Liquid aluminum is a Newtonian fluid, e.i., the viscosity, μ , (the ratio of shear stress to shear rate) is constant with a value of approximately ~ 0 . 002 Pa ⋅ s [ 3]. With semi-solid aluminum, the shear stress to shear rate ratio is not constant and the fluid belongs to the family of non-Newtonianfluids. High pressure die casting produces a large amount of shearing during die filling and the viscosity of the fluid therefore varies over a wide range. The fluid exhibits a thixotropic behavior whereby the non-Newtonian viscosity decreases with time under constant shearing. This is caused by disagglomeration and morphological changes in the solid particles  but this effect is difficult to quantify in HPDC since mold filling typically occurs in less than half a second. Semi-solid aluminum also exhibits a shear thinning behavior  where the non- Newtonian viscosity,η , decreases with shear rate and may be expressed with a power law model.
not a purely geometrical parameter depending only of the mixing system as assumed by Metzner and Otto (1957).
Then, it will be illustrated how the case of heat transfer with Newtonianfluids is analogous to the non-Newtonian flow under isothermal conditions in a stirred vessel considered here. Indeed, a spatial distribution of viscosity exists in both cases (respectively, induced by a tempera- ture distribution and a shear-rate distribution within the flow domain), and the common aim is to account for the effect of viscosity distribution on the output of the system (respectively, heat transfer coefficient and power con- sumption). This will allow us to discuss the need of determining effective quantities in order to model the process.
To cite this version :
Bentata, Omar and Brancher, Pierre and Anne-Archard, Dominique Generation and maturation of a vortex ring in non- Newtonianfluids . (2012) In: XVIth International Congress on Rheology, Aug. 5-10, 2012 , Lisbon, Portugal. (Unpublished)
 M. C. Delfour and J.-P. Zol´ esio. Shapes and geometries: metrics, analysis, differential calculus, and optimization, volume 22. SIAM, 2011.
 L. Diening, M. R˚ uˇ ziˇ cka, and J. Wolf. Existence of weak solutions for un- steady motions of generalized Newtonianfluids. Annali della Scuola Nor- male Superiore di Pisa. Classe di scienze, 9(1):1–46, 2010.
Following a procedure similar to ( 41 ), Séon et al. ( 42 ) investigated experimentally the mixing of two fluids with different properties in a long tube that could be tilted at different angles, 0°< θ < 90 °. An important finding was that by increasing θ, the macroscopic diffusion coefficient increases strongly. On the other hand, it was found that the dependency of the macroscopic diffusion coefficient on the Atwood number is not strong: for example, if the Atwood number increases from At = 4 × 10 −3 to At = 3.5 × 10 −2 , the macroscopic diffusion coefficient only increases by 30%. In addition, Séon et al. ( 43 ) studied the buoyancy driven miscible front dynamics in tilted tubes as a function of the Atwood number, pipe inclination, and fluid viscosity. They found that by increasing θ and keeping all the other parameters fixed, three different flow regimes are observed. As θ increases the front velocity increases sharply (θ changes between 10° to 65°). It is due to the Boycott effect (the heavy fluid is locally separated from the light fluid and therefore sedimentation velocity increases ( 69 )). Close to the front tip, the flow is turbulent and the transverse mixing is strong (regime 1). By increasing θ, 65°< θ < 82°, the front velocity reaches a plateau and remains at a maximum value (regime 2). In this regime, the segregation of the displacing and the displaced fluids is strong and the transverse mixing is not observed. In addition, the mean concentration profile is not diffusive. In the third regime, θ > 82°, the fluid layers are separated into two counter currents, which are almost parallel. The control parameter is the viscosity and therefore this regime is called the viscous regime. In this regime, by increasing the pipe inclination, the front velocity decreases. Owing to the interesting features of the plateau regime, Séon et al. ( 44 ) studied the buoyancy driven mixing in tilted tubes using Laser Induced Fluorescence (LIF) technique. They looked more precisely at the point-wise concentration field and studied the penetrating front inside the pipe. Fig. VII illustrates the images of the concentration field for the exchange flow obtained at different pipe inclinations using LIF. This figure proves the segregation induced by tilting the tube. The front velocity ( ˆ V f ), and the macroscopic diffusion coefficient ( ˆ D M ), in the strong mixing regime in a tilted tube were studied experimentally as a function of the Atwood number, the viscosity, and the tube diameter by Séon et al. ( 46 ). They found that the normalized front velocity, ( ˆ V f / ˆ V t), and the normalized macroscopic diffusion coefficient, ˆ D M /( ˆ V t D) ˆ , are scaled respectively as Re −3/4
We expose ways to model mixing phenomena for Newtonianfluids un- der unsteady stirring conditions in agitated vessels using helical ribbon impellers. A model of torus reactor including a well-mixed zone and a transport zone is considered. The originality of the arrangement of ideal reactors developed in [5, 6] lies in the time-dependent location of the boundaries between the two zones. Interestingly, this concept is applied to model the positive influence of unsteady stirring conditions on homog- enization process. It appears that this model allows the easy derivation of a control law, which is a great advantage when optimizing the dynamics of a mixing process. We now detail this model.
Returning to the historical perspective, it can be remarked that in the early decades of the twentieth century, only the occasional rheological study was undertaken. However, during and after the Second World War, rheology emerged to become very vital from a practical standpoint. It was discovered that when high molecular weight polymers like natural rubber are dissolved in gasoline (to make flamethrower liquids), they exhibit very unusual behavior. It was also observed that when these types of solutions are mixed in an agitated vessel, they climb up the agitating shaft and even out of the vessel. It was then found that the materials used in the flamethrowers are viscoelastic and rod climbing is caused by normal stresses generated by the shearing. This fact was the starting point for original research on material rheology during the War. For rheological measurements, materials are usually investigated under standard flows such as simple steady shear flow, small- amplitude oscillatory-shear flow, and extensional flow (Howard A. Barnes & Hutton, 1993). When viscosity decreases with increasing shear rate, we call this fluid shear-thinning. It should be mentioned that the shear-thinning type of fluid is the most commonly encountered time- independent fluid behavior. Almost all non-Newtonianfluids display shear-thinning behavior under appropriate conditions (Howard A. Barnes & Hutton, 1993; R. P. Chhabra, 2006; Velez- Cordero & Zenit, 2011). In the case of shear-thinning fluids, the power-law model has been widely used to predict the apparent viscosity and describe their rheological behaviour as follows:
In addition to the behavior of the fluid, the first crucial process at play during two-phase flow is deposition. Thus, we studied the analogous of the Bretherton problem for yield-stress fluids. We used glass capillaries (i.e. with a negligible roughness) chemically treated to avoid wall slip. With no-slip boundary conditions, the deposition process can be described by considering the flow inside the dynamic meniscus and the classical scaling arguments proven for Newtonianfluids. Therefore, the deposited thickness depends on a balance between friction inside the thin film, associated with the internal stresses of the fluid, and the capillary pressure gradient that develops inside the meniscus. Although, there exists no simple analytical expression of the film thickness for a yield stress fluid, the results obtained with this approach are in good agreement with the experiments. However, the approach breaks down when lubrication approximation does not hold any more, which is anticipated by the modeling. Finally, since the viscous stress, the capillary stress, and the yield stress are taken into account, the Bingham number, that quantifies the stress state of the system, or the capillary number can both be used to describe the results.
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-Newtonianfluids, 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.
Vortex rings are coherent vortical structures widely presents in geophysical flows and engineering applications. Numerous applications imply industrial processes including food processing, or petrol industry. Those applications are very often confronted with non-Newtonianfluids. Nevertheless, to the best of our knowledge, only few studies dealing with vortex dynamics in non-Newtonian shear-thinning fluids exist, and none with viscoelastic ones.
Next, by rapidly moving the top plate to a higher position, an axial step- strain is imposed onto the liquid bridge Fig. 7(b). The shape of the liquid bridge evolves under the combined action of several physical processes: the capillary pressure which here plays the natural role of a ”force transducer”, viscous dissipation that the necking of the fluid filament and, in the case of polymeric fluids, the elastic forces that equally oppose the necking process. Bazilevsky and coworkers were first to propose a theoretical framework to describe the thinning of a Newtonian and an Oldroyd-B fluid filament in terms of measurements of its radius, Bazilevsky et al. (1990). Their analysis has been extended by Entov and Hinch to account for both the effect of a spectrum of relaxation times and of the finite extensibility of the polymer chains, Entov and Hinch (1997).
The Rayleigh-Bénard Convection (RBC) is a buoyancy-driven in- stability, i.e. it occurs in a ﬂuid layer heated from below and submitted to a vertical temperature gradient. This conﬁguration is widely en- countered in nature and also in industrial processes, justifying the im- pressive volume of work devoted to its understanding since more than a century especially in the case of Newtonian ﬂuids, reviews are provided by Refs. [ 13–15 ] and some recent studies are done by Refs. [ 16–20 ]. Compared to the Newtonian case, the RBC in viscoplastic ﬂuids has
1.4 Fluid/gravity correspondence
The fluid/gravity correspondence is the limit of the AdS/CFT correspondence in the regime where the boundary strongly-coupled field theory is well-approximated by its long wavelength effective description. The corresponding bulk is given by black holes in classical general relativity, with the possibility of adding supersym- metries and potential string corrections. Two alternative expansions are available for finding the holographic fluid dual to an asymptotically AdS black holes. The first expansion consists in perturbing a gravity solution, such as a black hole or a black brane, by means of isometry transformations whose infinitesimal pa- rameters depend on the coordinates on the boundary of the AdS bulk space. The transformed expression is no longer a solution of the bulk equations of motion. In order for it to be a solution, the local boundary parameters should satisfy some dif- ferential equations which turn out to be, at first order, the linearized Navier-Stokes equations for relativistic fluids. The procedure can be made iterative and thus in principle it can be used to compute the expansion at any desired order. Details on this expansion can be found, for example, in  and .
parameters. (3) Particle migration; Gradients in shear rate and elastic stress, and nonzero streamline curvature promote particle migration; Most rheometric shear flows have circular streamlines. The measurement time should be short as compared to the time scale for particle migration. (4) Boundary effects; In addition to slip, rheometer boundaries can induce ordering and disturb the orientation of nonspherical particles ( 122 ). The geometry should be large as compared to the particle size, typically greater than 10 particle diameters ( 117 ). (5) Measurement at elevated temperature and pressure; Reproducing downhole temperature and pressure introduces problems in evaporation and sample containment, and is generally available only for steady measurements above 100°C. (6) Repeatability and mixing; Care must be taken to properly hydrate and mix additives in fluids ( 123 ), and also that the materials have not degraded or biologically decomposed when used over several days. In all cases the material should reflect the state of hydration/homogeneity used in the actual process.