel produced from a carbopol solution was the viscoplastic material employed. In order to compare the outputs of this experimental study with the Newtonian ones by meaningful dimensionless numbers, the guidelines provided by Thompson and Soares (2016) were employed, i.e. the yield stress

40

ACCEPTED MANUSCRIPT

was included when the characteristic viscosity of the viscoplastic material was needed. The main result is the comparison of the geometric mass fraction and

CR IP T

displacement efficiency with the corresponding Newtonian mass fractions, i.e. with the same viscosity ratio and capillary number, for a wide range of the plastic number. In this sense, we have isolated the effect of the yield

stress from the viscosity as a non-Newtonian parameter. We have found that increasing the plastic number makes the layer of attached displaced fluid

AN US

thinner, and in addition a higher amount of fluid is displaced, which makes the viscoplastic material a more efficient one. In other words, if one wants to displace a Newtonian material with another material with a similar viscosity

and capillary numbers, using a viscoplastic material would lead to a greater amount of displaced fluid per unit of time. These conclusions can be more

M

associated to me since mg is contaminated by a non-parallel interface. With respect to the front part of the injected fluid, we can say that lower values

ED

of P l lead to thinner injected radius. However, concerning the middle and rear parts, we have always obtained wavy shapes, especially at the center

PT

part. Viscoelastic effects associated to very low shear rates can play a role, but were not significant, since the same qualitative behavior was obtained

CE

by Freitas et al. (2011b). However, further investigation can try to split the plastic from the elastic effects. A front for future investigation, namely the fact that the viscoplastic–

AC

Newtonian displacement problem is more susceptible to the formation of wavy shape interfaces than the Newtonian–Newtonian one is in order. Maintaining the viscosity ratio and capillary number with the same values, we found this pattern for non-vanishing plastic numbers. The wavy shapes en-

41

ACCEPTED MANUSCRIPT

countered in the viscoplastic–Newtonian case exhibit a more solid-like fingerprint when compared with the Newtonian–Newtonian ones. These waves

CR IP T

travel with the same velocity as the injection front. A similar traveling behavior is found in Hormozi et al. (2014), although the viscoplastic material

is kept in the annular space in order to produce the so-called visco-plastic lubrication.

The present analysis did not considered viscoelastic effects that are present

AN US

in the pre-yielding state. We kept the characteristic deformation rates of the experiments at least two orders of magnitude higher than the one associated to the yield stress value associated to the flow curve of the material, so that these effects are less significant. However, these effects can play a role on the formation of these waves, something to be investigated.

M

References

ED

Alba, K., Frigaard, I. A., 2016. Dynamics of the removal of viscoplastic fluids from inclined pipes. J. Non-Newt. Fluid Mech. 229, 43–58.

PT

Alba, K., Taghavi, S., de Bruyn, J., Frigaard, I., 2013. Incomplete fluid–fluid displacement of yield-stress fluids. Part 2: Highly inclined pipes. J. Non-

CE

Newt. Fluid Mech. 201, 80–93.

Alexandrou, A. N., Entov, V. M., 1997. On the steady-state advancement of

AC

fingers and bubbles in a Hele-Shaw cell filled by a non-Newtonian fluid. Euro. Jnl of Applied Mathematics 8, 73–87.

Allouche, M., Frigaard, I., Sona, G., 2000a. Static wall layers in the displace-

42

ACCEPTED MANUSCRIPT

ment in two visco-plastic fluids in a plane channel. J. Fluid Mech. 424, 243–277.

CR IP T

Allouche, M., Frigaard, I. A., Sona, G., 2000b. Static wall layers in the

displacement of two visco-plastic fluids in a plane channel. J. Fluid Mech. 424, 243–277.

Bretherton, F. P., 1961. The montion of long bubble in tubes. J. Fluid Mech.

AN US

10, 166–188.

Coussot, P., 1999. Saffman–Taylor instability in yield-stress fluids. J. Fluid Mech. 380, 363–376.

de Souza Mendes, P. R., Dutra, E. S. S., 2004. Viscosity function for yield-

M

stress liquids. Applied Rheology 14 (6), 296–302.

de Souza Mendes, P. R., Dutra, E. S. S., Siffert, J. R. R., Naccache, M. F.,

ED

2007. Gas displacement of viscoplastic liquids in capillary tubes. J. NonNewt. Fluid Mech. 145, 30–40.

PT

Dimakopoulos, Y., Tsamopoulos, J., 2003. Transient displacement of a viscoplastic material by air in straight and constricted tubes. J. Non-

CE

Newt. Fluid Mech. 112, 43–75.

AC

Eslami, A., Taghavi, S., 2017. Viscous fingering regimes in elasto-visco-plastic fluids. J. Non-Newt. Fluid Mech. 243, 79–94.

Fairbrother, F., Stubbs, A., 1935. Studies in electro-endosmosis part vi the “bubble-tube” method of measurement. J. Chem. Society 1, 527–539.

43

ACCEPTED MANUSCRIPT

Feng, J. Q., 2009. A long gas bubble moving in a tube with flowing liquid. Int. J. Multiphase flow 35, 738–746.

CR IP T

Freitas, A. A., Soares, E. J., Thompson, R. L., 2011a. Immiscible Newtonian displacement by a viscoplastic material in a capillary plate channel. Rheol Acta. URL DOI 10.1007/s00397-011-0544-3

AN US

Freitas, J. F., Soares, E. J., Thompson, R. L., 2011b. Immiscible newtonian

displacement by a viscoplastic material in a capillary plane channel. Rheol Acta 50, 403–422.

Freitas, J. F., Soares, E. J., Thompson, R. L., 2013. Viscoplastic– viscoplastic displacement in a plane channel with interfacial tension effects.

M

Chem. Eng. Sci 91, 54–64.

ED

Goldsmith, H. L., Mason, S. G., 1963. The flow of suspensions through tubes. J. Colloid Sci. 18, 237–261.

PT

Govindarajan, R., Sahu, K. C., 2014. Instabilities in viscosity-stratified flow. Annu. Rev. Fluid Mech. 46, 331–353.

CE

Hasnain, A., Alba, K., 2017. Buoyant displacement flow of immiscible fluids

AC

in inclined ducts: A theoretical approach. Phys. Fluids 29 (052102).

Hodges, S. R., Jenseng, O. E., Rallison, J. M., 2004. The motion of a viscous drop through a cylindrical tube. J. Fluid Mech. 501, 279–301.

Hormozi, S., Dunbrack, G., Frigaard, I. A., 2014. Visco-plastic sculpting. Phys. Fluids 26 (093101). 44

ACCEPTED MANUSCRIPT

Jalaal, M., Balmforth, N. J., 2016. Long bubbles in tubes filled with viscoplastic fluid. J. Non-Newt. Fluid Mech. 238, 100–106.

Annu. Rev. Fluid Mech. 29, 65–90.

CR IP T

Joseph, D., Bai, R., Chen, K. P., Renardy, Y. Y., 1997. Core-annular flows.

Kamisli, F., 2006. Gas-assisted displacement of a viscoelastic fluid in capillary geometries. Chem. Engng. Sci. 61, 1203–1216.

AN US

Kouris, C., Tsamopoulos, J., 2001. Dynamics of axisymmetric core-annular flow in a straight tube. I. The more viscous fluid in the core, bamboo waves. Phys. Fluids 13, 841–858.

Lac, E., Sherwood, J. D., 2009. Motion of a drop along the centerline of a

M

capillary in a pressure-driven flow. J. Fluid Mech. 640, 27–54. Lee, G., Shaqfeh, E. S. G., Khomami, B., 2002. A study of viscoelastic free

ED

surface by finite element method Hele-Shaw and slot coating flows. J. NonNewt. Fluid Mech. 117, 117–139.

PT

Lindner, A., Coussot, P., Bonn, D., 2000. Viscous fingering in a yield stress

CE

fluid. Phys. Rev. Lett. 85, 314–317. Maleki-Jirsaraei, N., Lindner, A., Rouhani, S., Bonn, D., 2005. Saffman–

AC

Taylor instability in yield stress fluids. J. Phys. Condens. Matter 17 (S1219).

Malekmohammadi, S., Naccache, M. F., Frigaard, I. A., Martinez, D. M., 2010. Buoyance driven slump flows of non-newtonian fluids in pipes. J. Petr. Sci. Engng. 33, 410–243. 45

ACCEPTED MANUSCRIPT

Mois´es, G. V. L., Naccache, M. F., Alba, K., Frigaard, I. A., 2016. Isodense displacement flow of viscoplastic fluids along a pipe. J. Non-Newt. Fluid

CR IP T

Mech. 236, 91–103. Soares, E. J., Carvalho, M. S., de Souza Mendes, P. R., 2005. Immiscible liquid-liquid displacement in capillary tubes. J. Fluids Eng. 127, 24–31.

Soares, E. J., Thompson, R. L., 2009. Flow regimes for the immiscible liquid-

AN US

liquid displacement in capillary tubes with complete wetting of the displaced liquid. J. Fluid Mech. 641, 63–84.

Soares, E. J., Thompson, R. L., Niero, D. C., 2015. Immiscible liquid–liquid pressure-driven flow in capillary tubes: Experimental results and numerical

M

comparison. Phys. Fluids 27 (082105), 1–13.

Sousa, D., Soares, E. J., Queiroz, R. S., Thompson, R. L., 2007. Numerical

ED

investigation on gas-displacement of a shear-thinning liquid and a viscoplastic material in capillary tubes. J. Non-Newt. Fluid Mech. 144, 149–159.

PT

Taghavi, S., Alba, K., Moyers-Gonzalez, M., Frigaard, I., 2012. Incomplete fluid–fluid displacement of yield stress fluids in near-horizontal pipes: Ex-

CE

periments and theory. J. Non-Newt. Fluid Mech. 167–168, 59–74. Taghavi, S. M., Seon, T., Martinez, D. M., Frigaard, I. A., 2009. Buoyancy-

AC

dominated displacement flows in near-horizontal channels: the viscous limit. J. Fluid Mech. 639, 1–35.

Taylor, G. I., 1961. Deposition of a viscous fluid on the wall of a tube. J. Fluid Mech. 10, 161–165. 46

ACCEPTED MANUSCRIPT

Thompson, R. L., Soares, E. J., 2012. Motion of a power-law long drop in a capillary tube filled by a newtonian fluid. Chem. Eng. Sci 72, 126–141.

J. Non-Newt. Fluid Mech. 238, 57–64.

CR IP T

Thompson, R. L., Soares, E. J., 2016. Viscoplastic dimensionless numbers.

Wielage-Burchard, K., Frigaard, I. A., 2011. Static wall layers in plane channel displacement flows. J. Non-Newt. Fluid Mech. 166, 245–261.

AN US

Zamankhan, P., Helenbrook, B. T., Takayama, S., Grotberg, J. B., 2011. Steady motion of bingham liquid plugs in two-dimensional channels. J. Fluid Mech. 705, 258–279.

Zare, M., Roustaei, A., Alba, K., Frigaard, I. A., 2016. Invasion of fluids into

M

a gelled fluid column: yield stress effects. J. Non-Newt. Fluid Mech. 238,

AC

CE

PT

ED

212–223.

47