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Zami-Pierre, Frédéric and Loubens, Romain de and Quintard, Michel and Davit, Yohan Effect of disorder in the pore-scale structure on the flow of **shear**-**thinning** **fluids** through porous media. (2018) Journal of Non-Newtonian Fluid Mechanics, 261. 99-110. ISSN 0377-0257

2 Institut de M´ecanique des Fluides de Toulouse (IMFT), Universite´ de Toulouse, CNRS, Toulouse, France
The present work investigates experimentally the dynamics of vortex rings in **shear**-**thinning** **fluids** at low generalized Reynolds numbers, with a focus on the range from 300 down to 30. The experimental apparatus consists of a vertical cylinder-piston system with the lower part immersed in a tank filled with the liquid. Particle image velocimetry is used to analyze the influence of the non-Newtonian nature of the fluid on the generation, propagation, and eventual dissipation of vortex rings. The results show that **shear**-**thinning** controls the generation phase, whereas the vortex ring subsequent evolution is independent of the power-law index. In particular, it is found that the final dissipation stage is char- acterized by a flow dynamics which tends ultimately to a regime at a constant viscosity corresponding to the Newtonian plateau. This reveals the role of the Carreau number and of the Reynolds number based on this specific viscosity as relevant control parameters for this last stage.

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2 Institut de M´ecanique des Fluides de Toulouse (IMFT), Universite´ de Toulouse, CNRS, Toulouse, France
The present work investigates experimentally the dynamics of vortex rings in **shear**-**thinning** **fluids** at low generalized Reynolds numbers, with a focus on the range from 300 down to 30. The experimental apparatus consists of a vertical cylinder-piston system with the lower part immersed in a tank filled with the liquid. Particle image velocimetry is used to analyze the influence of the non-Newtonian nature of the fluid on the generation, propagation, and eventual dissipation of vortex rings. The results show that **shear**-**thinning** controls the generation phase, whereas the vortex ring subsequent evolution is independent of the power-law index. In particular, it is found that the final dissipation stage is char- acterized by a flow dynamics which tends ultimately to a regime at a constant viscosity corresponding to the Newtonian plateau. This reveals the role of the Carreau number and of the Reynolds number based on this specific viscosity as relevant control parameters for this last stage.

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2. Non-Darcian Flow of **Shear**-**Thinning** **Fluids** in Porous Media
The use of Newtonian assumption in equations (1–5) is not realistic in the case of **shear**-**thinning** ﬂuids, lead- ing to important errors. Indeed, l depends on Q for these ﬂuids, so this relationship must be included in the mentioned equations. In addition, although numerical and theoretical studies have stated that the iner- tial coefﬁcients b and c do not depend on the ﬂuid rheology, no experimental evidence has been pre- sented. In particular, Tosco et al. [2013] demonstrated through numerical experiments that the value of the inertial parameter b is independent of the viscous properties of the ﬂuid. Also, c was analytically and numer- ically shown to be a porosity-dependent parameter by Firdaouss et al. [1997] and by Yazdchi and Luding [2012]. Therefore, experimental validation of these results is most valuable. A potential experimental issue concerns elongational ﬂows, which are known to induce extra pressure losses as compared to those pre- dicted by pure **shear** ﬂow during the injection of solutions of polymers presenting a certain degree of ﬂexi- bility through changing cross-sectional area media such as porous media [Rodrıguez et al., 1993; M€ uller and S aez, 1999; Nguyen and Kausch, 1999; Seright et al., 2011; Amundarain et al., 2009]. This was attributed to the formation of transient entanglements of polymer molecules due to the action of the extensional compo- nent of the ﬂow. Measurement of extensional viscosity is tricky and several methods have been proposed for its determination [Schunk et al., 1990; Petrie, 2006; Collier et al., 2007]. One of the simplest methods con- sists in the use of an opposed-jet device [Willenbacher and Hingmann, 1994; Gonz alez et al., 2005]. However, important differences between the results obtained by the available methods were reported [Petrie, 2006]. The empirical Carreau model [Carreau, 1972] is one of the most popular models to represent the **shear**- **thinning** behavior of semidilute polymer solutions [Sorbie et al., 1989; L opez et al., 2003; Rodrıguez de Castro et al., 2016] commonly used in Enhanced Oil Recovery (EOR) and soil remediation. The Carreau equation is based on molecular network theory and is often presented as l2l 1

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part. The off diagonal component is also biased in an analogous manner and can grow to comparable magnitude, rendering the system strongly anisotropie. Anisotropicity can be asse;sed on the basis of a single scalar parameter, the anisotropy factor c5; whereas the variation of this factor is monotonie with respect to the two rheological constants, the behavior is non trivial when analyzecl with respect to the geometry of the porous medium (expressed via 0) and the inclination angle of the pressure gra <lient, a. This appears to represent an obstacle to the development of simplified models of the flow of **shear** **thinning** **fluids** in porous media. However, the deviation of the mean flow with respect to the macro scopie pressure gradient is consistently limited to a few degree; for a wide range of parameters, so that approximating it to zero should con stitute an acceptable approximation.

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Velocity flow patterns for the screw agitated vessel with and without draft tube are shown in Figure 4. In Figure 4(a), when the geometry does not include a draft tube, the Newtonian fluid is projected in a radial direction away from the impeller with some axial movement. Due to the absence of the draft tube, the radial flow generated by the impeller edge, continues in a radial direction towards the vessel wall. The flow is then reoriented in an upwards direction due to the presence of the vessel wall and by other upward moving fluid. This causes the formation of self-feeding zones at the impeller edge. This phenomenon is even more pronounced when mixing non-Newtonian **fluids**, characterising inefficient fluid circulation. In Figure 4(b), a dominant axial circulation is observed. The screw pushes the fluid downwards in the draft tube with a simultaneous strong radial component at the outer edge of the impeller. The presence of the vessel bottom induces a change in direction of the fluid motion and the fluid is pumped upwards in the annular region between the draft tube and the vessel wall. A circulation loop is formed as the liquid is pushed down once again, into the draft tube. For the non-Newtonian liquid with draft tube, Figure 4(c), the velocity flow patterns are identical to those observed in the same vessel mixing a Newtonian liquid.

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Maxime Chinaud, Thomas Delaunay, Sébastien Cazin, Emmanuel Cid, Philippe Tordjeman.. To cite this version:.[r]

Open Archive TOULOUSE Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web[r]

1. Introduction
**Fluids** whose apparent viscosity decrease; under **shear** strain are very comrnon, and are often found in polymer and foam solutions; also corn plex **fluids** and suspensions like ketchup, paints and blood exhibit such a property which goes by the name of **shear** **thinning** or pseudo plasticity. The oldest model used to describe the rheological properties of pseudo plastic **fluids** is the empirical power law equation (Ostwald, W. (1925, 1929 )) which relates the **shear** stress to the **shear** rate elevated to a certain power, say n, with n < 1, via a coefficient called the flow con sistency index. For n = 1 the Newtonian behavior is recovered. The sim ple power law behavior yields infinite effective viscosity as the applied stress vanishes, and this can cause numerical clifficulties in applications, which is why more elaborate models have later been proposed, such as the Carreau, Carreau Y asuda, Cross or Powell Eyring models (Bird, R.B., Armstrong, R.C., Hassager, O. (1987); Tanner, R.I. (2000 )) . Ail of these models are reasonably simple to implement (for example in a numerical code), requiring no more than three empirical constants; they all yield rather good results, provided the fitting parameters are well chosen, as shown in Fig. 1 for a repre;entative engine oil in solution with viscosity index improver polymers (Marx, N., Fernândez, L, Barcel6, F., Spikes, H. (2018) .

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disregarded in the stability analysis, since at the critical flow rates for the onset of elastic flow instabilities these are still very weak. However, we have shown here that for **shear**-**thinning** **fluids** the instability onset shifts towards higher applied flow rates. At these largest flow rates secondary flows may be relatively strong, and it remains unknown whether they have an impact on flow stability (for example by advecting flow perturbations towards regions with a smaller flow curvature). Nevertheless, experiments performed by Larson et al. 16 in Taylor–Couette flows using **shear**-**thinning** polystyrene solutions also exhibited flow stabilization. Note that in the laminar large aspect ratio Taylor–Couette flows no secondary flows develop along the transverse direction (nor in cone-plate flows). We can therefore infer that secondary flows in serpentine channels, even though they cannot be disregarded, cannot be seen as the only mechanism responsible for flow stabilization.

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Beside intrinsic quality and characterisation clarity, computational speed is another important matter. As a low level image processing operator, skeletonisation can take advantage of parallelism. The most common way to do so is to implement skeletonisation as a **thinning** procedure that iteratively peels objects. Then massive data parallelism [2] proves particularly suitable due to the locality of the **thinning** procedure: only the values of the closest neighbours are necessary to decide whether a pixel can be removed or not. The number of iterations is determined by the maximal object thickness (the radius of the largest ball contained in the image). Another key advantage of data parallelism is that it allows an intrinsic isotropy of the **thinning** procedure, which is most important to ensure mediality. This is a point where many sequential algorithms fail.

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examples are presented in Section 4. We are not aware of any other method for **thinning** experimental designs that is applicable to data streaming; nevertheless, in Section 5 we compare our algorithm with an exchange method and with the IBOSS algorithm of Wang et al. (2019) in the case where the N design points are available and can be processed simultaneously. Section 6 concludes and suggests a few directions for further developments. A series of technical results are provided in the Appendix.

4 Discussion
Comparison between STANDFIRE and FFE-FVS results is challenging as the two models have very different out- puts; FFE uses ROS and fire intensity in internal calcula- tions of flame length but does not report those values. However, assuming that flame length increased with ef- fective wind speed, the models qualitatively agree on this point, as ROS and QSRF both increased (QSRF only slightly) in STANDFIRE. Mortality, one of the few met- rics they have in common, however, showed very differ- ent outcomes between the two models, with predicted mortality decreasing following **thinning** for all sites with STANDFIRE, while increasing in two of the three sites with FFE. This difference is important, as reduced fire- induced mortality relates to resilience, and increasing re- silience is an important goal of fuel treatment efforts (Larson and Churchill 2012 ). Most likely, the driving fac- tor between these differences is the dependency in FFE on stand level values, such as CBH, in mortality calculations. These single stand values are often highly volatile,

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T o get a sound denition of su
h topologi
al proper- ty in our
ubi
grid, spe
ial
are must be taken in the
hoi
e of the
onne
tivity . In parti
ular, an ob- je
t may be
rossed by a
onne
ted
omponent of the ba
kground only if there is a hole through it ! In this respe
t, it is usually
hosen the strongest
onne
tivity for the ba
kground (i.e. fa
e sharing), and a weaker one for the obje
t itself (i.e. edge or vertex sharing). The
onne
tivity model that is used in this paper is (26,6)-
onne
tivity , whi
h means 26-
onne
tivity for the image and 6-
onne
tivity for the ba
kground. Our **thinning** pro
ess works by iterative deletion of sets of points. The
entral notion around the
hara
- terization of the deleted points is simpli
ity. A point is simple if its deletion does not
hange the topology. As in 2D, the
omputation of simpli
ity
an be done within a nite neighborhood of the point. The most
on
ise
hara
terization is provided by Bertrand and Malandain in [1℄:

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The first investigation of the squeeze phenomenon dates from 1874, when professor Josepf Stephan drafts an analytical solution to the normal force necessary to distance two immersed plates, between which a fluid film is placed. This formula ( eq. 2.45 ), was obtained in the hypothesis of a constant contact surface between fluid and plates (see Figure 2.12.a ) for a purely viscous incompressible fluid. After a short while (1886), O. Reynolds deduces the “Reynolds approximation for lubrication” from the Navier-Stokes system and confirms Stephan’s theory [ 172 ]. J.R. Scott makes the transition from purely viscous **fluids** to **fluids** with a generalized Newtonian behavior in 1931, investigating the squeeze phenomenon of polymeric materials (resins) by means of an innovative experimental device called “parallel plate platometer”. He formulated the expression of the squeeze force for a generalized Newtonian fluid using the power-law model, a model used to confirm the experimental data in several studies of that time (Peek, 1932). The first dynamic squeeze test was performed in

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I. INTRODUCTION
In recent years, there have been many developments in both the basic formulation of Large Amplitude Oscillatory **Shear** (LAOS), as well as in its applications to the study of complex **fluids**. On the theoretical side, there are still issues to be addressed concerning the most appropriate way to capture and represent material data, and for these reasons LAOS remains an active area of research. From a practical standpoint, the utility of LAOS as an experimental methodology is that it allows both linear and nonlinear behavior of an unknown material to be probed within a single test protocol. Moreover, the ability to independently modify the frequency and strain amplitude of the oscillations enables the mapping of an entire phase space (commonly termed the Pipkin space (Pipkin, 1972)) and provides a “rheological fingerprint” of the material behavior.

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The remainder of the paper is organized as follows. Section 2 presents the equipments, as well as the polymer used and more precisely its bulk behavior. Section 3 introduces the leveling process and the experimental results demon- strating that improved models have to be considered to well describe the polymer reflow. Section 4 extends the code already used in a previous publication [22] to non isothermal flows and shows the impact of thermal diffusion and **shear**- **thinning** on the leveling process for a Newtonian behavior. The additional effect of **shear**-**thinning** is developed in section 5 where the relevance of the leveling method and the material parameters derivation are also discussed.

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6. Discussion
6.1. Potential limitations and solutions for NNSC 6.1.1. Selection bias
The aim of this paper is to show the performance of NNSC in calibrating **shear** measurement bias. To this end, we applied the following catalogue selection in order to remove any selection bias from the data. Any galaxy whose detection or shape mea- surement failed in any of the processes was removed from the catalogue, together with all its **shear** versions. This included not only the **shear** versions that were simulated with G alsim , but also the images derived from the M eta C alibration processing so that M eta C alibration has no selection bias either. More- over, when the data were split into bins of a measured variable, all the **shear** versions were removed as well if a galaxy fell in di fferent bins for different **shear** versions. With this, the selected data of our results were identical for all **shear** versions, and selec- tion bias was forced to be zero.

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Pro- ceedings of IEEE Computer Society Conference on Pattern Recognition and Image Processing, 1981, pp. H., An improved parallel algorithm for thinning digi- tal patterns[r]

Although our results are derived in a mean field set- ting, we expect that they describe accurately the critical exponents associated to the **shear** jamming line in finite dimensions, as it is the case for γ = 0 [24, 26]. Indeed, the **shear** jamming line has the same critical properties of the isotropic jamming transition [25]. On the contrary, even at the mean field level, the critical properties of the **shear** yielding line are not fully understood, because this line falls in a region where the glass is marginally stable and a full replica symmetry breaking scheme is needed [34]. Moreover, because the yielding transition is a spinodal point in presence of disorder, it cannot be strictly described by mean field in any dimension [59, 60]. A detailed characterization of the yielding transition is thus a very difficult task and it is certainly a very im- portant line for future research. However, at the crit- ical point ( ϕ b c , γ c ), we conjecture that the system sizes

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