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We examine the buoyancy-driven rise of a bubble in a Newtonian or a viscoplastic fluid assuming axial symmetry and steady flow. Bubble pressure and rise velocity are determined, respectively, by requiring that its volume remains constant and its centre of mass remains fixed at the centre of the coordinate system. The continuous constitutive model suggested by Papanastasiou is used to describe the viscoplastic behaviour of the material. The flow equations are solved numerically using the mixed finite-element/Galerkin method. The nodal points of the computational mesh are determined by solving a set of elliptic differential equations to follow the often large deformations of the bubble surface. The accuracy of solutions is ascertained by mesh refinement and predictions are in very good agreement with previous experimental and theoretical results for Newtonian fluids. We determine the bubble shape and velocity and the shape of the yield surfaces for a wide range of material properties, expressed in terms of the Bingham Bn=$\tau_y^{\ast}/\rho^{\ast}g^{\ast} R_b^{\ast}$ Bond Bo =$\rho^{\ast}g^{\ast} R_b^{\ast 2}/\gamma^{\ast}$ and Archimedes Ar=$\rho^{\ast2}g^{\ast} R_b^{\ast3}/\mu_o^{\ast2}$ numbers, where ρ* is the density, μ*o the viscosity, γ* the surface tension and τ*y the yield stress of the material, g* the gravitational acceleration and R*b the radius of a spherical bubble of the same volume. If the fluid is viscoplastic, the material will not be deforming outside a finite region around the bubble and, under certain conditions, it will not be deforming either behind it or around its equatorial plane in contact with the bubble. As Bn increases, the yield surfaces at the bubble equatorial plane and away from the bubble merge and the bubble becomes entrapped. When Bo is small and the bubble cannot deform from the spherical shape the critical Bn is 0.143, i.e. it is a factor of 3/2 higher than the critical Bn for the entrapment of a solid sphere in a Bingham fluid, in direct correspondence with the 3/2 higher terminal velocity of a bubble over that of a sphere under the same buoyancy force in Stokes flow. As Bo increases allowing the bubble to squeeze through the material more easily, the critical Bingham number increases as well, but eventually it reaches an asymptotic value. Ar affects the critical Bn value much less.

Finite volume methods (FVMs) constitute a popular class of methods for the numerical simulation of fluid flows. Among the various components of these methods, the discretisation of the gradient operator has received less attention despite its fundamental importance with regards to the accuracy of the FVM. The most popular gradient schemes are the divergence theorem (DT) (or Green-Gauss) scheme, and the least-squares (LS) scheme. Both are widely believed to be secondorder accurate, but the present study shows that in fact the common variant of the DT gradient is second-order accurate only on structured meshes whereas it is zeroth-order accurate on general unstructured meshes, and the LS gradient is second-order and first-order accurate, respectively. This is explained through a theoretical analysis and is confirmed by numerical tests. The schemes are then used within a FVM to solve a simple diffusion equation on unstructured grids generated by several methods; the results reveal that the zeroth-order accuracy of the DT gradient is inherited by the FVM as a whole, and the discretisation error does not decrease with grid refinement. On the other hand, use of the LS gradient leads to second-order accurate results, as does the use of alternative, consistent, DT gradient schemes, including a new iterative scheme that makes the common DT gradient consistent at almost no extra cost. The numerical tests are performed using both an in-house code and the popular public domain PDE solver OpenFOAM. This is the accepted version of the article published in:

The sedimentation of a single particle in materials that exhibit simultaneously elastic, viscous and plastic behavior is examined in an effort to explain phenomena that contradict the nature of purely yield-stress materials. Such phenomena include the loss of the fore-and-aft symmetry with respect to an isolated settling particle under creeping flow conditions and the appearance of the "negative wake" behind it. Despite the fact that similar observations have been reported in studies involving viscoelastic fluids, researchers conjectured that thixotropy is responsible for these phenomena, as the aging of yield-stress materials is another common feature. By means of transient calculations, we study the effect of elasticity on both the fluidized and the solid phase. The latter is considered to behave as an ideal Hookean solid. The material properties of the model are determined under the isotropic kinematic hardening framework via Large Amplitude Oscillatory Shear (LAOS) measurements. In this way, we are able to predict accurately the unusual phenomena observed in experiments with simple yield-stress materials, irrespective of the appearance of slip on the particle surface. Viscoelasticity favors the formation of intense shear and extensional stresses downstream of the particle, significantly changing the entrapment mechanism in comparison to that observed in viscoplastic fluids. Therefore, the critical conditions under which the entrapment of the particle occurs deviate from the well-known criterion established theoretically by Beris et al. (1985) and verified experimentally by Tabuteau et al. (2007) for similar materials under conditions that elastic effects are negligible. Our predictions are in quantitative agreement with published experimental results by Holenberg et al. (2012) on the loss of the fore-aft symmetry and the formation of the negative wake in Carbopol with well-characterized rheology. Additionally, we propose simple expressions for the Stokes drag coefficient, as a function of the gravity number, Yg (related to the Bingham number), for different levels of elasticity and for its critical value, under which entrapment of particles occurs. These criteria are in agreement with the results found in the recent work by Ahonguio et al. (2014). Finally, we propose a method to quantify experimentally the elastic effects in viscoplastic particulate systems.

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Blood plasma has been considered a Newtonian fluid for decades. Recent experiments (Brust et al., Phys. Rev. Lett., 2013, 110) revealed that blood plasma has a pronounced viscoelastic behavior. This claim was based on purely elastic effects observed in the collapse of a thin plasma filament and the fast flow of plasma inside a contraction-expansion microchannel. However, due to the fact that plasma is a solution with very low viscosity, conventional rotational rheometers are not able to stretch the proteins effectively and thus, provide information about the viscoelastic properties of plasma. Using computational rheology and a molecular-based constitutive model, we predict accurately the rheological response of human blood plasma in strong extensional and constriction complex flows. The complete rheological characterization of plasma yields the first quantitative estimation of its viscoelastic properties in shear and extensional flows. We find that although plasma is characterized by a spectrum of ultra-short relaxation times (on the order of 10-3-10-5 s), its elastic nature dominates in flows that feature high shear and extensional rates, such as blood flow in microvessels. We show that plasma exhibits intense strain hardening when exposed to extensional deformations due to the stretch of the proteins in its bulk. In addition, using simple theoretical considerations we propose fibrinogen as the main candidate that attributes elasticity to plasma. These findings confirm that human blood plasma features bulk viscoelasticity and indicate that this non-Newtonian response should be seriously taken into consideration when examining whole blood flow.

We study the displacement of a viscous fluid by highly pressurized air in a straight or a suddenly constricted cylindrical tube of finite length. In contrast to previous efforts, the transient situation is examined. A long, narrower than the tube and round-ended bubble is created during the process. This is sometimes called “fingering instability” and is often encountered in several applications, but we will focus on process parameters that are relevant to the gas-assisted injection molding. For our numerical simulations we have combined the mixed finite element method with an appropriate system of elliptic partial differential equations and boundary conditions, capable of generating a boundary-fitted finite element mesh. The bubble front and the thickness of the deposited film on the tube wall are affected by the properties of the fluid being displaced and the flow conditions. Specifically, in straight tubes, the bubble keeps accelerating due to the decreasing fluid mass ahead of it. Increasing the Reynolds number decreases the film thickness and makes the bubble front steeper. When inertia becomes significant a tip-splitting instability arises. For sufficiently low Reynolds numbers, but still large applied pressures, a steady bubble shape is attained even in a straight tube and the fraction of the liquid deposited on the wall of the tube reaches the asymptotic value of 0.60, as observed by Taylor [J. Fluid Mech. 10, 161 (1961)] and Cox [J. Fluid Mech. 14, 81 (1962)]. In a constricted tube, the bubble temporarily attains a nearly constant velocity. When its front approaches the tube constriction it becomes pointed, due to the extensional flow that prevails there, but it reassumes its finger-like profile, after it goes through the constriction.