fluids
A road towards better understanding
Susanna Formenti
Dissertation presented in partial
fulfillment of the requirements for the
degree of Doctor of Engineering
Science (PhD): Chemical
Engineering
February 2020
Supervisors:
Dr. eng. F. Briatico Vangosa
Prof. dr. habil. C. Clasen
A road towards better understanding
Susanna FORMENTI
Examination committee: Prof. dr. ir. , chair
Dr. eng. F. Briatico Vangosa, supervisor Prof. dr. habil. C. Clasen, supervisor Prof. dr. ir. P. Van Puyvelde
Prof. dr. ir. J. Degrève Prof. dr. P.D. Anderson
(Eindhoven University of Technology ) Prof. dr. C. Marano
(Politecnico di Milano) Prof. dr. G.H. McKinley
(Massachusetts Institute of Technology)
Dissertation presented in partial fulfillment of the requirements for the degree of Doctor of Engineering Science (PhD): Chemical Engineer-ing
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The present PhD thesis focusses on exploring current open-ended issues in capillary break-up rheometry. Exploiting the natural phenomenon of capillarity instability, capillary break-up rheometry represents a very powerful technique to extract material information in free-surface, uniaxial extensional flows, which are of the utmost importance in understanding and controlling processes and applications involving the formation of liquid ligaments, jets or sheets. Throughout the present manuscript some of the challenges in capillary break-up rheometry have been investigated. The first challenge to be dealt with is the definition of criteria for the identification of dynamical regimes. Contrarily to the already existing criteria, which were proven to be often difficult to deploy for the interpretation of experimental data, a novel set of non-dimensional groups based on the experimentally relevant axial length scale of the filament is introduced. On the basis of the newly introduced groups, experimental limits, which depend on the set-up configuration, can be employed to identify the observable regimes and their transitions, resulting in a 2D operating map in which the boundaries are set by fluid properties and a single geometrical parameter and shedding light on the relevance of the axial length scale of the filament in determining the transitions between dynamical regimes. The theoretical developed framework is subsequently used to face the challenge of capillary break-up in two-fluid systems. Here, experiments, encompassing combinations of Newtonian fluids and a wide range of viscosity ratios, are employed to explore the interplay of the inner and outer fluids in determining complex thinning behaviours. Besides the interpretation of the different cases and their assignment in a viscosity -viscosity ratio map, it is highlighted that, when performing experiments in a liquid-liquid configuration, the combination of fluids must be carefully selected, as using an outer fluid with too high viscosity could potentially forbid the extraction of material properties.
Next, the issues related to the choice of geometrical boundaries in CaBER experiments is tackled for Newtonian cases and concentrated polymer solutions.
The apparent effect of the step height on the thinning behaviour seems to be more related to the high curvature of the filaments at low step height and a step ratio of 3 to 3.5 seemed to be the optimal range for the tested samples. The behaviour of concentrated solutions, though, is a still open challenge, since no clear elastic behaviour was though identified, nor the thinning could be ascribable to a purely viscous regime. Their behaviour seems the result of a complex interplay of viscous and elastic behaviour and no straightforward analytical model is expected to capture it.
Lastly, the comparison of two experimental techniques, ROJER and CaBER is presented. A systematic study on diluted polymer solutions allows for the validation of a new scaling law, and the versatile jetting set-up developed confirmed that the observation of the scaling is not affected by the parameters used to induce the perturbation. Additionally, the investigation also brings to light the presence of polymer degradation at high molecular weights due in ROJER experiments, to the sudden extension to which the fluid in contact with the walls of the nozzle can be subjected to at the nozzle exit. In terms of measured relaxation times, this effect reflects in systematic changes of the relaxation times calculated from ROJER experiments, which show a significant reduction compared to the results obtained in CaBER experiments.
abbreviations
Symbols in arabic letters
Symbol Description, units
a Non-dimensional radius [-]
aH Mark-Houwink-Sakurada exponent [-]
acrit Critical non-dimensional radius [-]
ares Critical non-dimensional radius at the resolution limit [-]
aup Critical non-dimensional radius at the upper limit [-]
b Dimensionless constant for two-fluid Stokes similarity solution [-]
B Dimensionless coefficient [-]
Bo Bond number [-]
c Concentration [wt%]
c∗ Critical overlap concentration [wt%]
c∗∗ Critical concentration for the concentrated regime [wt%] ce Entanglement concentration [wt%]
C Correction factor [-]
Ci Amplitude of cosine i
Ca Capillary number [-] D Rate of strain tensor [1/s] De Deborah number [-]
Dec Critical Deborah number [-]
Ec Elastocapillary number [-]
Ecc Critical Elastocapillary number [-]
f Frequency of the imposed jetting perturbation [Hz]
fmax Frequency corresponding to the most unstable wavelength [Hz]
Symbol Description, units
f(t) Temporal function describing minimum radius evolution [m] Fbond Bond strength [N]
Fmax Maximum drag force on a polymer chain [N]
Fz Axial force [N]
H Pre-factor of minimum radius evolution equation [-] k Disturbance wavenumber [m]
K Filament axial curvature [1/m]
KH Mark-Houwink-Sakurada coefficient [(ml g−1)(mol g−1)aH]
˜
K Consistency constant [-] l Contour length [m]
¯L Finite extensibility [-]
L Axial filament length scale [m] Lf Final plate separation distance [m]
Li Initial plate separation distance [m]
Lmax Distance between points of minimum and maximum axial
velocity [m]
˙Lη Axial viscous velocity [m/s]
˙Lρ Axial inertia velocity [m/s]
L/D Nozzle aspect ratio [-] M Molar mass [kg/mol]
Me Entanglement molecular weight [kg/mol]
Mw Weight average molecular weight [kg/mol]
n Power-law exponent [-] N Number of molecules [-]
Oh Ohnesorge number [-]
Ohc Critical Ohnesorge number [-]
OhL Axial Ohnesorge number [-]
OhL,res Critical axial Ohnesorge number for the resolution limit [-]
OhL,up Critical axial Ohnesorge number for the upper limit [-]
p Viscosity ratio [-]
pres Critical viscosity ratio for the resolution limit [-]
pup Critical viscosity ratio for the upper limit [-]
Q Volumetric flow rate [m3/s]
r Normalized filament radius [-] R Filament radius [m]
R0 Initial filament radius[m]
R0,jet Unperturbed jet radius [m]
RBo<0.1 Critical radius for gravity effect [m]
Rcrit Critical filament radius [m]
Rcrit Critical filament radius [m]
Symbol Description, units
Rp CaBER plate radius [m]
Rout Radius of the jet at the nozzle exit [m]
Rres Critical filament radius for the resolution limit [m]
Runperturbed Unperturbed cylindrical radius [m]
Rup Critical filament radius for the upper limit [m]
˙Rη Radial viscous velocity [m/s]
˙Rρ Radial inertia velocity [m/s]
Re Reynolds number [-] Reloc Local Reynolds number [-]
T Temperature [C]
t Time [s]
tb Time to break-up [s]
tEC Time of the IC to EC transition [s]
tL,η Axial viscous time scale [s]
tL,ρ Axial inertia time scale [s]
ts CaBER step-stretch time [s]
tη (Radial) viscous time scale [s]
tη,out Viscous time scale of the outer fluid [s]
tρ (Radial) inertia time scale [s]
Tr Trouton ratio [-]
u Local axial velocity [m/s] ua Average axial velocity [m/s]
uCa Capillary speed [m/s]
umax Maximum axial velocity [m/s]
Ujet Jet velocity [m/s]
Unozzle Velocity in the nozzle [m/s]
Up Plate separation velocity [m/s]
Vcumul Cumulative volume [m3]
w Differential distribution of the molar mass [-]
We Weber number [-] Wi Weissenberg number [-]
z0 Axial position of the minimum radius [m]
zmax Axial position of the point of maximum axial velocity [m]
Greek symbols
Symbol Description, units
α1 Similarity exponent 1 [-]
α2 Similarity exponent 2 [-]
β Similarity exponent [-] γ Surface tension [N/m]
˙γ Shear rate [1/s]
δ Small disturbance amplitude [m]
∆P Pressure difference [Pa] ∆ρ Density difference [kg m3] Slenderness parameter [-] Perturbation amplitude [-] 0 Initial perturbation [-] εH Hencky strain [-] ˙ε Strain rate [1/s]
˙εexit Strain rate at the nozzle exit [1/s]
˙εf Fracture strain rate [1/s]
ζ Similarity variable [-] η Shear viscosity [Pa s]
[η] Intrinsic viscosity [Pa s]
η0 Zero shear viscosity [Pa s]
ηE Extensional viscosity [Pa s]
ηE,app Apparent extensional viscosity [Pa s]
ηin Inner fluid viscosity [Pa s]
ηout Outer fluid viscosity [Pa s]
ηs Solvent viscosity [Pa s]
ηsp Specific viscosity [-]
λw Disturbance wavelength [m]
λw,max Disturbance wavelength corresponding to the maximum
instability growth rate [m]
λ Relaxation time [s]
Λ Relaxation time ratio [-]
λd Disengagement (reptation) relaxation time [s]
λR Rouse relaxation time [s]
λterm Terminal relaxation time [s]
λz Zimm relaxation time [s]
ν Scaling exponent [-] ρ Density [kg m3]
σ Conductivity [S]
τw Shear stress at the wall [Pa]
Symbol Description, units
ψ Similarity function [-] ω Instability growth rate [1/s] ω*
exp Normalized instability growth rate [-]
ωmax Maximum instability growth rate [1/s]
Abbreviations
Abbreviation Description AR Aspect ratio
CaBER Capillary Break-up Extensional Rheometer DoS Dripping-on-Substrate
EC Elastocapillary regime
GPC Gel Permeation Chromatography IC Inertiacapillary regime
IV Inertiaviscous regime LV Two-fluid viscous regime MW Nominal molecular weight PDMS polydimethylsiloxane PEO polyethylene oxide PDI Poly Dispersity Index PL Power-law
SEC Size Exclusion Chromatography
ROJER Rayleigh-Ohnesorge Jetting Extensional Rheometer V Viscous regime
Abstract i
List of symbols and abbreviations iii
Contents ix
List of Figures xiii
List of Tables xix
1 Introduction 1
1.1 Why capillary break-up? . . . 1 1.2 Challenges in capillary break-up and approach of the thesis . . 4
2 State of the art: Dynamics of capillary instability 7
2.1 Free-surface flows and capillary instability . . . 8 2.1.1 Non-dimensional description . . . 10 2.2 Capillary instability . . . 15 2.2.1 Slender filaments and one-dimensional approximation . 15 2.2.2 Away from break-up: linear stability analysis . . . 16 2.2.3 Approaching break-up: non-linear effects and similarity
solution . . . 18
2.2.4 Newtonian fluids: a multitude of scaling laws . . . 19
2.2.5 Effect of a different medium: the outer fluid . . . 21
2.2.6 Non-Newtonian effects . . . 23
2.2.7 Transitions between capillary thinning regimes . . . 26
2.3 Concluding remarks . . . 30
3 Capillary break-up rheometry 31 3.1 Introduction . . . 31
3.2 Fundamentals of capillary break-up rheometry . . . 33
3.2.1 Extensional flow . . . 33
3.2.2 Force balance and apparent extensional viscosity . . . . 34
3.2.3 Experimental configurations . . . 36
3.3 Experimental set-ups . . . 37
3.3.1 CaBER - Capillary Break-up Extensional Rheometer . . 38
3.3.2 DoS - Dripping-onto-Substrate . . . 41
3.3.3 ROJER - Rayleigh Ohnesorge Extensional Rheometer . 43 3.3.4 Other techniques . . . 45
3.4 Concluding remarks . . . 45
4 Novel non-dimensional groups to map transitions in capillary break-up experiments 47 4.1 Introduction . . . 48
4.2 Viscous-dominated capillary thinning in air . . . 49
4.2.1 Deviation of the velocity profile . . . 51
4.3 Capillary thinning in a different outer fluid . . . 55
4.3.1 Large viscosity ratios (p 1) . . . . 55
4.3.2 Small viscosity ratios (p 1) . . . . 57
4.5 The map in action: experimental cases . . . 61
4.6 Conclusion . . . 62
5 Capillary break-up of liquid-liquid interfaces: on scalings and transitions 65 5.1 Introduction . . . 66 5.2 Theoretical background . . . 67 5.3 Experimental . . . 69 5.3.1 Simulations . . . 70 5.4 Results . . . 71
5.4.1 Capillary thinning in air . . . 71
5.4.2 Capillary thinning at liquid-liquid interfaces . . . 71
5.5 Identification of regimes . . . 74
5.5.1 Thinning rates . . . 74
5.5.2 "Classic" transition criteria and local Reynolds number analysis . . . 78
5.6 The OhL-p approach . . . 81
5.6.1 Defining Lmax. . . 81
5.6.2 Setting the experimental limits . . . 83
5.6.3 Prediction and data comparison . . . 84
5.7 Concluding remarks and opportunities . . . 90
6 Newtonian fluids, concentrated solutions and the effect of step stretch parameters 91 6.1 Problem statement . . . 92 6.2 Experimental . . . 94 6.2.1 Newtonian samples . . . 94 6.2.2 Polymer solutions . . . 94 6.2.3 CaBER experiments . . . 97
6.3 Determination of the characteristic length . . . 99
6.4 Viscous Newtonian fluids . . . 99
6.5 Polymer solutions . . . 104
6.6 Concluding remarks . . . 111
7 Jetting of weakly viscoelastic fluids: jetting parameters and polymer degradation 113 7.1 Introduction . . . 114
7.2 CaBER vs ROJER: a mere scaling issue? . . . 116
7.3 Experimental . . . 117
7.3.1 Polymer solutions . . . 118
7.3.2 CaBER experiments . . . 119
7.3.3 Jetting experiments . . . 120
7.4 A study on the effect of jetting parameters . . . 123
7.5 A study on polymer degradation . . . 129
7.6 Conclusion and future work . . . 138
8 Conclusion and outlook 139
Bibliography 145
Curriculum vitae 157
1.1 List of materials, processes and outcomes in free-surface flows,
and their relevant performance indices. . . 2
1.2 Examples of different free-surface flow operations. . . 2
1.3 From spinning to spraying. . . 3
2.2 Evolution of a capillary instability. . . 9
2.3 Work-flow of the dimensional analysis tool. . . 11
2.4 Evolution of characteristic speeds and filament features at break-up. 12 2.5 Operational maps based on dimensionless quantities. . . 13
2.6 Instability on a jet. . . 16
2.7 Dispersion relations of the instability growth rate. . . 18
2.8 Similarity functions for the V regime. . . 19
2.9 Similarity functions for the IV regime. . . 21
2.10 Capillary thinning of a two-fluid system. . . 22
2.11 Capillary break-up at low viscosity ratios. . . 23
2.12 Comparison of an inviscid and a viscoelastic jet; example of thinning curve for a polymer solution. . . 24
2.13 Shape near break-up for different fluids. . . 27
2.14 Possible transitions between dynamical regimes. . . 28
3.1 Working range of the most common extensional rheometers. . . 32
3.2 Fluid element under uniaxial extension. . . 33
3.3 Example of the outcome of capillary break-up experiments for a viscous and a viscoelastic fluid. . . 35
3.4 Experimental configurations used to study free-surface flows. . 36
3.5 A CaBER in a laboratory. . . 38
3.6 Issues during CaBER testing of low viscoelastic samples. . . 39
3.7 The Slow Retraction Method. . . 40
3.8 Layout of a DoS set-up. . . 42
3.9 Substrate effect and DoS-CaBER comparison. . . 43
3.10 "Flying" CaBER. . . 44
3.11 Pros and cons of capillary break-up rheometry techniques. . . . 46
4.1 Capillary thinning evolution of a Newtonian fluid with η = 0.365 mPa s in air and and comparison of the ideal and the real filament profiles. . . 52
4.2 Temporal evolution of R(z) and ua(z) in a capillary break-up experiment of a glycerol-water mixture in air. . . 53
4.3 Temporal evolution of ψ as a function of inacapillarybreak − upexperimentof aglycerol − watermixtureinair. . . . 54
4.4 Direct comparison between the temporal evolution of the minimum radius and the characteristic axial length scale for different fluids. . . 56
4.5 High-resolution images from CaBER experiments performed with a surrounding immiscible outer liquid and schematics of the inner and outer flow profiles. . . 59
4.6 2D operating map as a function of the dimensionless groups OhL and p. . . . 60
4.7 Minimum radius evolution in time, and critical transition radii Rcrit for different fluid combinations. . . 63
5.1 Overview of fluid combinations. . . 69 5.2 Liquid-liquid set-up. . . 70 5.3 Rmintemporal evolution of a glycerol 95 wt% solution in air and
in PDMS 50. . . 72 5.4 Rmintemporal evolution for a glycerol 99 wt% solution as function
of p. . . . 73 5.5 Temporal evolution of the minimum position for a glycerol 99
wt% solution as function of p. . . . 74 5.6 Snapshots of a glycerol 99 wt% solution in different outer media. 75 5.7 Exponential instability growth rate as a function of viscosity ratio. 77 5.8 H for the viscous, inertia-viscous and two-fluid viscous regimes. 77 5.9 Comparison between observed transitions and transition criteria
predicted by theory. . . 79 5.10 Evolution of the local Reynolds number. . . 80 5.11 Evolution of measured Lmax as a function of p. . . . 83
5.12 Comparison of the velocity profiles obtained with the different methods. . . 84 5.13 Inner vs outer viscosity map and expected regimes. . . 85 5.14 Radius evolution, limits and critical transition radius for the case
of p = 218 (left) and p = 18.7 (right). . . . 86 5.15 Simulation velocity profiles in the inner and outer liquid (p = 18.7). 86 5.16 Similarity functions for the W-G 99, p = 18.7 case. . . . 87 5.17 Similarity functions for the W-G 85, p = 20.5 case. . . . 88 5.18 Similarity functions for the p = 0.9 and p = 0.2 cases. . . . 89 5.19 Simulation velocity profiles in the inner and outer liquid (p = 0.2). 89 6.1 Effect of step height on PEO 1M solutions. . . 93 6.2 Specific viscosity vs. normalized concentration and Graessley’s
6.3 Representation of CaBER geometry parameters before and after the step stretch. . . 98 6.4 Dual-camera set-up. . . 98 6.5 Length scale as a function of the aspect ratio. . . 100 6.6 OhL as function of Oh for the Newtonian samples and filament
profile comparison. . . 100 6.7 Thinning curves of PDMS 350 and 500. . . 101 6.8 Comparison of the thinning curves of a PDMS and a glycerol
solution with similar Oh. . . 102 6.9 Comparison of the minimum radius position evolution of a PDMS
and a glycerol solution with similar Oh. . . 102 6.10 Comparison of the thinning curves of the glycerol 90 wt% solution
for different AR. . . 103 6.11 Thinning curves (log-lin) of PEO 35k at varying aspect ratio. . 104 6.12 Strain rate and extensional viscosity evolutions as function of
aspect ratio. . . 105 6.13 Thinning curves (lin-lin) of PEO 35k at varying aspect ratio. . 106 6.14 comparison of different step velocities. . . 106 6.15 High resolution images of concentrated polymer solutions at
different AR. . . 107 6.16 Comparison of thinning curves for PEO 35k and PEO 203k as
function of the concentration. . . 108 6.17 Comparison of radius evolution curves for different samples with
similar Oh. . . 108 6.18 Wi and local Ec for PEO 203k 9 wt%. . . 110 7.1 Graessley diagram of the weakly viscoelastic PEO solutions. . . 120 7.2 Lay-out of the jetting set-up at KUL. . . 121 7.3 Comparison of the perturbation growth on two jets. . . 122 7.4 Effect of different frequencies and flow rates on jetting thinning
7.5 Direct comparison of CaBER and jetting thinning dynamics. . 125 7.6 Jetting and CaBER scaled profiles. . . 126 7.7 Scaled relaxation times as function of reduced concentration. . 127 7.8 Comparison of jetting and CaBER images. . . 128 7.9 Scheme of the steps followed for the assessment of polymer
degradation in ROJER experiments. . . 129 7.10 Comparison of thinning curves for PEO 1M and 5M for ROJER
and CaBER on fresh and jetted samples. . . 130 7.11 Evolution of the relaxation time ratio as a function of molecular
weight. . . 131 7.12 Molar mass distribution curves obtained by GPC measurements. 133 7.13 Nozzle geometry. . . 134 7.14 Illustrative representation of an extended bead-rod model chain. 135 7.15 Comparison of the maximum shear drag force and the bond
strength. . . 136 7.16 Comparison of the fracture strain rate and the strain rate at the
nozzle exit. . . 137 8.1 Liquid-liquid experiments on viscoelastic PEO solutions. . . 141 8.2 An air bubble surrounded by a silicon oil (ηout = 500 Pa s)
undergoing break-up in a CaBER-like experiment. . . 142 8.3 Thinning curves of a 300k Pullulan-water solution for aspect
4.1 Physical properties and AR of the explored cases. . . 61 5.1 List of the calculated lengths, OhL and resolution limits for each
inner and outer liquid combination. . . 85 6.1 Physical properties of Newtonian samples. . . 95 6.2 Physical properties of the prepared polymer solutions. . . 97 6.3 Stretch parameters. . . 98 6.4 Global Ec for concentrated polymer solutions. . . 110 7.1 Physical properties of the PEO solutions. . . 118 7.2 Average molar mass distributions obtained from GPC analysis
for PEO samples. . . 132 7.3 Calculated flow characteristics. . . 134
Introduction
1.1
Why capillary break-up?
Capillary break-up rheometry is an experimental technique, which exploits the observation of the capillary driven break-up dynamics of thin fluid threads to assess the rheological response of fluids in uniaxial extensional free-surface flows. The reason for which this technique is the focus this thesis resides in its crucial role played in understanding and controlling a wide number of technological processes and natural phenomena, such as spraying of paints and sneezing. These processes belong to the class of free-surface flows, and they share one important feature: they are intrinsically subjected to the formation, elongation and possible break-up of liquid threads, jets or sheets, or, in other words, to capillary instabilities. Depending on the type of process, the use of different materials can lead to substantially different outcomes; some examples are illustrated in Figure 1.2, while a list of relevant materials, applications/processes and potential outcomes is displayed in Figure 1.1. For instance, dosing of commercial products often involves pumping through a nozzle high viscous fluids, which tend to form long lasting threads, as shown for fabric softener dispensing in the central photograph in Figure 1.2. In terms of process optimization, modifying the extensional response of the material in order to obtain a faster break-up would lead to a shorter dosing time, particularly interesting for speeding production up. On the other hand, longer lasting filaments connected droplets might be required in applications where a high control of the droplet size distribution is required.
Although determining the properties of the fluids controlling capillary break-up
Figure 1.1: Examples of materials, processes and common outcomes in free-surface flow applications. In grey, some of the abstract concepts historically employed to define the degree of performance.
Figure 1.2: From left to right, top down: jetting of a polymer solution to achieve controlled droplet size at moderate dispensing rates; dispensing of a commercial suspension (fabric softener) where a long, uniform and continuous jet is achieved at moderate dispensing rates [1]; air-assisted atomization process using low elastic fluids and fast flow conditions [2]; in a microfluidic device, production at low flow rates of uniform micro-droplets suspended in an immiscible liquid for a double template emulsion [3].
Figure 1.3: The effect of fluid elasticity on the electrospinning process outcome: decreasing the level of elasticity, assessed by means of relaxation time measurements in capillary break-up experiments allows to transition from uniform fibres, to ill-formed fibres with beads-on-a-string morphology, to droplets (i.e. electrospraying). Images from [6], electrospraying image courtesy of R.
Koekoekx.
is evidently of the utmost importance, it appears not as straightforward to be determined experimentally, and, historically, descriptive concepts such as "stringiness" or "tackiness" have been employed as performance indices instead of actual material characteristics. In this scenario, capillary break-up rheometry provides an effective tool, which mimics the same instability phenomenon to quantify the dynamics of the extensional response, thereby offering the possibility to control processing parameters and design fluids with a required processability. At the same time, it can also be readily employed to investigate natural phenomena such as the formation of mucus droplets during sneezing [4] or the peculiar properties of hangfish slime [5].
Figure 1.3 illustrates an example of the advantages of correctly evaluating extensional properties with capillary break-up techniques in electrospinning applications. Here, by evaluating the degree of elasticity of the fluid, usually expressed in terms of the relaxation time, one can switch between an electrospinning operation, where a continuous jet allows to produce non-woven mats of nanofibers, for instance used for production of scaffoldings or filtration purposes, and a spraying operation, where a discontinuous jet allows to produce droplets, usually employed in the pharmaceutical industry.
Other important industrial examples encompass the formulation of inks: for ink-jet printing, flexo and gravure printing, it is essential to achieve a clean,
consistent filament break-up. A high printing quality depends on avoiding the formation of secondary droplets and phenomena like cobwebbing; to achieve this, small amount of flexible polymers are often added to tune the formulation. Evaluating the capillary extensional response of such inks is essential to optimizing the material: the polymer content, for instance, has to be corrected to provide sufficient elasticity to prevent secondary droplets, but it must at the same time not exceed to avoid phenomena like cobwebbing. In the food industry, a proper estimation of the capillary break-up response can be used to mimic the stringiness of dairy products with plant-based replacements. The study of extensional properties is also essential to the process optimization when dosing foodstuff, but also commercial products such as detergents or cosmetics, from a nozzle: in order to increase the productivity of the filling lines and avoid waste of material, the break-time in combination with avoiding misting place a crucial role.
1.2
Challenges in capillary break-up and approach
of the thesis
As aforementioned, capillary break-up rheometry is based on the observation of the thinning dynamics of liquid filaments; more in details, it consists in the evaluation of the interplay between capillary forces, which drive the thinning, and the resistance provided by viscous, inertial and elastic forces intrinsic to the fluid. Unlike most shear and extensional rheometers, this technique does not involve active stretch but exploits only surface tension to create a uniaxial extensional flow. Moreover, except for a very limited amount of studies where the implementation of force sensors was attempted, no active measurement of stress or strain is performed during experiments, but simply the evolution of the filament diameter in time is recorded. From this perspective, using the term indexers rather than rheometry would surely be a more appropriate description. Even when classified as indexers, this technique still allows to get quantitative information about extensional rheological response of the fluids that goes beyond the aforementioned mere heuristic concepts can be obtained by monitoring the temporal evolution of the minimal radius of the filament. To start with, being no external forcing exerted, the fluid thread can spatially rearrange and select its own time scales, allowing to determine a characteristic break-up time. Even though not a material property, this information can be of great utility for industrial applications, for instance when benchmarking the easiness of dispensing different products. Quantitative observations about strain rate, along with an apparent extensional viscosity and the break-up time of the fluid,
can be also estimated directly by the evolution of the filament diameter in time. However, real material properties cannot be directly inferred, but they have to be derived after some theoretical considerations based on the balance of the forces acting in the liquid filament. Practically speaking, this implies that desired material information, such as the extent of non-Newtonian behaviour or the relaxation time, are obtained by choosing an appropriate constitutive model or scaling law and by fitting to the diameter temporal evolution data. Despite the apparent simplicity of this technique, its characteristic features also entail a series of great challenges. Amongst them, the most important is how to determine which constitutive model or scaling law has to be considered to obtain material properties. Moreover, a fluid can potentially transition through behaviours ascribable to different constitutive models during a capillary thinning experiments, and so the issue becomes also which part of thinning evolution is representative of a certain behaviour. Although some insights can be obtained by looking at the shape of the curve representing the temporal evolution of the minimum radius of the filament and by the characteristic shapes of the threads, quantitative would surely be more beneficial. In the last three decades, considerable findings have been achieved in this sense, nonetheless, there are still nowadays important gaps to be filled when it comes to quantitative criteria to identify different dynamical regimes.
The possibility of using various experimental configurations brings other challenges: indeed, although the different techniques are supposed to be interchangeable, they do entail small differences and so far too little validation has been carried out to compare results obtained using different instruments. Additionally, there is often scant of understanding on how and which specific experimental parameters influence the measurements and the obtained results. Lastly, as the complexity of the tested system increases, additional uncertainties sum up to the already listed points of concern, making sometimes hardly impossible to have enough confidence in the interpretation of the experimental data to extract quantitative material properties, hence reducing this powerful technique to the source of rather descriptive evaluations.
In the described framework, the aim of this thesis is to solve some of the current challenges in capillary break-up rheometry, by combining different fluids and techniques, as well as developing a theoretical toolbox based on dimensionless analysis. After a short overview on the present advances in describing capillary break-up dynamics, the manuscript proceeds as follows: first, in Chapter 3 the existing different techniques are critically reviewed, pointing out the main advantages and disadvantages of each one of them.
existing challenges in terms of the correct interpretation of the results and the identification of dynamical regimes. In the same chapter, the theoretical framework is applied to some case studies with a static bridge configuration as a proof of concept of the newly developed approach.
In Chapter 5, the challenge of static-bridge capillary break-up experiments involving a liquid filament thinning surrounded by another immiscible liquid rather than by air is dealt with, also by applying the concepts developed in Chapter 4.
Follows in Chapter 6 the study of the effect of stretch parameters in static bridge experiments for Newtonian fluids and a subclass of non-Newtonian fluids, namely concentrated polymer solutions, whose behaviour interpretation constitutes a great challenge.
At last, in Chapter 7 one of the issues related to the confidence level of results obtained from different techniques is explored, by direct comparison of the static bridge and jetting configurations for testing of weakly viscoelastic polymer solutions, another material type most complex to assess. Moreover, in the same chapter the effect of jetting parameters on polymer samples is investigated.
State of the art: Dynamics of
capillary instability
Capillary break-up experiments do not involve controlling or measuring either stress or strain, rather exclusively rely on the action of the capillary pressure inherent to the fluid to drive the process. As a consequence, quantitative estimations of the extensional material properties have to be obtained by interpreting the dynamics underlying an experiment and by fitting the experimental data to the appropriate scaling laws, based on the material constitutive models. Comprehending the dynamics underlying the capillary instability phenomenon, and how to describe them according to scaling laws, is thus an essential step on the route not only towards a general better understanding of capillary break-up rheometry, but also to properly and accurately implement this technique in practice. In this chapter, a short overview about this topic is presented: starting from the general definition of free-surface flows and capillary instability in Section 2.1, the discussion continues with the introduction of the tool of dimensional analysis in Section 2.1.1, to then dive into the actual description of the physics of capillary instability throughout Section 2.2, where a selection of the most common approaches and scaling laws is examined, ultimately followed by a brief synopsis of the transition criteria between different dynamics (Section 2.2.7).
Figure 2.1: Examples of free-surface flows in daily life and common processes. From the top down, and left to right: a spider weaving a web, getting honey from a jar, a ’thumb-and-forefinger’ test with saliva, breaking of a ladyfinger, ink-jet printing, threading of sweet-and-sour sauce, sneezing, drop-on-demand printing, fuel nozzle spraying, paint spraying, mud fracking, 3D printing, electrospun webs, dosing of cleaning products, printed electronics, printing of biological material.
2.1
Free-surface flows and capillary instability
According to Wikipedia: “In fluid mechanics a free-surface flow, also called open channel flow, is the gravity driven flow of a fluid under a free-surface, typically water flowing under air in the atmosphere” [7]. Despite representing a niche of fluid mechanics, free-surface flows are ubiquitous in every day life: indeed, a classic and unambiguous example of free surface flow is a drop of water hanging and, eventually, falling from a faucet. Some commonly encountered applications are displayed in Figure 2.1, amongst which technological operations, such as fuel spray injection, ink-jet printing and dosing, are found together with ordinary life processes, such as biological material threading, food dosing and sneezing. Looking at the examples in Figure 2.1, one might immediately recognize a common feature of free-surface flows, namely the formation, elongation and possible break-up into droplets of liquids, whether filaments, jets or sheets. Triggered originally by the observation of the tendency of falling liquid streams to spontaneously break into droplets, this phenomenon has been a captivating subject for many scientists for centuries, going all the way back to Leonardo Da
Figure 2.2: Evolution of a capillary instability from the orginal cylindrical fluid column (a), to the perturbed configuration (b), to the final break-up into droplets. In (b), λw indicates the perturbation wavelength, R1 and R2 the
curvature radii, P0 and P1 the minimum and maximum pressure. The black
arrows indicate the direction of the ’squeezing’ uniaxial extensional flow. Vinci’s studies [8]. Pioneering experimental works on free-surface flows focussed on dripping and jetting of low viscous liquids and deformation of droplets in an immiscible liquid [9, 10, 11, 12]. In particular, Savart [11] was the first scientist concluding that the break-up of a liquid jet into several droplets is an intrinsic feature of free-surface flow dynamics, as this happens spontaneously and regardless of the external forces. Following Savart’s work, Plateau [10] proved the crucial role played by surface tension, by pointing out that any liquid filament spontaneously breaks up into separate droplets only when subjected to a perturbation that can lead to a reduction of surface area. The real turning point occurred thanks to Lord Rayleigh [9], who employed the equations of fluid motion and linear stability analysis to determine the perturbation wavelength to which a jet is the most unstable, and, consequently, the size of the resulting droplets. Initially developed by Lord Rayleigh for inviscid fluids, the study of surface-tension-driven phenomena has continued until nowadays, including the description of complex problems such secondary satellite droplets, blistering and fragmentation; an extensive review can be found in [13].
In other words, free-surface flows are non confined flows, i.e. characterized by the absence of closed streamlines, and, most interestingly for the topic of this thesis, they are inherently unstable to perturbations, which leads to the phenomenon just described, commonly known as capillary instability, capillary break-up, or Rayleigh-Plateau instability. As made evident by its name, the
main force driving the instability is the surface tension of the liquid itself: assuming to start from a long and thin cylindrical column fluid (Fig. 2.2(a)), in quiescent conditions the free-surface is stabilized by the surface pressure
P = γ/R, where γ is the surface tension of the fluid and R the radius of the
column. If a sinusoidal disturbance is induced on the free-surface, the local fluctuation in the column radius occurs, creating a necked region, or filament, within two swollen regions, or bulges (Fig. 2.2(b)); the pressure gradient created across the free surface can be described by the Young-Laplace equation:
P1− P0= γ 1 R1 + 1 R2 , (2.1) where R1and R2are the principal radii of curvature, P0the pressure within the
bulges and P1the pressure in the filament. Generally, the external phenomena
triggering capillary instability can vary from gravity, to thermal fluctuations, air flows, mechanical and electromechanical stimuli, etc. Once the perturbation has been initiated, according to Plateau’s and Rayleigh’s findings [10, 9, 14], the surface tension spontaneously drives the free-surface to minimise its surface energy, and, if the wavelength of the disturbance λw is sufficiently large,
the evolution in separate droplets is energetically favoured over the original configuration. Therefore, the surface tension drives a progressive thinning of the filament, and, thanks to the pressure difference (P1> P0), the fluid is squeezed
out from necked central region towards the bulges (Fig. 2.2(b)). As a matter of fact, this phenomenon create an actual free-surface unidirectional extensional flow in the filament. On the other side of the coin, fluid properties such as density, viscosity and elasticity resist the squeezing action of the surface tension, delaying, stabilizing, or even arresting the progressive filament thinning. Whenever the fluid does not provide enough resistance in contrasting the capillary action, the liquid column ultimately breaks into separate spherical droplets (Fig. 2.2(c)).
2.1.1
Non-dimensional description
The dynamics of capillary thinning phenomena are determined by the interplay of capillary, inertial, viscous and elastic stresses, which are expressed by material properties in terms of surface tension γ, density ρ, viscosity η and relaxation time λ, respectively. Dimensional analysis can be used as a very effective tool to identify the relative importance of each mechanism, and, consequently, also the dominating one. Moreover, it can be successfully employed to evaluate a free-surface process outcome starting from given material and process parameters (see the scheme in Fig. 2.3). In free-surface flows, dimensional analysis can be carried out either comparing intrinsic characteristic scales, or taking into considerations external forces.
Figure 2.3: Schematics of the work-flow leading to the assessment of the outcome of a capillary break-up process, starting from known material and process parameters by means of dimensional analysis tools.
The first approach is particularly appropriate when dealing with ’self-thinning’ processes, which, as described by McKinley [15], are an important subclass of free-surface flows, in which no external driving force is imposed or at least the internal dynamics are much faster than any external input. In self-thinning processes, the fluid filament thins down and breaks up being naturally driven by capillary forces, counteracted by the fluid viscoelasticity. The filament thinning dynamics are determined by the balance of surface tension with the dominant resisting stress, or, in other words, by the mechanism with the longest characteristic time (or equivalently the lowest characteristic velocity). For a viscous dominated fluid, the resistance to the capillary flow is due to internal friction, and is dependent on the fluid viscosity η via the characteristic time scale
tη = ηR/γ and velocity Uη = γ/η. On the other hand, the effects associated
with the inertia of accelerated fluid elements are characterized by the Rayleigh time scale tR = ρR3/γ and velocity UR= γ/ρR. The relative importance of
viscous and inertial stresses can be determined by comparing the viscous and Rayleigh time scales, yielding the first dimensionless quantity, the Ohnesorge number:
Oh =√η
ργR . (2.2)
Accordingly, high Oh values mean a viscosity dominated balance. Likewise, the importance of elasticity, which is highly dependent on the fluid microstructure, is determined via the fluid characteristic relaxation time tλ = λ and velocity
Uλ = R/λ. According to the dominant mechanism between tλ and tR, the
(a) (b)
Figure 2.4: (a) Evolution of the characteristic speeds as the filament radius decreases [16]. To be noted that Uη does not depend on R. (b) Photographs
taken in proximity of break-up show dramatically different features for different fluids: from left to right, symmetric pinching of a viscosity dominated fluid, beads-on-a-string structure for an elasticity dominated one, and asymmetrical break-up of an inertia dominated liquid.
number:
De = s
λγ
ρR3 , (2.3)
comparing elastic and inertial effects, or with the Elastocapillary number: Ec = λγ
ηR , (2.4)
expressing the relative magnitude of elasticity and viscosity.
The three dimensionless numbers Oh, De and Ec depend solely on material properties and a characteristic length scale, and form the material-property-based dimensionless group. The value assigned to R is, however, extremely important, and will be extensively discussed throughout the manuscript. A possibility certainly consists in choosing as R a characteristic length scale relevant to the considered process; for capillary thinning, this is often identified as a radial dimension, for instance the diameter of the nozzle in a spraying operation. This highlights that the overall dominant behaviour assessed with the dimensionless quantity can potentially vary based on the considered length scale, which in self-thinning processes can vary up to an order of magnitude. Vice versa, one could also use as length scale the radius of the filament itself: in this case R is though a function of time, meaning that different mechanisms might become progressively relevant through the thinning process. Clasen et al. [16] refer to the dimensionless numbers determined with the first definition
(a) (b)
Figure 2.5: (a) A 3D map for jetting operations, showing the combinations of dimensionless groups leading to dripping or jetting outcomes. Adapted from [16]. (b) A 2D operational map for ’self-thinning’ processes [15].
as ’global’ quantities, since they determine the overall thinning dynamics, and the one determined with the second approach as ’local’ quantities, which give an indication of the dominating mechanism in a specific time of the thinning phenomena. An example is given in Figure 2.4a, displaying the evolution of the characteristic speeds: as it can be observed, the dynamics are initially governed by inertial effects, but, as the filament radius progressively decays, viscosity and subsequently elasticity start to dominate the process. Depending on the mechanism governing the latest stages of the capillary phenomenon, the break-up (and post break-up) features appear substantially different, as shown in Figure 2.4b for an example of viscosity, elasticity ans inertia dominated break-up. As shown later in this chapter, the non-dimensional numbers can thus be employed to determine criteria that set the transitions between regimes characterized by different force balances within the same capillary break-up event. Another significant observation is the fact that only two of the three numbers are independent (Ec = De/Oh), which means that any mechanism can be described by a set of 2 dimensionless quantities, normally chosen depending on the most important stresses.
A similar route can be followed to derive a second set of dimensionless quantity for free-surface processes in which an external driving force is applied. For many processes, the external forcing is related of external imposed kinematics, i.e. an external velocity U. The first important group to consider is the Reynolds number, Re, which expresses the relative contribution of inertial and viscous
forces:
Re = ρU R
η . (2.5)
Similarly to the previous dimensionless groups, a local Reynolds number Reloc
can be defined using the local minimum filament radius Rmin. The relative
importance of viscosity, elasticity and inertia compared to the processing forcing can be then expressed, respectively, by the Capillary, Ca, Weissenberg, Wi, and numbers, Weber, We, defined as:
Ca = ηU gamma , (2.6) Wi = λU R , (2.7) We =ρRU2 γ . (2.8)
Clasen et al. [16] summarized the two groups into a single operational diagram, displayed in Figure 2.5a. The map present by the authors allows to identify the process outcome based on Ca, Wi or We depending on the material-property-based Oh, Ec, De. Since the authors specifically address dispensing operations, the external forcing is here the speed of the jet leaving the nozzle Ujet and
the dimensionless quantities define the transition between jetting or dripping regimes. Nonetheless, the same type of approach can be extended to describe other free-surface processes; for instance, in case of a self-thinning process, the map can be conveniently translated in a 2D diagram, as originally proposed by McKinley [17] (see Fig. 2.5b).
The Bond number is the last relevant dimensionless group, and expresses the relevance of gravitational effects compared to capillarity:
Bo = ρgR2
γ , (2.9)
with g the gravity constant. In capillary break-up phenomena, the effect of gravity is usually assumed to be negligible, due to the very small dimensions of the threads. Nonetheless, gravity can sometimes become important, for instance when considering capillary break-up experiments with large sample volumes, and hence the value of the Bo number must be carefully considered.
2.2
Capillary instability
2.2.1
Slender filaments and one-dimensional approximation
Dimensional analysis undoubtedly provides an effective strategy to examine the physical mechanisms underlying a capillary break-up phenomenon. The dynamics of such a free-surface flow can also be described employing the Navier-Stokes equations: ρ ∂u ∂t + (u · ∇) u = −∇P + ∇ · σ + ρg , (2.10) ∂ρ ∂t + ∇ · (ρu) = 0 . (2.11)
The terms on the left-hand side in Eq. 2.10 are the temporal and convective inertial terms, which are balanced on the right-hand side by the terms expressing, respectively, the contributions of capillarity (via the pressure gradient P), the extra stress tensor (∇ · σ), which includes the relevant constitutive equations of the fluid, and gravity. However, solving analytically Eq. 2.10 and 2.11 to get quantitative predictions and extraction of material properties is far too complex even for the most simple cases in which only a few of the aforementioned mechanisms are relevant to the process. Sometimes, even solving numerically the full Navier-Stokes equations for a free-surface flow might demand unfeasible computational times [13]. The approach presented by Eggers and Dupont [18] allows to reduce the 3D problem to 2D, by considering the axisymmetry of the liquid filaments. If the filament is assumed as slender (i.e. the filament is assumed as slender and its axial dimension is significantly larger in magnitude than the radial one), long-wave models allow to decouple the two dimensions and, by carrying out a radial expansion of the pressure and velocity terms, to solve the equations to the lowest radial coordinate order. By further simplifications and a radial expansion of the boundary conditions, Eggers and Dupont [18] gave a one-dimensional approximation of the momentum balance equation for a slender filament in a free-surface flow as:
ρ ∂u ∂t + u ∂u ∂z = −γ∂K ∂z + 3η R2 ∂ ∂z R2∂u ∂z + 1 R2 ∂ ∂z R 2(σ zz− σrr)−ρg , (2.12) ∂R2 ∂t + ∂uR2 ∂z = 0 , (2.13)
where the curvature K is expressed as a function of the filament radius and its first and second order axial derivatives (R0 and R00, respectively) by:
K= 1
R(1 + R02)1/2 −
R00
Figure 2.6: A photograph of a viscoelastic jet shows the progressive growth of the instability as the fluid is convected downstream (left to right), with the initial linear stages and the following achievement of large perturbations. The one-dimensional description lays the ground for the derivation of the analytical solutions of different thinning regimes, depending on which terms of Eq. 2.12 can be neglected according to the nature of the fluid and, as mentioned earlier, to the radial scale of the problem. In the following sections, the description is firstly illustrated for the initial stages of the capillary thinning, when the amplitude of the instability is still small, and secondly for the stages close to the break-up event, where the large deformations require the evaluation of non-linear terms. More detailed information can be found in the reviews by Eggers [19], Eggers and Villermaux [13] and McKinley [17].
2.2.2
Away from break-up: linear stability analysis
When a sinusoidal disturbance is generated on the free surface of a liquid cylindrical jet (or filament) of radius R0, jet, the amplitude of the perturbation
is initially small (see Fig. 2.6). The observations of Plateau based on thermodynamic arguments [10]determined that only perturbations with a wavelength λw > 2πR0,jet leads to a reduction of the free surface and can
grow in time. Rayleigh employed linear stability analysis to determine the most unstable mode and the maximum rate at which a small sinusoidal disturbance of the form R = R0,jet+ sin(kz) evolves on the surface of an inviscid jet [9].
In the following, the jet subscript notation is dropped for conciseness. Here,
k= 2π/λwis the wave-number and the amplitude 1. By linearising the
equations of motion and assuming an infinite slender cylindrical jet, and looking for a temporal evolution of of the form = 0e−iωt, the following dispersion
relation for the instability growth rate ω is found:
ω2= − γ ρR3 0 (kR0) 1 − (kR0)2 I1(kR0) I0(kR0) , (2.15)
where I1, I0 are the modified Bessel functions. For the non-dimensional
wave-number kR0<1, ω2<0 and −1ω is real and positive, hence the perturbation
grows exponentially in time according to:
Rmin= R0,jet− δeωt . (2.16)
The maximum growth rate is found from Eq. 2.15 to be at kR0≈0.7, as it can
be observed in Figure 2.7a, or, equivalently, at λw,max≈9R0.
A non negligible viscous contribution significantly alters the dynamics of the perturbation growth, as demonstrated by Rayleigh in [20], and the dispersion relation becomes: ω= 1 tη 1 − (kR0)2 1 + (kR0)2 " 1 − I1(kR0) I0(kR0) 2# . (2.17) The presence of significant viscous stresses (Oh»1) induces both a reduction of the maximum growth rate, as well as an increase of λw,max, as shown in Figure
2.7c. For the limit η = ∞, the maximum instability growth rate is found at a wavenumber kR0= 0, corresponding to the wavelength being infinite.
Tomotika [21] used a similar approach to introduce the effect of an immiscible outer fluid on the evolution of the perturbation. Using linear stability analysis, he introduced the viscosity ratio p = ηin/ηout into the general equation for ω,
and derived the analytical solutions for an arbitrary value of p, for p = ∞, which returns Rayleigh’s solution for viscous jets, and for p = 0. For the two limiting cases of zero and infinite viscosity ratio (a hollow jet in a viscous medium (Oh«1) and viscous jet in air (Oh»1)), the maximum growth rate is obtained as the disturbance wavelength goes to infinity, as shown in Figure 2.7b. For all intermediate cases, the maximum growth rate occurs at a finite wavelength; furthermore, Stone and Brenner directly derived the solution for the case p = 1 [22].
The presence of non-Newtonian effects was analysed for various constitutive models and was shown to be rather limited. The additional stresses contribute for little to the overall balance, but a small increase of the growth rate is observed (see Fig. 2.7c). An expression for the dispersion relation of ω was derived by Brenn et al. [23] using an Oldroyd model, from which a simplified version for the Oldroyd-B model can be obtained. The non-Newtonian terms, and particularly the elastic stresses if present, on the other hand, do play a very significant role as soon as large perturbation amplitudes are reached.
(a) (b) (c)
Figure 2.7: (a) Comparison between experimental [24] and predicted [9] non-dimensional growth rates (ω∗, rescaled with respect to t
η) as a function of
kR0for an inviscid jet (adapted from [13]). (b) Comparison of the dispersion
relation of the non-dimensional growth rate (ω∗, rescaled with respect to t η) for
different viscosity ratios p = ηin/ηout. (c) Effect of the fluid nature on the shape
of the dispersion relation: both the presence of non-Newtonian effects and an increase in viscosity shift the maximum instability growth rate towards lower wave-numbers compared to the inviscid case; at the same time, non-Newtonian effects result in a higher maximum growth rate compared to the equivalent Newtonian case (adapted from [23]).
2.2.3
Approaching break-up: non-linear effects and similarity
solution
As the instability grows and strains become progressively larger, non-linear effects are no longer negligible and linear stability analysis fails to capture further the evolution of the instability. Yet, when the perturbation has grown sufficiently large, the liquid threads connecting two adjacent instability bulges become long and thin enough to apply the one-dimensional approximation of Section 2.2.1. These slender liquid filaments display another remarkable feature: when approaching break-up, the filaments of similar fluids are observed to retain the same shape, independently of the initial conditions. This phenomenon results from the progressive localization of the break-up in the spatial and temporal domains, caused by the increasing thinning rates and the subsequent inability of the fluid elements far from the singularity to continue following the fluid motion. At this point, the fluid motion is solely influenced by internal length, lv= η2/γρ, and tv= η3/γ2ρtime scales, which depend exclusively on
the fluid properties. This fingerprint of the capillary break-up phenomenon suggested that the behaviour approaching the break-up singularity can be captured using self-similarity and, consequently, the set of 1D equations of motion was solved by looking for similarity solutions for a variety of cases. Thanks to the independence on external length scales, the self-similar behaviour
Figure 2.8: Similarity functions for the radius and velocity profiles derived by Papageorgiou for the V regime [19, 25].
allows for rescaling the filament radius and velocity profiles with respect to the internal scales, and the similarity functions φ(ξ) and ψ(ξ). The radius and velocity profiles can thus be expressed, in terms of dimensionless time t0 and
axial coordinate z0 as:
R(z, t) = φ(ξ)lv|t0|α1 (2.18)
u(z, t) = ψ(ξ)lv tv
|t0|α2 , (2.19)
where xi is the similarity variable, defined as ξ = z0/|t0|β by the dimensionless
coordinates: z0= z − z0 lv , t0= t − tb tv (2.20) with tband z0the time at which the singularity occurs and the origin of the
coordinate system. The value of the similarity exponents α1, α2and β depends
on the governing momentum balance. Depending on the relative magnitude of inertial, elastic and viscous stresses, different scaling laws based on self-similar considerations were derived to describe the temporal evolution of the filament profile [19]. The following sections give an overview of the most commonly encountered solutions for Newtonian and non-Newtonian fluids, surrounded by air or by different outer media.
2.2.4
Newtonian fluids: a multitude of scaling laws
When dealing with Newtonian fluids undergoing capillary break-up in air, the fluid motion is determined by the balance of capillarity, inertia and viscous friction (assuming a negligible effect of gravity). For purely viscous fluids,
characterized by a large Oh number, the similarity solution is obtained neglecting the inertial terms in 2.12, hence reducing it to the one-dimensional Stokes equation. Papageorgiou [25] derived a symmetric similarity solution to the Stokes equation, with its minimum at φ = 0.0709 (see Fig. 2.8); according to Papageorgiou’s solution, in a purely visco-capillary regime (V) the minimum filament radius decays linearly in time, following:
Rmin(t) = 0.0709
γ
η(tb− t) , (2.21)
with tbthe time corresponding to the break-up event. Next to the radius linear
decay, a capillary instability governed by a visco-capillary balance exhibits the persistence of a long living filament, with a symmetrical shape about a centrally localized minimum. It must be noted, however, that the present similarity solution is still partially dependent on the external length scales, via a single parameter setting the width of the solution.
In the case both the viscous and inertial contributions of the fluid play a significant role, Eggers employed a self-similar approach to obtain a universal solution to the one-dimensional Navier-Stokes flow [26]. The similarity solution found by Eggers for the inertia-visco-capillary regime (IV) is independent of the initial conditions and, likewise 2.21, describe a linear decay of the minimum filament radius in time. Conversely to the one determined by Papageorgiou, the minimum of the solution is here φ = 0.0304, leading to a different pre-factor and a solution of the form:
Rmin= 0.0304
γ
η(tb− t) . (2.22)
Although also linear, the Navier-Stokes solution in 2.22 reveals an asymmetric pinching of the filament, where the minimum is located at the edges of the thread and near the droplets, with a shallow slope side belonging to the filament and a steep one to the bulges [26]. Figure 2.9 shows the similarity functions derived by Eggers. The solution was also later revealed the one, out of an infinite family of solutions, to be least susceptible to finite perturbations, while every other destabilized alternative self-similar profile returns to 2.21 [27]. Lastly, for very low Oh, the thinning dynamics are set by the balance of the inertial and capillary terms, reducing 2.12 to a Euler (or potential) flow. In this inertia-capillary regime (I), the solution follows a power law behaviour of the form [28, 29, 30]: Rmin= 0.64(γ/ρ) 1 3(tb− t) 2 3 . (2.23)
The exact value of the numerical front factor in 2.23 is, however, not unequivocally determined in the literature: on the one hand, Tirtaatmadja et al.[30] state having obtained the value 0.7 from [28] and Eggers suggests
Figure 2.9: Similarity functions for the radius and velocity profiles derived by Eggers for the IV regime [26].
the same correlation to have been proven experimentally by Burton [31]. On the other hand, the authors in [32] calculate a pre-factor of 0.67, although the number 0.64 appears in [33], [16], referring again to [28], [29]. Regardless of the numerical pre-factor value, as already apparent from the Rayleigh time tR,
and here recalled in the temporal exponent 2/3, the inertia-capillary balance is characterized by considerably faster thinning rates, which do not allow for the formation of a long slender thread, but rather a strongly asymmetrical filament shape, which resembles a conical wedge, displaying angles of 112.8°and 18.1° [29] (see Fig. 2.13).
2.2.5
Effect of a different medium: the outer fluid
The effect of the medium surrounding the capillary instability has been until now not taken into considerations. Nevertheless, on small scales an significant additional contribution might be given by the viscous friction of the surrounding fluid [34], setting a balance between capillarity, inner and outer viscous drag. While in the case of air this is most probably happening at atomic length scales, for higher viscous fluids the effect can be visible already at length scales that can be resolved experimentally, and described by continuum mechanics. Based on scaling arguments, Lister and Stone [34] obtained a solution for a viscous liquid surrounded by a fluid of non-negligible viscosity (viscous-viscous regime (LV)), which takes on the form:
Rmin= H(p)
γ ηout
(tb− t) , (2.24)
(a) (b)
Figure 2.10: Photographic images of dripping experiments in two-fluid systems for different fluid combinations in (a); rescaled radius profile and radius linear temporal evolution (inset) in (b) [35].
again to follow a linear decay in time, yet the filament exhibits asymmetric conical features, with the necked region located in the proximity of the droplets. This regime was proven to be self-similar in nature by Zhang, Sierou and co-workers, who also demonstrated a good agreement of simulations with the observed thinning rates, as well as filament cone angles [36, 37]. Additionally, the velocity field around the break-up point was numerically and analytically shown not to be local, thereby determining a competition between the conical region of the filament and the droplet side and an axial shift of the location of the minimum radius [34, 36]. In a series of papers following these conclusions, the viscous-viscous problem was revisited with the addition of experimental observations [38, 35, 39], where the expected linear radius decrease is observed for a range of viscosity ratios 10−3<(η/η
out) < 10 (see Fig. 2.10). The numerical
pre-factor H was predicted and experimentally verified to be a function of p, although no analytical expression for the determination of H is found in the literature. In an attempt to explain this trend, Tomotika’s results from the linear stability analysis [21] were employed, and the maximum exponential growth rate ω was shown to form an upper bound for the radius decrease [36, 38] and the slope of the shallow cone to be relatively well correlated with the wavelength at maximum growth λmax. Nonetheless,s significant discrepancies were noted
for both small and large viscosity ratios [35], reason for which the dependency of the front factor H on the inner fluid viscosity remains still unclear.
(a) (b)
Figure 2.11: (a) Images of dripping experiments showing the features of the non-universal behaviour at low viscosity ratios (here p = 10−4) [39]. (b) Photographs
of a bubble pinching in silicon oil, showing the characteristic hyperboloidal shape [40].
Approaching zero viscosity: bubbles and low viscosity ratios
The capillary break-up of inviscid fluid filaments in very viscous fluids, such as breaking of air bubbles in honey, behaves significantly different than larger viscosity ratios. Doshi and co-workers [41] discovered that the thinning of an inviscid fluid retains memory of the boundary and initial conditions, an exceptional phenomenon that infringes the results of the previous studies, and that the filament maintains a characteristic quadratic profile down to the break-up event. The authors further prove that this peculiar form of singularity is cut off in presence of a finite inner viscosity, even if very small, and the formation of a surprisingly long and thin filament is observed, as shown in Figure 2.11a. Nonetheless, for extremely small viscosity ratios, this phenomenon might happened only at subatomic scales, and hence not be visible in experiments or simulations. Following the observation of this non-universal behaviour, Burton at al. [31] showed that the break-up of an inviscid fluid follows a power-law scaling, of the form Rmin∝ tα, where α ≈ 0.57, as further observed
experimentally by [40, 42], and the filament shape resembles an hyperboloid of rotation (see Fig. 2.11b).
2.2.6
Non-Newtonian effects
Elastic fluids
While the non-Newtonian effects were stated in 2.2.2 not to be particularly relevant in the early stages of capillary thinning phenomena, as the deformations in the filament become progressively larger, they start playing a very significant
(a) (b)
Figure 2.12: (a) Adding a flexible polymer in solution (b) allows to stabilize the filament form between subsequent beads compared to the neat solvent (a) [43]. (b) Example of a thinning curve of a dilute polymer solution obtained from a
CaBER experiment. After the initial I regime, the radius follow an exponential decay (linear in the represented log-lin scale), correspondent to the EC regime, here indicated with the dashed line. The cartoon in the inset illustrates the effect of the extensional flow on the polymer coils, as the chain are perturbed from their coil state and progressively stretched.
role. Taking for example a solution containing flexible polymer chains, the extensional flow in the necking filament is very effective (much more than shear flow) in unravelling and stretching the chains from their unperturbed coil state. The deformation of the polymer chains, on the other hand, leads to a rapid built up of additional stresses in the filament, which stabilize the thinning and form a long cylindrical thread between the droplets. The stabilization of the filament depends on several parameters, amongst the most important are the polymer molecular weight and its concentration, as described in [43]. An example of the effect of polymer stabilization can be seen in Figure 2.12, which compares the jet break-up of a Newtonian (a.a) and a viscoelastic (a.b) solution. Throughout the years, different constitutive models were used to model the elastic contribution to the momentum equation, nonetheless, Bousfield et al. [44] first employed the Oldroyd-B model to show, by means of numerical analysis the elastic stresses follow an exponential growth, which takes place as soon as the extensional flow achieve a Weissenberg number Wi > 1/2, corresponding to the strain rate necessary to perturb a polymer chain from the coil to a stretched state. Thereby, when the strain rates have grown large enough in the necking filament, the thinning dynamics are determined by an elasto-capillary balance (EC), with a resulting temporal evolution of the minimum radius of the form:
Rmin∼exp − t 3λ , (2.25)
