Flow Of Viscoplastic Fluid Research Articles

The isolated effect of yield stress in viscoplastic turbulent flow

Turbulent flows of viscoplastic fluids are present in several industrial and natural applications. The effects of yield stress on this problem have always been studied as a part of a larger physical context, because real viscoplastic materials have many properties that cannot be easily isolated. Direct numerical simulations have recently emerged as a viable tool for investigating non-Newtonian fluid flow in turbulent regimes. In the present work, we solve the turbulent flow of an ideal Bingham fluid, focusing on the isolated effect of yield stress. A numerical scheme for viscoplastic flows was implemented based on the lattice Boltzmann method. An outstanding characteristic of this scheme is the possibility of representing infinite viscosity by setting the relaxation frequency to zero, enabling the representation of the Bingham constitutive equation without artifacts, and producing a more accurate representation of the yield surfaces. In the turbulent channel flow simulations, the friction Reynolds number was fixed at 180, while the Bingham number varied from 0 (Newtonian) to 0.15. It is shown that unyielded portions of material travel along with the flow near the centerline. These unyielded spots do not disappear quickly, but rather have a significant lifetime. Another interesting outcome is that the yield stress increases the turbulence anisotropy, by lowering the spanwise and normal velocity fluctuations, while the streamwise component becomes higher. Reynolds stresses and budgets of turbulent kinetic energy have been analyzed regarding the increased bulk velocities that were found by increasing the yield stress.

Lattice Boltzmann simulations of unsteady Bingham fluid flows

Transient flows of viscoplastic fluids have very peculiar characteristics. The startup and cessation flows of viscoplastic materials have been subject to many theoretical and numerical investigations. The most challenging aspect of numerical solutions of viscoplastic fluids is the viscosity singularity during the transition from yielded to unyielded material. Hence, the proper representation of yield surfaces is the most critical aspect of numerical methods in viscoplastic fluid flow. In the present work, we use a lattice Boltzmann scheme to solve an ideal Bingham fluid’s startup and cessation flows. This numerical scheme advantage is that can represent infinite viscosity without noticeable numerical instabilities, producing yield surfaces with more accuracy and quality. Theoretical solutions for the startup flow are available in the literature. However, it is unclear which is more accurate and what their validity ranges are. Nonetheless, these solutions served as a reference for the present simulations. The overall aspect of the numerical solutions agreed with the theoretical models. The cessation flow of the Bingham fluid was also simulated. Unlike a Newtonian fluid, this type of flow is known to have a finite period until cessation. The simulations correctly reproduced this behavior. The transient yield surfaces matched very well with augmented Lagrangian solutions.

Derivation and numerical resolution of 2D shallow water equations for multi-regime flows of Herschel–Bulkley fluids

This paper presents mathematical modelling and simulation of thin free-surface flows of viscoplastic fluids with a Herschel–Bulkley rheology over complex topographies with basal perturbations. Using the asymptotic expansion method, depth-averaged models (lubrication and shallow water type models) are derived for 3D (three-dimensional) multi-regime flows on non-flat inclined topographies with varying basal slipperiness. Starting from the Navier–Stokes equations, two flow regimes corresponding to different balances between shear and pressure forces are presented. Flow models corresponding to these regimes are calculated as perturbations of the zeroth-order solutions. The classical reference models in the literature are recovered by considering their respective cases on a flat-inclined surface. In the second regime case, a pressure term is non-negligible. Mathematically, it leads to a corrective term to the classical regime equations. Flow solutions of the two regimes are compared; the difference appears in particular in the vicinity of sharp changes of slopes. Nonetheless, both regime models are compared with experiments and are found to be in good agreement. Furthermore, numerical examples are shown to illustrate the robustness of the present shallow water models to simulate viscoplastic flows in 3D and over an inclined topography with local perturbations in basal elevation and basal slipperiness. The derived models are adequate for direct (engineering and geophysical) applications to real-world flow problems presenting Herschel–Bulkley rheology like lava and mud flows.

Viscosity-model-independent generalized Reynolds number for laminar pipe flow of shear-thinning and viscoplastic fluids

Understanding the flow dynamics of non-Newtonian fluids is crucial in various engineering, industrial, and biomedical applications. However, the existing generalized Reynolds number formulations for non-Newtonian fluids have limited applicability due to their dependencies on their specific viscosity models. In this study, we propose a new viscosity-model-independent generalized Reynolds number formulation for laminar pipe flow. The proposed method is based on the direct adaptation of the measurement principles of rotational viscometers for wall shear rate estimation. We assess the accuracy of this proposed formulation for power-law and Carreau-Yasuda viscosity models through robust friction factor experiments. The experimental results demonstrate the applicability and effectiveness of the proposed viscosity-model-independent Reynolds number, as the measured friction factor data align closely with our Reynolds number predictions. Furthermore, we compare the accuracy of our Reynolds number formulation against established generalized Reynolds formulations for pure shear-thinning (Carreau-Yasuda) and viscoplastic (Herschel-Bulkley-extended) models. The results of the comparative analysis confirm the reliability and robustness of this generalized Reynolds number in characterizing and interpreting flow behavior in systems with visco-inelastic non-Newtonian fluids. This unified generalized Reynolds number formulation presents new and significant opportunities for precise pipe flow characterization and interpretation as it is applicable to any visco-inelastic (time-independent) viscosity model without requiring additional derivations.

On the efficient non-linear solver for hydraulic fracturing and well cementing simulations based on Anderson acceleration

The aim of this study is to create a fast and stable iterative technique for numerical solution of a quasi-linear elliptic pressure equation. We developed a modified version of the Anderson acceleration (AA) algorithm to fixed-point (FP) iteration method. It computes the approximation to the solutions at each iteration based on the history of vectors in extended space, which includes the vector of unknowns, the discrete form of the operator, and the equation's right-hand side. Several constraints are applied to AA algorithm, including a limitation of the time step variation during the iteration process, which allows switching to the base FP iterations to maintain convergence. Compared to the base FP algorithm, the improved version of the AA algorithm enables a reliable and rapid convergence of the iterative solution for the quasi-linear elliptic pressure equation describing the flow of particle-laden yield-stress fluids in a narrow channel during hydraulic fracturing, a key technology for stimulating hydrocarbon-bearing reservoirs. In particular, the proposed AA algorithm allows for faster computations and resolution of unyielding zones in hydraulic fractures that cannot be calculated using the FP algorithm. The quasi-linear elliptic pressure equation under consideration describes various physical processes, such as the displacement of fluids with viscoplastic rheology in a narrow cylindrical annulus during well cementing, the displacement of cross-linked gel in a proppant pack filling hydraulic fractures during the early stage of well production (fracture flowback), and multiphase filtration in a rock formation. We estimate computational complexity of the developed algorithm as compared to Jacobian-based algorithms and show that the performance of the former one is higher in modelling of flows of viscoplastic fluids. We believe that the developed algorithm is a useful numerical tool that can be implemented in commercial simulators to obtain fast and converged solutions to the non-linear problems described above.

Natural convection of viscoplastic fluids in a triangular enclosure

This study extends the analysis of natural convection in a viscoplastic fluid to flow within a triangular enclosure. The finite-element approach provided a numerical solution to the continuity, momentum, and energy equations. The governing parameters for this problem are the Rayleigh number, (Ra=104−106), yield number, (Y=0−Ymax), aspect ratio, (HL=0.5−2.5), and slope angle, (∅=0−π2). The influence of these parameters on the heat and mass transfer, morphology of yielded/unyielded regions, and fluid flow were thoroughly examined. The results show that two opposing factors influence the flow behavior and heat transmission within the triangular enclosure. The proximity of the walls restricts the convective movement, leading to reduced heat transfer. However, the proximity of hot and cold sources increases the temperature gradient and heat transfer. The unique influence of the viscoplastic material properties, particularly the yield stress, further distinguishes the heat transfer in this triangular enclosure from other geometries. The results indicate that an increase in the Rayleigh number mitigates the effects of yield stress to some extent. However, the accumulation of unyielded material at the triangular apex hinders convection flow. Furthermore, the viscoplastic fluid flow and heat transfer changed significantly with changes in triangle height. In particular, the maximum yield stress increased by more than 100 % as the aspect ratio increased from 0.5 to 2.5. A change in the slope angle causes a continuous transition from subcritical to supercritical bifurcation, significantly affecting the morphology of the yielded and unyielded areas, and the maximum (critical) yield number. Finally, correlations were developed to predict the Nusselt number and maximum yield stress in all cases.

General hydrodynamic features of elastoviscoplastic fluid flows through randomised porous media

A numerical study of yield-stress fluids flowing in porous media is presented. The porous media is randomly constructed by non-overlapping mono-dispersed circular obstacles. Two class of rheological models are investigated: elastoviscoplastic fluids (i.e. Saramito model) and viscoplastic fluids (i.e. Bingham model). A wide range of practical Weissenberg and Bingham numbers is studied at three different levels of porosities of the media. The emphasis is on revealing some physical transport mechanisms of yield-stress fluids in porous media when the elastic behaviour of this kind of fluids is incorporated. Thus, computations of elastoviscoplastic fluids are performed and are compared with the viscoplastic fluid flow properties. At a constant Weissenberg number, the pressure drop increases both with the Bingham number and the solid volume fraction of obstacles. However, the effect of elasticity is less trivial. At low Bingham numbers, the pressure drop of an elastoviscoplastic fluid increases compared to a viscoplastic fluid, while at high Bingham numbers we observe drag reduction by elasticity. At the yield limit (i.e. infinitely large Bingham numbers), elasticity of the fluid systematically promotes yielding: elastic stresses help the fluid to overcome the yield stress resistance at smaller pressure gradients. We observe that elastic effects increase with both Weissenberg and Bingham numbers. In both cases, elastic effects finally make the elastoviscoplastic flow unsteady, which consequently can result in chaos and turbulence.Graphical abstract

Numerical simulations of dam-break flows of viscoplastic fluids via shallow water equations

This paper presents simulations of dam-break flows of Herschel–Bulkley viscoplastic fluids over complex topographies using the shallow water equations (SWE). In particular, this study aims to assess the effects of rheological parameters: power-law index (n), consistency index (K), and yield stress (τc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _{c}$$\\end{document}), on flow height and velocity over different topographies. Three practical examples of dam-break flow cases are considered: a dam-break on an inclined flat surface, a dam-break over a non-flat topography, and a dam-break over a wet bed (downstream containing an initial fluid level). The effects of bed slope and depth ratios (the ratio between upstream and downstream fluid levels) on flow behaviour are also analyzed. The numerical results are compared with experimental data from the literature and are found to be in good agreement. Results show that for both dry and wet bed conditions, the fluid front position, peak height, and mean velocity decrease when any of the three rheological parameters are increased. However, based on a parametric sensitivity analysis, the power-law index appears to be the dominant factor in dictating fluid behaviour. Moreover, by increasing the bed slope and/or depth ratio, the wave-frontal position moves further downstream. Furthermore, the presence of an obstacle is observed to cause the formation of an upsurge that moves in the upstream direction, which increases by increasing any of the three rheological parameters. This study is useful for an in-depth understanding of the effects of rheology on catastrophic gravity-driven flows of non-Newtonian fluids (like lava or mud flows) for risk assessment and mitigation.Graphical abstract

Viscoplastic flows in superhydrophobic channels with oriented grooves: From anisotropic slip to secondary flow

In this work, Poiseuille flows of viscoplastic fluids in typically thin channels equipped with a superhydrophobic groovy wall are numerically studied. The orientation of the groove relative to the applied pressure gradient can vary, and this orientation is measured via the groove orientation angle (θ). In particular, longitudinal (θ=0), oblique (0<θ<90∘), and transverse (θ=90∘) flow configurations are considered. The Bingham constitutive equation is employed to model the viscoplastic rheology, within the framework of the Papanastasiou regularization method. Assuming that air (gas) fills the groove completely and that the formed liquid/air interface remains flat while pinned at the groove edges, the viscoplastic fluid slippage is modeled on the liquid/air interface using the Navier slip law. Due to the anisotropic slip dynamics for the oblique flow configuration, a secondary flow is generated normal to the direction of the pressure gradient, offering unique flow features. Our work systematically analyzes the effects of the flow parameters, i.e., the groove orientation angle (θ), the Bingham (B) and slip (b) numbers, the groove periodicity length (ℓ), and the slip area fraction (φ) on the flow variables of interest, i.e., the main and secondary velocity fields, the unyielded center plug zone, the effective slip length tensor (χ), the secondary flow index (IS), the slip angle difference (θ−s), and the pressure drop (ΔP). It is demonstrated that χ’s shear component, IS, and θ−s are maximum at intermediate θ, the value of which generally decreases with B. In addition, the center plug is unbroken for the longitudinal flow while it breaks with an increase in θ for sufficiently large b.

A case study on the evolution of rigid zones within the permanent two-dimensional Herschel-Bulkley fluid flow

The formation and development of undeformed regions during the flow of viscoplastic fluids can significantly affect fluid flow behaviour. They can impair the efficiency and effectiveness of industrial operations that use viscoplastic fluids. This paper uses numerical analysis to examine the emergence of rigid zones during non-stationary Herschel-Bulkley fluid flow across a square plate. The impact of pressure and yield stress on the behaviour of rigid zones over time is explored, and the development of the rigid zone area is shown. This work produced mathematical correlations that describe the relationship between the area of rigid zones and both the time of flow and the yield stress, as well as the area of rigid zones and the stagnation time.

Identification of rheological parameters for shallow water flows of viscoplastic fluids using elevation hydrographs

In this paper, rheological parameters, in particular yield stress and consistency index, for viscoplastic fluids are inferred from elevation hydrographs derived from experiments. The direct model consisting of shallow water equations with a Herschel–Bulkley rheology is used to simulate a fluid flowing down an inclined plane and past a cylindrical occlusion. Numerical simulations are validated with experimental and related results from the literature. The aim is to infer the unknown rheological parameters using hydrograph measurements in the contact line region between the fluid and the occlusion. The rheological identification problem is formulated to minimize an objective functional that measures the discrepancy between the elevation hydrographs from the model output and experimental data. The inverse solver is tested on both synthetic and laboratory data. The set of rheological parameters inferred is compared with the values measured on a rheometer for the fluid used in the experiments. Inference of the unknown flow quantities from the wetting free-surface data has direct applications not only in industrial settings, to predict the wetting dynamics, but also in geophysical ones for risk assessments and management plans.

Numerical Simulation of Thermally Radiative and EMHD Influenced Viscoplastic Fluid Flow: A Second Law Investigation

"This work rigorously presents the findings of entropy generation in Casson fluid flow on a Riga plate, exposed in a radiative environment. The conceptual basis for the investigation of fluid flow is the system of governing boundary layer equations. By means of implicit finite difference procedure the solutions of boundary layer equations are acquired. Primary goal of this study manifests the optimization of entropy generation in Casson fluid flow associated with EMHD (Electro Magneto Hydro Dynamics) Lorentz force and thermal radiation. Additionally, the contours of entropy generation and Bejan number are demonstrated. Validation of present results with previously obtained results from literature is accomplished conscientiously."

Dynamics of displacement flows of viscoplastic fluids in obstructed eccentric annuli at moderate Reynolds numbers

As a sequel to Mitishita et al. [“Turbulent displacement flows of viscoplastic fluids in obstructed eccentric annuli: Experiments,” Phys. Fluids 34, 053114 (2022)], we present an experimental study of laminar displacement flows in obstructed eccentric annuli. Xanthan gum (XG) solutions (0.35%, 0.50%, or 0.75%) are used to displace a 0.15% viscoplastic Carbopol solution. The eccentricity of the annulus section is set to near 0.5. We study the effect of a solid obstruction in the narrow side of the annulus, similar to that provided by a consolidated residual cuttings bed, and compare the results to unobstructed displacement flows. While we predicted that all displacements would be in the laminar regime, we actually observe mixed regimes where the initial displacement of Carbopol can be transitional or turbulent. With the obstruction on the narrow side of the annulus, we observe the formation of cavities in the Carbopol layer, both upstream and downstream of the obstruction. We believe that the cavities are formed because the obstruction behaves like an abrupt contraction/expansion. This geometric irregularity affects the velocity profiles of the displacing fluid near the obstruction. Once the cavities reach the bottom of the pipe, we observe that the remaining Carbopol layer is more easily eroded. The dynamics of the Carbopol removal also share similarities to cleaning of soil layers in pipes, as described by Palabiyik et al. [“Flow regimes in the emptying of pipes filled with a Herschel–Bulkley fluid,” Chem. Eng. Res. Des. 92, 2201–2212 (2014)].

The role of Cheeger sets in the steady flows of viscoplastic fluids in pipes: A survey

Given a convex or a Jordan domain Ω, let Ω′ be a subset of this domain, with P(Ω′) denoting its perimeter and A(Ω′) its area. If a subset Ωc exists such that h=P(Ωc)/A(Ωc) is a minimum, the subset Ωc is called the Cheeger set of Ω and h, the Cheeger constant of the given domain. If one considers the reciprocal of this minimum or the maximum ratio of the area of the subset to its perimeter, t∗=1/h. It follows from the work of Mosolov and Miasnikov that the minimum pressure gradient G to sustain the steady flow of a viscoplastic fluid in a pipe, with a cross section defined by Ω, is given by G&amp;gt;τy/t∗, where τy is the constant yield stress of the fluid. In this survey, we summarize several results to determine the constant h when the given domain is self-Cheeger or a Cheeger-regular set that touches each boundary of a convex polygon and when the Cheeger-irregular set does not do so. The determination of the constant h for an arbitrary ellipse, a strip, and a region with no necks is also mentioned.

A CFD examination of free convective flow of a non-Newtonian viscoplastic fluid using ANSYS Fluent

In this work, a laminar steady-state investigation of free convection in a square cavity with differential heated side walls is examined. The cavity is immersed with a viscoplastic liquid the Bingham prototype. The horizontal walls are assumed as adiabatic and the vertical wall has two spatial differing sinusoidal temperature profiles with diverse phases and amplitudes. The hydro-thermal features are systematically analyzed via a broad choice of Rayleigh numbers Ra (103-106), Bingham numbers Bn, Prandtl numbers Pr (0.1, 1, 10), amplitude ratio ε (0–––1), phase difference ϕ (0-π) and flow index n (0.3–2). The governing equations are treated computationally utilizing a commercial computational simulation code CFD: FLUENT. It has been observed that average Nusselt numbers grow with growing Rayleigh numbers and drop with increasing Bingham quantity Bn, since heat transition occurs primarily due to thermic conductivity. In general, a higher Rayleigh number promotes convection and increases heat transfer efficiency, leading toa higher Nusselt number, whereas a higher Bingham number, indicating a higher degree of viscous or yield-stress behavior, may inhibit convection and decrease heat transfer efficiency, which ended in a lower Nusselt number. These connections frequently appear in heat transport and fluid dynamics simulations. The rise in the phase difference suggests an upsurge in heat transference, as the impact of the phase shift on the Nusselt is perpetually enhanced due to the amount of Rayleigh boosts for all phase differences. Heat transition rate forϕ=πis boosted because of the values. Moreover, when the amplitude ratio goes up, so does heat transfer. The gap between average Nusselt and Rayleigh amounts rose as the Rayleigh rose. The thermal flow rate is bigger in ε=1 than in the other cases. The rise in phase difference suggests a rise in heat transference, as this effect of phase shift on Nusselt continues to be enhanced due to the amount of Rayleigh boosts for all phase differences. Heat transition rate is enhanced forϕ=πbased on the values. Plus, when the amplitude ratio grows, so does heat transmission. The rate of heat transport for ε=1 is larger than in the other cases.

Characteristics of Viscoplastic Fluid Flow at Various Heat Transfer Regimes on the Walls of a Sudden Contraction Channel

Characteristics of Viscoplastic Fluid Flow at Various Heat Transfer Regimes on the Walls of a Sudden Contraction Channel

Pipe Flow of Viscoplastic Fluids and Analytical Predictions of Concrete Pumping Based on the Shear-Stress-Dependent Parabolic Model

This study investigates the Hagen–Poiseuille pipe flow of viscoplastic fluids, focusing on analytical predictions of concrete pumping using the shear-stress-dependent parabolic model, extending analytical studies to a nonlinear rheological model with easily accessible experimental parameters. Research novelty and highlights encompass solving the steady laminar pipe flow for viscoplastic fluids described by the parabolic model, presenting detailed results for the two-fluid parabolic model, and introducing a computational app implementing theoretical findings. The parabolic model outperforms linear models, such as the Bingham model, in accuracy by accounting for the nonlinearity in the flow curves (i.e., shear stress and shear rate relations) of pumped concrete. The influence of rheological parameters on these relations is analyzed, and their versatility is demonstrated by a Wolfram Mathematica-based application program. The analytical approach developed in this work is adaptable for other models with shear stress as the independent variable, offering valuable insights into viscoplastic fluid flows.

On the flow of viscoplastic fluids in non-circular tubes

Steady flow of the viscoplastic Bingham and Herschel–Bulkley (H–B)​ fluids in tubes of non-circular cross-section is investigated analytically. The solution methodology is general in scope, does not put any restrictions on the Bingham number and thus allows the mapping of the flow field for high Bingham numbers in straight tubes with non-circular axially-symmetric but otherwise arbitrary cross-sectional contours. The circular tube contour is mapped onto an arbitrary non-circular contour on which the no-slip condition is satisfied via a one-to-one and continuous mapping. Governing equations are solved for the full spectrum of axially symmetrical cross-sectional shapes and a specific example is developed for the viscoplastic H–B fluid flow in a tube with an equilateral triangular cross-section. The shape and the extent of the plug zones centered on the tube axis and the stagnant zones in the corners are predicted for both Bingham and H–B fluids. The effect of the shear rate dependent viscosity of the H–B fluids, leading to either shear-thickening or shear-thinning behavior, on the formation of the plug and stagnation zones is examined.

Hydrothermal performance for forced convective flow of viscoplastic fluid through a backward facing step channel

The hydrothermal performance for steady, laminar forced convective flow of viscoplastic fluid through a backward facing step channel is studied. The effect of Bingham number (0 ≤ Bn ≤ 1), Reynolds number (50 ≤ Re ≤ 300), and power law index (0.2 ≤ n ≤ 1) on the thermal and flow characteristics are analyzed. A recirculatory zone is generated at the step corner and the findings depict a decrement in the length of reattachment as n and Bn increases. The temperature drop in the vicinity of the step increases as Bn increases. Also, it is found that the local Nusselt number (Nu) attains a peak in the vicinity of the step and the maximum Nu in this region increases with the increase in Bn. Moreover, for any particular value of Bn, Nu decreases with the increase in n. Also, the average Nusselt number increases significantly for viscoplastic fluids in comparison to shear-thinning and Newtonian fluids, and the heat transfer rate increases as Bn increases. Furthermore, there exists a critical value of Bn that alters the variation of average Nusselt number with the power-law index in case of separated flows.

Rheological characterization of viscoplastic fluid flow in a pipe with wall slip using in situ particle image velocimetry

Rheological characterization of viscoplastic fluid flow in a pipe with wall slip using in situ particle image velocimetry