Bulkley Fluid Research Articles

Self-similar solution of the first Stokes problem for non-Newtonian fluids with power-law viscosity

The first Stokes problem is considered for flows of non-Newtonian fluids with effective viscosity varying in accordance with the power law. Self-similar solutions are constructed for the fluids, whose mechanical behavior is described by the Ostwald – de Waele and Herschel–Bulkley rheological models with the corresponding restrictions on the nonlinearity degree exponent n. It is shown that for the Ostwald – de Waele fluid self-similar solutions exist only for 0 n 1, which corresponds to the pseudoplastic behavior. At the same time, self-similar solutions for the Herschel–Bulkley fluid can be obtained only at n 1, when this fluid exhibits the properties of dilatancy.

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.

Bingham and herschel-bulkley fluids flow regimes in rough-walled rock fractures

Flow of typical non-Newtonian fluids such as cement grouts can experience different regimes as the Reynolds number (Re) changes, understanding of which is important for design and operation of rock grouting in various rock engineering applications. Here, flow regimes of representative non-Newtonian fluids, i.e., Bingham and Herschel–Bulkley (H–B) fluids, are numerically investigated with experimental validations. Three tensile rock fracture surfaces originated from a fine-grained sandstone, a medium-grained sandstone and a medium-grained granite samples are used to create rough-walled fracture models with variable aperture structures. Flow of groundwater, Bingham and H–B fluids through these fractures is numerically simulated respectively, by solving the full mass and momentum conservation equations with the Re ranging from 0.01 to 1000. The regimes for these fluids flowing through the fractures are characterized. Laboratory flow tests are conducted in a cylindrical granite fracture sample to verify the characterized flow regimes. The results reveal important differences of flow regimes between Newtonian and non-Newtonian fluids. Specifically, the transmissivity for water flow is constant when Re is relatively small until the Re reaches certain critical values; the transmissivity for Bingham and H–B fluids flow increases with increasing Re until asymptotically reaches certain peak values, followed by a descending stage when Re is relatively large. The critical (water) and peak (Bingham and H–B fluids) values are affected by surface roughness, that is, a rougher surface results in smaller critical and peak values as well as greater discrepancies compared to the analytical solutions based on the smoothed parallel plates model. These results show that a peak or optimum transmissivity is achievable in a specific range of Re (Re = 10–100 for the fractures studied). This new finding can potentially help optimize the injection pressure or flow rate in rock grouting practices.

Optimisation of Synchronous Grouting Mix Ratio for Shield Tunnels

During shield construction in underground spaces, synchronous grouting slurry is poured between the surrounding rock and tunnel lining to ensure stability. For synchronous grouting slurries, few studies have investigated the relationship between the rheological parameters and physical properties, grout-segregation mechanism, and anti-segregation performance. Therefore, we explored the relationships between the slurry rheological parameters, segregation rate, and bleeding rate. Cement, sand, fly ash, and bentonite were used to prepare the slurry, and the effects of different polycarboxylate water-reducing agents and dispersible latex powder dosages were studied. The rheological parameters of 16 groups of uniformly designed slurries were tested, and the data were fit using the Herschel–Bulkley model. The optimal mix ratio lowered the slurry segregation rate, and its rheological behaviour was consistent with the Herschel–Bulkley fluid characteristics. High-yield-shear-stress synchronous grouting slurries with high and low viscosity coefficients were less likely to bleed and segregate, respectively. The optimised slurry fluidity, 3 h bleeding rate, 24 h bleeding rate, segregation rate, coagulation time, and 28 days compressive strength were 257.5 mm, 0.71%, 0.36%, 3.1%, 6.7 h, and 2.61 MPa, respectively, which meet the requirements of a synchronous grouting slurry of shield tunnels for sufficiently preventing soil disturbance and deformation in areas surrounding underground construction sites.

Torsional parallel plate flow of Herschel–Bulkley fluids with wall slip

The effect of wall slip on the apparent flow curves of viscoplastic materials obtained using torsional parallel plate rheometers is analyzed by considering Herschel–Bulkley fluids and assuming that slip occurs above the slip yield stress τc, taken to be lower than the yield stress, τ0. When the rim shear stress τR is below τc, the exerted torque is not sufficient to rotate the disk. When τc<τR≤τ0 the material is still unyielded but exhibits wall slip and rotates as a solid at half the angular velocity of the rotating disk. Finally, when τR>τ0, the material exhibits slip everywhere and yields only in the annulus r0≤r≤R, where r0 is the critical radius at which the shear stress is equal to the yield stress and R is the radius of the disks. In the general case, the slip velocity, which varies with the radial distance, can be calculated numerically and then all quantities of interest, such as the true shear rate, and the two branches of the apparent flow curve can be computed by means of closed form expressions. Analytical solutions have also been obtained for certain values of the power-law exponent. In order to illustrate the effect of wall slip on the apparent flow curve and on the torque, results have been obtained for different gap sizes between the disks choosing the values of the rheological and slip parameters to be similar to reported values for certain colloidal suspensions. The computed apparent flow curves reproduce the patterns observed in the experiments.

Characteristics of a Particle’s Incipient Motion from a Rough Wall in Shear Flow of Herschel–Bulkley Fluid

Characteristics of a Particle’s Incipient Motion from a Rough Wall in Shear Flow of Herschel–Bulkley Fluid

Rheology-based wall function approach for wall-bounded turbulent flows of Herschel–Bulkley fluids

Modeling fully developed turbulent flow for Herschel–Bulkley (HB) fluids in pipes is a long-standing challenge. Existing semi-empirical, theoretical, and numerical methods are either inconsistent with experimental data or are validated for low Reynolds numbers. This study focuses on validating a novel approach using rheology-based wall functions within Reynolds-averaged Navier–Stokes solvers. Simulations of wall shear stress and velocity profiles were conducted across a wide range of Reynolds numbers using a single-phase HB fluid, with measurements taken both upstream and downstream of a 90° pipe bend. Two turbulence closure models, the k–ε model and the Reynolds stress model, were employed with the wall function implemented as a specified shear boundary condition. Results demonstrate significant improvements over the Newtonian-based models, such as standard wall function by Launder–Spalding or with available semi-empirical models, achieving strong statistical correlations and minimal deviation (from the experimental findings) at high Reynolds numbers. The study also examines the utility of the wall viscosity Reynolds number and assesses the reliability of semi-empirical models for HB fluids. These findings offer valuable insights for enhancing modeling accuracy in complex fluid flow scenarios, with potential applications spanning industries like mining, chemical processing, petroleum transportation, and sanitation systems, providing practical alternatives to costly experimental procedures in pipe systems.

Wellbore Pressure Modeling for Pumping and Tripping Simultaneously to Avoid Severe Pressure Swab

A pumping-while-tripping method is proposed to mitigate pressure swabs during tripping out in wells with a narrow mud density window and extended reach. In the proposed tripping-out process, the fluid circulation is started by using a special pump from a customized circulation line before tripping is initiated. During the tripping out, drilling fluid is circulated in the wellbore simultaneously while the drilling string is moving. A model to simulate the dynamic pressure changes in this process is developed based on the Navier–Stokes (N-S) equations and a damped free vibration system. The model was initially developed for Herschel–Bulkley (H-B) fluid; however, it can be applied to other fluid models by eliminating the non-existing terms. An analysis was conducted to investigate the effect of tripping velocity and circulation pumping rate on the pressure changes. The results show that pumping-while-tripping is effective in mitigating the pressure swab during tripping out, which is especially useful for extended-reach wells. It can also help to increase tripping out velocity and save tripping time for drilling operations.

Analyzing Local Shear Rate Distribution in a Dual Coaxial Mixing Bioreactor Handling Herschel–Bulkley Biopolymer Solutions through Computational Fluid Dynamics

For the aeration of highly viscous non-Newtonian fluids, prior studies have demonstrated the improved efficacy of dual coaxial mixing bioreactors fitted with two central impellers and a close clearance anchor. Evaluating the effectiveness of these bioreactors involves considering various mixing characteristics, with a specific emphasis on shear rate distribution. The study of shear rate distribution is critical due to its significant impact on the mixing performance, gas dispersion, and homogeneity in aerated mixing systems comprising shear-thinning fluids. Although yield-pseudoplastic fluids are commonly employed in various industries, there is a research gap when it comes to evaluating shear rate distribution in aerated mixing bioreactors that utilize this fluid type. This study aims to investigate shear rate distribution in an aerated double coaxial bioreactor that handles a 1 wt% xanthan gum solution, known as a Herschel–Bulkley fluid. To achieve this goal, we employed an experimentally validated computational fluid dynamics (CFD) model to assess the effect of different mixing configurations, including down-pumping and co-rotating (Down-Co), up-pumping and co-rotating (Up-Co), down-pumping and counter-rotating (Down-Counter), and up-pumping and counter-rotating (Up-Counter) modes, on the shear rate distribution within the coaxial mixing bioreactor. Our findings revealed that the Up-Co system led to a more uniform local shear distribution and improved mixing performance.

A Review of the Settling Law of Drill Cuttings in Drilling Fluids

During the drilling process, cuttings settle under the action of gravity, which easily results in the formation of a cuttings bed, which then results in wellbore cleaning problems. The settling law of cuttings in drilling fluid is essentially a problem of solid–liquid two-phase settling. This study analyzes and summarizes the effects of the wall effect, the rheology of the fluid, particle shape irregularity, and particle concentration on the settling rate of particles and clarifies the problems faced by current research on the settling rate of particles and the development direction. Studies have shown that walls exert additional blocking effects on particles, thus reducing their settling velocity. The shear thinning effect of non-Newtonian fluids such as power-law fluids and Herschel–Bulkley fluids will reduce the viscosity of the liquid, thus increasing the settling velocity of the particles. Compared with spherical particles, irregular particles will obtain higher resistance in the fluid, leading to a decline in the particle settling velocity. The mutual interference between particles will result in an increase in the drag force on the particles and a decline in the settling velocity. However, when the particles are aggregated, the settling velocity will increase. This study can provide theoretical guidance for predicting the migration law of cuttings during the drilling of horizontal wells, and it has important significance for enriching the theory of solid–liquid two-phase flow.

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)].

Analysis of entropy generation and Brownian motion for the pulsatile flow of Herschel–Bulkley fluid in a diseased curved artery

The influence of Thermophoresis and Brownian motion on the pulsing flow of nano-fluid (blood) through a curved artery with stenosis and post-stenotic dilatation in its interior is investigated numerically. The Herschel-Bulkley model of fluid dynamics accurately represents the fluid's rheology. By applying the mild stenosis condition, the highly coupled momentum, energy, and mass concentration equations can be modelled and reduced. Explicit finite differences methods are used to discretize and solve the non-dimensionalized governing equations associated with the boundary condition. Entropy creation in blood flow due to loss and magnetohydrodynamic processes is also investigated. Graphs and discussions of the effects of changing pertinent geometric and rheological factors on key flow characteristics, such as temperature, velocity, and mass concentration, are provided. Because of the small changes in blood temperature and mass concentration caused by the artery's curvature, it is observed that the radius of the curved channel has a significant impact on blood velocity. An increase in Brownian motion also resulted in a drop in the nano-temperature, a fluid's but the thermophoresis parameter exhibited the reverse behavior.

Flow modeling and in situ viscosity monitoring for a combined rotational and pressure-driven flow of a non-Newtonian fluid through a concentric annulus

Based on the energy dissipation rate, we proposed a systematic method for quantifying the effective viscosity, effective shear rate, and flow characteristics in a drilling flow of non-Newtonian fluids in a concentric annulus subjected to a combined pressure-driven and rotational flow. Two flow parameters, i.e., the energy dissipation rate coefficient and the effective shear rate coefficient, were introduced to quantify flow characteristics, such as the relationship between pressure drop, flow rate, torque, and rotational speed, which are nearly independent of rheological behaviors. In this work, we began with flow quantification of the individual flow problem in a concentric annulus, i.e., pressure-driven flow and rotational flow, and derived expressions of two flow parameters analytically in each case. Then, we established the flow quantification method for the combined pressure-driven and rotational flow. The proposed flow modeling scheme was derived analytically with a power-law fluid and was validated for various non-Newtonian fluids, such as Carreau and Herschel–Bulkley fluids, through extensive numerical simulations. The method can be employed for the in situ viscosity measurement of drilling muds in terms of shear rate, as well as for the estimation of torque, pressure drop, and power consumption. Maximum errors between theoretical prediction and numerical simulation results in estimating torque, pressure drop, and shear-dependent viscosity were found to be 5.5%, 7.3%, and 6%, respectively.

Gravity flow in rock fractures with substrate and edge drainage

We investigate the influence of fluid rheology on flow in a finite rock fracture with vertically varying aperture and subject to competing drainage mechanisms due to a permeable substrate and a draining edge. The flow is due to the release of a finite volume of fluid, and the rheology of the fluid is either Newtonian, Ostwald–deWaele, or Herschel–Bulkley. The Hele–Shaw analogy between lubrication and seepage flows allows extending our results to a porous medium with permeability and porosity varying in the vertical direction. The general solution is numerical, except for a self-similar solution derived for Newtonian fluids in a constant aperture fracture and another for Ostwald–deWaele fluids without substrate drainage. Results for the profile of the current and the volume remaining within the fracture, or drained at the substrate and edge, depend on a dimensionless parameter λ incorporating fluid rheology, fracture geometry, and ambient depth; drainage times exhibit order of magnitude variations depending on λ. A second dimensionless parameter, λ′, intervenes for Herschel–Bulkley fluids, with λ′→∞ for Ostwald–deWaele fluids. The theoretical model is validated with a series of experiments conducted with a novel experimental apparatus, accurately reproducing the condition of substrate drainage and allowing the experimental determination of λ and λ′. The agreement between theory and experimental results for both configurations with constant and V-shaped aperture is quite good, considering model approximations and experimental uncertainties. The present analysis shows how domain anisotropy, though simply schematized, and fluid rheology are relevant for the correct estimation of all integral variables, such as the residual fluid volume in the fracture as a function of time.

Experimental and Numerical Analysis of the Effect of Rheological Models on Measurements of Shear-Thinning Fluid Flow in Smooth Pipes

The aim of this research is to investigate the effects of rheological models of shear-thinning fluids and their estimated parameters on the predictions of laminar, transitional, and turbulent flow. The investigation was carried out through experimental and computational fluid dynamics (CFD) studies in horizontal pipes (diameters of 19.1 mm and 76.2 mm). Six turbulent models using Reynolds averaged Navier–Stokes equations in CFD_ANSYS Fluent 19.0 were examined in a 3D simulation followed by comparison studies between numerical and experimental results. Regarding results of laminar regions in power-law rheology models, Metzner and Reed presented the best fit for the pressure loss and transitional velocity. For the turbulent region, correlations observed by Wilson and Thomas as well as Dodge and Matzner had good agreement with the experimental results. For Herschel–Bulkley fluids, pressure losses and transitional regions based on a yielded region were examined and compared to the experimental results and the modified Slatter Reynolds number, where the results provided good estimation. For both pipe diameters, the Slatter model was the best fit for pressure losses of Herschel–Bulkley fluids in the turbulent regime. Furthermore, when comparing k-omega and k-epsilon turbulence models to the power-law behaviour, numerical studies delivered the most accurate results with fluids that have a higher behaviour index. However, the error percentage significantly increased at a higher shear rate in the Herschel–Bulkley fluids with a greater yield stress effect. Moreover, the modified Herschel–Bulkley viscosity function by Papanastasiou was implemented in the current CFD study. This function was numerically stabilized, devoid of discontinuity at a low strain rate, and more effective in transitional regions.

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.

3D printing of honeycomb Si3N4 ceramic

AbstractIn this paper, a honeycomb Si3N4 ceramic was fabricated by 3D printing with a well‐preserved structure. The effects of Si3N4 content on the rheological properties of Si3N4/sol–silica ink and the printing resolution of products were investigated. The microstructure, phase composition, liner shrinkage rate, and fracture behavior of printed samples before and after sintering were systematically characterized in detail. The results showed that the modified inks had the optimized rheological properties, and the stress–shear rate curves corresponding to each slurry could be well described by Bingham and Herschel–Bulkley fluid models. The corresponding slump rates of the printed samples with different Si3N4 to sol–silica mass ratios were all lower than 4%, and the linear shrinkage rate of all of the samples after sintering was below 20%. The fracture behavior under compressive loading of the honeycomb Si3N4 ceramics tended to be non‐catastrophic fractures both before and after sintering. The compressive strengths of all of the printed samples decreased with the increase of the Si3N4 content, and the highest compressive strength of the honeycomb ceramics could reach 131.2 MPa after sintering at 1600°C, which was about 366.9% higher than that of the samples in green state prior to the sintering.

Impact of unsteadiness on the non-Newtonian flow of blood in a vascular tube with stenosis and aneurysm: analytical solution

A theoretical analysis is conducted to explore the time-dependent problem of blood flow streaming in a tapered vascular tube having post-stenotic dilatation followed by symmetric arterial stenosis. Blood is considered a homogeneous incompressible fluid whose rheological behavior is estimated by Herschel–Bulkley fluid model. By exploiting the prescribed boundary conditions, unique expressions of velocity, flow rate, plug core radius and resistance to the flow have been made by handling the complexity of the governing equations with the standard perturbation technique. The steady-state component of the pressure gradient originating due to the functioning of the heart is computed numerically. The effects of various parameters involved in the problem in accordance with the stenosis and aneurysm on the different flow variables are elaborated with the help of graphs. It has been investigated that the rheological parameters have considerable effects on the dynamic of the flow throughout the constricted region of the vascular tube.

Experimental footprints of a water-rich depletion layer in the Herschel–Bulkley pipe flow of solidifying polyelectrolytes

A fundamental understanding of the transition from fluid-like to gel-like behavior is critical for a range of applications including personal care, pharmaceuticals, food products, batteries, painting, biomaterials, and concrete. The pipe flow behavior of a Herschel–Bulkley fluid is examined by a combination of rheology, ultrasound imaging velocimetry, and pressure measurements together with modeling. The system is a solution of 0.50 wt. % polyelectrolytes of sulfated polysaccharides in water that solidifies on cooling. Fluids with different ionic strengths were pumped at various rates from a reservoir at 80 °C into a pipe submerged in a bath maintained at 20 °C. The fluid velocity, pressure drop ΔP, and temperature were monitored. The same quantities were extracted by solving continuity, energy, and momentum equations. Moreover, the modeling results demonstrate that the local pressure gradient along the pipe dPdx|x is related to the local yield stress near the pipe wall τywall|x, which explains the variations of dPdx|x along the pipe. Experimental results show much lower values for ΔP compared to those from modeling. This discrepancy is exacerbated at higher ionic strengths and smaller flow rates, where fluid shows a higher degree of solidification. The tabulated experimental ΔP data against the solidification onset length Lonset (where the fluid is cool enough to solidify) along with the ultrasound imaging velocimetry associate these discrepancies between experiments and models to a depletion layer of ∼1 μm, reflecting the lubrication effects caused by the water layer at the wall.

Study on the Influence of Tripping Operation on Annular Transient Surge-Swab Pressure of Herschel–Bulkley Fluid

Abstract The accurate prediction of transient surge-swab pressure, during the tripping operation under a narrow pressure window, is an important premise to ensure the safety of tripping operation and drilling. Different rheological modes of drilling fluid will have a great influence on the surge-swab pressure in wellbore, while the Herschel–Bulkley mode can simulate the rheological characteristics of actual drilling fluid in a better way. Therefore, the influence of drilling fluid flow friction under different rheological modes was mainly considered according to the transient flow model. Meanwhile, the influence of pumpage on transient surge-swab pressure during tripping operation was also considered with Herschel–Bulkley fluid-based tripping operation bottomhole transient surge-swab pressure calculation model established and model verification carried out. The following research results were obtained. First, compared with the Herschel–Bulkley mode, the power-law mode and Bingham mode will overestimate the flow friction of drilling fluid. Second, the surge-swab pressure peak values of different mud systems are water-based mud > polymer mud > oil-based mud. Third, the peak value of bottomhole transient surge-swab pressure increases with the increase of drilling fluid yield value, consistency coefficient, and flow behavior index, and the influence law is similar. Fourth, the tripping velocity and pumpage will directly affect the peak value and variation rule of bottomhole transient surge-swab pressure. Herschel–Bulkley fluid-based tripping operation bottomhole transient surge-swab pressure study helps control the surge-swab pressure during tripping operations and it is of great significance for lean, safe, and efficient drilling in narrow pressure window formations.