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We examine computationally the two-dimensional flow of elastoviscoplastic (EVP) fluids around a cylinder symmetrically placed between two plates parallel to its axis. The Saramito-Herschel-Bulkley fluid model is solved via the finite-volume method using the OpenFOAM software. As in viscoplastic materials, unyielded regions arise around the plane of symmetry well ahead or behind the cylinder, as two small islands located above and below the cylinder and as polar caps at the two stagnation points on the cylinder. Most interestingly, under certain conditions, an elongated yielded area around the midplane is predicted downstream of the cylinder, sandwiched between two unyielded areas. This surprising result appears, for example, with Carbopol 0.1% when considering a blockage ratio of 0.5 (the ratio of the cylinder's diameter to the channel's width) and above a critical elastic modulus (G>30 Pa). An approximate semi-analytical solution in the region mentioned above, reveals that it is caused by the intense variation of the stress magnitude there, which may approach the yield stress asymptotically either from above or below, depending on material elasticity. The drag coefficient on the cylinder increases with yield stress and blockage ratio but decreases with material elasticity. The unyielded regions expand as the yield stress increases. They also expand when material elasticity increases because this allows the material to elastically deform more before yielding. Furthermore, by decreasing the elastic modulus or increasing the yield stress beyond a critical value, the yield surface may exhibit damped oscillations, or irregular shapes even without a plane of symmetry, all under creeping flow conditions.

We consider the buoyancy-driven rise and interaction between two gravity-aligned bubbles of wide radii ratio and constant volume in an elasto-visco-plastic (EVP) material, extending our previous work on equal bubbles [Kordalis et al., Phys. Rev. Fluids 8, 083301, (2023)]. Primarily we consider a 0.1% aqueous Carbopol solution and model it with the Saramito-Herschel-Bulkley model. Initially, we investigate the dynamics for a specific initial separation distance in a wide range of bubble radii and we determine the conditions leading to three distinct patterns: bubble approach, bubble separation and establishment of a constant distance between them. Specifically, when the leading bubble (LB) is smaller than the trailing bubble (TB), the bubbles approach each other due to the smaller buoyancy of the LB. Strong attraction also occurs when the ratio of buoyant over viscous force of both bubbles is considerable. On the other hand, when the size of the TB is such that this ratio is moderate or small, the pattern is dictated by the size of the LB: A significantly larger LB compared to the TB causes separation of the pair. On the contrary, an only slightly larger LB may result in the bubbles rising with the same terminal velocity establishing a constant distance between them, the magnitude of which is mainly determined by the elastic response of the surrounding medium. The coupling of a negative wake behind the LB with a slight modification of the stresses exerted at its rear pole generates this dynamic equilibrium. The same equilibrium may be achieved by other specific pairs of bubble sizes for different initial distances of the pair, if a critical initial distance is exceeded. Below this critical value, the bubbles approach each other. Finally, we construct maps of the three patterns with TB radius versus bubble radii ratio for different initial separation distances and material properties., Comment: 36 pages, 18 figures

We investigate theoretically the buoyancy-driven motion of a viscous drop in a yield-stress material, incorporating elastic effects represented by the Saramito-Herschel-Bulkley constitutive equation. We solve the governing equations using an open-source finite volume solver and utilizing the volume of fluid technique to accurately capture the interface between the two fluids. To validate our numerical approach, we compare our results with data from previous experimental and numerical studies. We find quantitative agreement in terms of terminal velocities and drop shapes, affirming the accuracy of our model and its numerical solution. Notably, we observe that incorporating elastic effects into the modelling of the continuous phase is essential for predicting phenomena reported in experiments, such as the inversion of the flow field behind the sedimenting drop (i.e., the negative wake) or the formation of a teardrop shape. Due to the elastoviscoplastic nature of the continuous phase, we observe that small drops remain entrapped because the buoyancy force is insufficient to fluidize the surrounding material. We investigate entrapment conditions using two different protocols, which yield different outcomes due to the interplay between capillarity and elasto-plasticity. Finally, we conduct an extensive parametric analysis to evaluate the impact of rheological parameters (yield stress, elastic modulus, and interfacial tension) on the dynamics of sedimentation., Comment: 37 pages, 24 figures in the main article and 3 figures in the appendix

We present predictions for the flow of elastoviscoplastic (EVP) fluids in the 4 to 1 planar contraction geometry. The Saramito-Herschel-Bulkley fluid model is solved via the finite-volume method with the OpenFOAM software. Both the constitutive model and the solution method require using transient simulations. In this benchmark geometry, whereas viscoelastic fluids may exhibit two vortices, referred to as lip and corner vortices, we find that EVP materials are unyielded in the concave corners. They are also unyielded along the mid-plane of both channels, but not around the contraction area where all stress components are larger. When the Bingham or the Weissenberg numbers are lower than critical values, and then, a steady state is reached. When these two dimensionless numbers increase while they remain below the respective critical values, which are interdependent, (a) the unyielded regions expand and shift in the flow direction, and (b) the maximum velocity increases at the entrance of the contraction. Increasing material elasticity collaborates with increasing the yield stress, which expands the unyielded areas, because it deforms the material more prior to yielding compared to stiffer materials. Above the critical Weissenberg number, transient variations appear for longer times in all variables, including the yield surface, instead of a monotonic approach to the steady state. They may lead to oscillations which are damped or of constant amplitude or approach a flow with rather smooth path lines but complex stress field without a plane of symmetry, under creeping conditions. These patterns arise near the entrance of the narrow channel, where the curvature of the path lines is highest and its coupling with the increased elasticity triggers a purely elastic instability. Similarly, a critical value of the yield stress exists above which such phenomena are predicted.

We propose a family of gradient reconstruction schemes based on the solution of over-determined systems by orthogonal or oblique projections. In the case of orthogonal projections, we retrieve familiar weighted least-squares gradients, but we also propose new direction-weighted variants. On the other hand, using oblique projections that employ cell face normal vectors we derive variations of consistent Green-Gauss gradients, which we call Taylor-Gauss gradients. The gradients are tested and compared on a variety of grids such as structured, locally refined, randomly perturbed, unstructured, and with high aspect ratio. The tests include quadrilateral and triangular grids, and employ both compact and extended stencils, and observations are made about the best choice of gradient and weighting scheme for each case. On high aspect ratio grids, it is found that most gradients can exhibit a kind of numerical instability that may be so severe as to make the gradient unusable. A theoretical analysis of the instability reveals that it is triggered by roundoff errors in the calculation of the cell centroids, but ultimately is due to truncation errors of the gradient reconstruction scheme, rather than roundoff errors. Based on this analysis, we provide guidelines on the range of weights that can be used safely with least squares methods to avoid this instability.

A new discretisation scheme for the gradient operator, suitable for use in second-order accurate Finite Volume Methods (FVMs), is proposed. The derivation of this scheme, which we call the Taylor-Gauss (TG) gradient, is similar to that of the least-squares (LS) gradients, whereby the values of the differentiated variable at neighbouring cell centres are expanded in truncated Taylor series about the centre of the current cell, and the resulting equations are summed after being weighted by chosen vectors. Unlike in the LS gradients, the TG gradients use vectors aligned with the face normals, resembling the Green-Gauss (GG) gradients in this respect. Thus, the TG and LS gradients belong in a general unified framework, within which other gradients can also be derived. The similarity with the LS gradients allows us to try different weighting schemes (magnitudes of the weighting vectors) such as weighting by inverse distance or face area. The TG gradients are tested on a variety of grids such as structured, locally refined, randomly perturbed, and with high aspect ratio. They are shown to be at least first-order accurate in all cases, and are thus suitable for use in second-order accurate FVMs. In many cases they compare favourably over existing schemes.

We investigate the radial oscillations of small gas bubbles trapped in yield-stress fluid and driven by an acoustic pressure field. We model the rheological behavior of the yield-stress fluid using the recently developed elasto-visco-plastic (EVP) constitutive equation that takes into account the elastic and visco-plastic deformations of the material [P. Saramito, J. NonNewton. Fluid Mech. 158 (1-3) (2009) pp.154-161]. Assuming that the bubble remains spherical during the pressure driving, we reduce the problem to a set of ODEs and an integrodifferential equation, which we solve numerically for the case of two yield-stress fluids, a sot Carbopol gel and a stiffer Kaolin suspension. We find that, depending on the amplitude and frequency of the pressure field, the radial oscillations of the bubble produce elastic stresses that may or may not suffice to yield the surrounding material. We evaluate the critical amplitude of the acoustic pressure required to achieve yielding and we find a good agreement between numerical simulations and an analytical formula derived under the assumption of linear deformations. Finally, we examine the bubble oscillation amplitude for a very wide range of applied pressures both below and above the critical value to assess the impact of yielding on the bubble dynamics. This analysis could be used to identify a signature of yielding in experiments where the radial dynamics of a bubble is measured. More generally, these results can be used to rationalize the optimal conditions for pressure-induced bubble release from yield-stress fluids, which is relevant to various biomedical and industrial applications, including oil industry and food processing.

We propose a Finite Volume Method for the simulation of elastoviscoplastic flows, modelled after the extension to the Herschel-Bulkley model by Saramito [J. Non-Newton. Fluid Mech. 158 (2009) 154-161]. The method is akin to methods for viscoelastic flows. It is applicable to cell-centred grids, both structured and unstructured, and includes a novel pressure stabilisation technique of the "momentum interpolation" type. Stabilisation of the velocity and stresses is achieved through a "both sides diffusion" technique and the CUBISTA convection scheme, respectively. A second-order accurate temporal discretisation scheme with adaptive time step is employed. The method is used to obtain benchmark results of lid-driven cavity flow, with the model parameters chosen so as to represent Carbopol. The results are compared against those obtained with the classic Herschel-Bulkley model. Simulations are performed for various lid velocities, with slip and no-slip boundary conditions, and with different initial conditions for stress. Furthermore, we investigate the cessation of the flow, once the lid is suddenly halted.

The flow inside a fluid damper where a piston reciprocates sinusoidally inside an outer casing containing high-viscosity silicone oil is simulated using a Finite Volume method, at various excitation frequencies. The oil is modelled by the Carreau-Yasuda (CY) and Phan-Thien \& Tanner (PTT) constitutive equations. Both models account for shear-thinning but only the PTT model accounts for elasticity. The CY and other generalised Newtonian models have been previously used in theoretical studies of fluid dampers, but the present study is the first to perform full two-dimensional (axisymmetric) simulations employing a viscoelastic constitutive equation. It is found that the CY and PTT predictions are similar when the excitation frequency is low, but at medium and higher frequencies the CY model fails to describe important phenomena that are predicted by the PTT model and observed in experimental studies found in the literature, such as the hysteresis of the force-displacement and force-velocity loops. Elastic effects are quantified by applying a decomposition of the damper force into elastic and viscous components, inspired from LAOS (Large Amplitude Oscillatory Shear) theory. The CY model also overestimates the damper force relative to the PTT, because it underpredicts the flow development length inside the piston-cylinder gap. It is thus concluded that (a) fluid elasticity must be accounted for and (b) theoretical approaches that rely on the assumption of one-dimensional flow in the piston-cylinder gap are of limited accuracy, even if they account for fluid viscoelasticity. The consequences of using lower-viscosity silicone oil are also briefly examined.

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 second-order 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., Comment: Minor corrections compared to the previous version

Numerical simulations of the flow in an extrusion damper are performed using a finite volume method. The damper is assumed to consist of a shaft, with or without a spherical bulge, oscillating axially in a containing cylinder filled with a viscoplastic material of Bingham type. The response of the damper to a forced sinusoidal displacement is studied. In the bulgeless case the configuration is the annular analogue of the well-known lid-driven cavity problem, but with a sinusoidal rather than constant lid velocity. Navier slip is applied to the shaft surface in order to bound the reaction force to finite values. Starting from a base case, several problem parameters are varied in turn in order to study the effects of viscoplasticity, slip, damper geometry and oscillation frequency to the damper response. The results show that, compared to Newtonian flow, viscoplasticity causes the damper force to be less sensitive to the shaft velocity; this is often a desirable damper property. The bulge increases the required force on the damper mainly by generating a pressure difference across itself; the latter is larger the smaller the gap between the bulge and the casing is. At high yield stresses or slip coefficients the amount of energy dissipation that occurs due to sliding friction at the shaft-fluid interface is seen to increase significantly. At low frequencies the flow is in quasi steady state, dominated by viscoplastic forces, while at higher frequencies the fluid kinetic energy storage and release also come into the energy balance, introducing hysteresis effects.