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Purpose The purpose of this study is to solve convective diffusion equation analytically by considering appropriate boundary conditions and using the Taylor-Aris method to determine the solute concentration, the effective and relative axial diffusivities. Design/methodology/approach >An analysis has been conducted on how body acceleration affects the dispersion of a solute in blood flow, which is known as a Bingham fluid, within an artery. To solve the system of differential equations analytically while validating the target boundary conditions, the blood velocity is obtained. Findings The blood velocity is impacted by the presence of body acceleration, as well as the yield stress associated with Casson fluid and as such, the process of dispersing the solute is distracted. It graphically illustrates how the blood velocity and the process of solute dispersion are affected by various factors, including the amplitude and lead angle of body acceleration, the yield stress, the gradient of pressure and the Peclet number. Originality/value It is witnessed that the blood velocity, the solute concentration and also the effective and relative axial diffusivities experience a drop when either of the amplitude, lead angle or the yield stress rises.

In this work, the momentum and heat transfer characteristics of a time-dependent flow of Bingham plastic fluid over a heated isothermal sphere have been investigated numerically over wide ranges of conditions: Reynolds number, 5 ≤ Re ≤ 120, Bingham number, 0.1 ≤ Bn ≤ 100, Prandtl number, 0.7 ≤ Pr ≤ 100, frequency, π/4 ≤ ω* ≤ π and amplitude, 0 ≤ A ≤ 0.8. The influence of the fluid yield stress and inertia due to the imposed flow pulsations on the flow and thermal fields have been examined in detail. Detailed structure of the flow and temperature fields are analysed in terms of the instantaneous streamlines, isothermal contours, yielded/unyielded zones, surface pressure profiles, time-average drag coefficient and surface- and time-average Nusselt number. The influence of the frequency and amplitude of pulsations on the size of yielded (fluid-like) and unyielded (solid-like) zones is considered in order to understand convective heat transport. The size of yielded zones is seen to be in phase with the imposed pulsating velocity. However, the yield stress effects suppress the influence of flow pulsations. The temporal evolution of the drag coefficient and Nusselt number lag the imposed pulsating flow by different degrees thereby indicating the inherently different evolution of the momentum and thermal boundary layers. Broadly, the pulsating flow conditions may lead to up to 20% augmentation in heat transfer provided there is a moderate degree of advection and/or when the fluid yield stress effects are weak, i.e., small Bingham numbers. Thus, the maximum benefits of pulsating flow accrue in Newtonian fluids only. Finally, the present values of the time-average Nusselt number have been consolidated in the form of a predictive expression.

Grouting is a commonly used technique in rock engineering to enhance the joint strength and improve the stability of surrounding rock. Grout penetration characteristic is controlled by grouting parameters and has a significant role on practice. In the study, a numerical manifold method (NMM) for grout penetration process simulation in fractured rock mass is firstly proposed. The fluid flow behaviour of the grout is assumed to be a Bingham fluid and control equations are established using discrete fracture network model. The global discretization equation, element sub-matrixes and NMM simulation algorithm for grouting are presented. Then, numerical tests for grouting process in a single fracture and a regular fracture network are conducted firstly to verify the proposed NMM grouting model by comparing with analytical solutions and experimental results. Furthermore, the effects of mesh size, fracture and slurry parameters on the grouting performance are systematically investigated using a random fracture network example. The numerical results indicate that the grouted zone and propagation depth decrease as the mesh size of numerical model and yield strength increases, while it increases as initial fracture aperture and grouting pressure increases.

This paper discusses the well posedness of an initial value problem describing the motion of a Bingham fluid in a basin with a degenerate bottom topography. A physical interpretation of such motion is discussed. The system governing such motion is obtained from the Shallow Water-Bingham models in the regime where the Froude number degenerates, i.e taking the limit of such equations as the Froude number tends to zero. Since we are considering equations with degenerate coefficients, then we shall work with weighted Sobolev spaces in order to establish the existence of a weak solution. In order to overcome the difficulty of the discontinuity in Bingham's constitutive law, we follow a similar approach to that introduced in [G. DUVAUT and J.-L. LIONS, Springer-Verlag, 1976]. We study also the behavior of this solution when the yield limit vanishes. Finally, a numerical scheme for the system in 1D is furnished.

The paper is devoted to the study of convective motions of viscoplastic fluids in closed two-dimensional rectangular domains with different aspect ratios under lateral heating. The problem was solved numerically using the ANSYS Fluent software. The Herschel-Bulkley model was chosen for describing the rheological behavior of the fluid. Under certain rheological parameters, this model was transformed into a Newtonian fluid model, whose behavior was also simulated as a limiting case. Based on the calculated results, the dependences of the maximum value of the stream function in the cavity on the Rayleigh number are plotted. It was found that at small Rayleigh numbers the intensity of motion is close to zero. At a certain threshold value of the Rayleigh number there is a sharp increase in the intensity of motion, and a further increase in the Rayleigh number causes an almost linear growth of the maximum value of the stream function. For each of the considered ratios of cavity sides, the "threshold" values of the Rayleigh number, at which a sharp increase in the intensity of motion of liquid is observed, were determined. The obtained values of Rayleigh numbers turned out to be close to the "threshold" values of the Rayleigh number for the Bingham fluid found in the works of other authors earlier. The fields of the stream function and the square root of the second invariant of the viscous stress tensor were obtained for different values of the Rayleigh number and different aspect ratios. The scenarios of rearrangement of quasisolid motion zones with increasing Rayleigh number were compared with the available results for the Bingham fluid.

Modeling of fluids with complex rheology in the lattice Boltzmann method (LBM) is typically realized through the introduction of an effective viscosity. For fluids with a yield stress behavior, such as so-called Bingham fluids, the effective viscosity has a singularity for low shear rates and may become negative. This is typically avoided by regularization such as Papanastasiou’s method. Here we argue that the effective viscosity model can be re-interpreted as a generalized equilibrium in which no violation of the stability constraint is observed. We implement a Bingham fluid model in a three-dimensional cumulant lattice Boltzmann framework and compare the direct analytic effective viscosity/generalized equilibrium method to the iterative approach first introduced by Vikhansky which avoids the singularity in viscosity that can arise in the analytic method. We find that both methods obtain similar results at coarse resolutions. However, at higher resolutions the accuracy of the regularized method levels off while the accuracy of the direct method continuously improves. We find that the accuracy of the proposed direct method is not limited by the singularity in viscosity indicating that a regularization is not strictly necessary.

A mathematical study is developed for the electro-osmotic flow of a non-Newtonian fluid in a wavy microchannel in which a Bingham viscoplastic fluid model is considered. For electric potential distributions, a Poisson-Boltzmann equation is employed in the presence of an electrical double layer (EDL). The analytical solutions of dimensionless boundary value problems are obtained with the Debye-Huckel theory, the lubrication theory, and the long wavelength approximations. The effects of the Debyelength parameter, the plug flow width, the Helmholtz-Smoluchowski velocity, and the Joule heating on the normalized temperature, the velocity, the pressure gradient, the volumetric flow rate, and the Nusselt number for heat transfer are evaluated in detail using graphs. The analysis provides important findings regarding heat transfer in electroosmotic flows through a wavy microchannel.

The linear stability of channel flows driven by pressure drops is carried out for fluids that exhibit a yield stress, the mechanical response of which is prescribed by the Bingham constitutive relation or two of its regularizations: the model due to Allouche and co-workers that is usually referred to as the “simple model”, and Papanastasiou model. Despite the fact that these two regularized models provide a good approximation of the steady Poiseuille flow of a Bingham fluid, they fail to predict the stability characteristics of the exact Bingham model. The critical thresholds for the onset of turbulence predicted by using the simple and Papanastasiou models are essentially the same, but they differ significantly from that of the exact Bingham model. This discrepancy is shown to be due to the absence of energy dissipation in the rigid core of a Bingham fluid.

The aiding-buoyancy mixed convection heat transfer in Bingham plastic fluids from an isothermal cylinder of elliptical and circular shape in a vertical adiabatic channel is numerically investigated. For a fixed shape of the elliptical cylinder E = 2 (ratio of major to minor axes), the effect of confinement is studied for three values of blockage ratio, B, defined as the ratio of the channel width to the circumference of the cylinder/π, as 6.5, 2.17 and 1.3. In order to delineate the role of cross-section of the cylinder, results are also presented here for a circular cylinder of the same heat transfer area as the elliptical cylinder. The results presented herein span the range of conditions as: Bingham number, 0 ≤ Bn ≤ 100, Reynolds number, 1 ≤ Re ≤ 40, and Prandtl number, 1 ≤ Pr ≤ 100 over the range of Richardson number Ri = 0 (pure forced convection) to Ri = 10. Extensive results on drag coefficient, local and surface averaged values of the Nusselt number and yield surfaces are presented herein to elucidate the combined effects of buoyancy, blockage ratio and fluid yield stress. The morphology of the yield surfaces shows that the unyielded plug regions formed upstream and downstream of the cylinder grow faster at low Reynolds numbers with the increasing yield stress effects under the weak buoyancy forces, i.e., small values of Grashof or Richardson number. The heat transfer enhancement is observed with the increasing channel-confinement due to the sharpening of the temperature gradients near the surface of the cylinder. The average Nusselt number shows a positive dependence on the Reynolds number, Prandtl number and Richardson number irrespective of the shape of the cylinder or the type of fluid. By employing the modified definitions of the dimensionless parameters (based on the two choices of the overall effective fluid velocity), predictive correlations have been established for estimating the value of the average Nusselt number in a new application.

We are concerned with a system of partial differential equations (PDEs) describing internal flows of homogeneous incompressible fluids of Bingham type in which the value of activation (the so-called yield) stress depends on the internal pore pressure governed by an advection–diffusion equation. After providing the physical background of the considered model, paying attention to the assumptions involved in its derivation, we focus on the PDE analysis of the initial and boundary value problems. We give several equivalent descriptions for the considered class of fluids of Bingham type. In particular, we exploit the possibility to write such a response as an implicit tensorial constitutive equation, involving the pore pressure, the deviatoric part of the Cauchy stress and the velocity gradient. Interestingly, this tensorial response can be characterized by two scalar constraints. We employ a similar approach to treat stick-slip boundary conditions. Within such a setting we prove long-time and large-data existence of weak solutions to the evolutionary problem in three dimensions.

It is well known that the Non-Newtonian Bingham fluid flow in porous media does not obey the conventional linear Darcy’s law due to the yield stress for the Bingham fluid: There exists a threshold pressure gradient, which means that the seepage flow only happens when the threshold pressure gradient is overcome. The principle of non-Darcy seepage flow with the threshold pressure gradient is also applicable into the situation of the fluid flow in the low-permeable porous media. Here, a nonlinear moving-boundary mathematical model is built for the semispherical centripetal non-Darcy seepage flow with the threshold pressure gradient in a three-dimensional infinite heavy oil reservoir with the type of Bingham fluid; wherein the moving boundary conditions are incorporated for describing the effect of the threshold pressure gradient. In consideration of the strong nonlinearity of the model, the similarity transformation method is applied into obtaining the exact analytical solution of the model. In order to keep full self-similarity for the model, the inner boundary condition is set as variable flow rate that increases linearly with the time. As a result, an exact analytical solution for the nonlinear moving-boundary mathematical model of semispherical centripetal non-Darcy seepage flow with the threshold pressure gradient is obtained. The existence and the uniqueness of the exact analytical solution are also strictly proved. It is also theoretically proved that as the threshold pressure gradient tends to zero, the exact analytical solution can be reduced to that of a mathematical model of semispherical centripetal Darcy’s seepage flow. The presented exact analytical solution can be used for strictly verifying the validity of the numerical methods for solving the three-dimensional moving boundary models of non-Darcy seepage flow with the threshold pressure gradient in the actual engineering problems. From the exact analytical solution, it is also revealed that when the threshold pressure gradient exists, the spatial pressure distribution exhibits an instructive feature of compact support; as the threshold pressure gradient tends to zero, the sensitivity of its effect on the transient distance of the moving boundary and the transient pressure will grow, which reveals the difficulty in accurately determining the position of the moving boundary by the numerical methods and the serious uncertainty problem in the interpretation of the threshold pressure gradient by the pressure transient analysis method in engineering as the threshold pressure gradient is rather small. Through the comparison of the two different exact analytical solutions that corresponds to the two different models with and without incorporating the moving boundary conditions for describing the effect of the threshold pressure gradient, it is demonstrated that when the moving boundary conditions are not incorporated in the modeling, the effect of the threshold pressure gradient on the spatial pressure distribution, the transient pressure and the productivity index can be overestimated largely. Therefore, it is very necessary to incorporate the moving boundary conditions in the modeling of non-Darcy seepage flow with the threshold pressure gradient. The study in the paper definitely provides solid theoretical basis of fluid mechanics for the relevant engineering applications in the development of heavy oil reservoirs and low-permeable reservoirs in petroleum engineering and in the development of water resources in low-permeable formations in hydraulic engineering.

In this paper, laminar flow of non-Newtonian (Bingham) fluid is studied numerically in a two-dimensional lid-driven skewed cavity that incorporates Papanastasiou exponential regularization approach...

This article evaluates the slurry viscosity of radioactive waste. Hanford waste is a motley mixture of particle sizes and densities composed of metal and oxide particles with constituents spanning much of the periodic table stored at high ionic strength and high pH. Hanford wastes have traditionally been represented as simply an ideal Bingham plastic, but this rheological approximation often provides a poor fit from 0 to 200 1/s, precisely the range of interest for engineering design of waste transport and mixing systems. A modified Cross formulation inclusive of a yield stress is used to fit stress-strain rate data for reconstituted Hanford REDOX sludge waste, measured by increasing shear-rate (up-ramp) and decreasing shear-rate (down-ramp) sweeps. This continuous single equation formulation quantitatively matches experimental data better than other rheological models including the Bingham plastic and Herschel-Bulkley approximations. The increase of viscosity and yield stress from up ramp to down ramp suggests the formation of particle microstructure during shear due to attractive forces.

International audience; We study the steady incompressible flow of a Bingham fluid in a thin T-like shaped domain, under the action of given external forces and with no-slip boundary condition on the whole boundary of the domain. This phenomenon is described by non linear variational inequalities. By letting the parameter describing the thickness of the thin domain tend to zero, we derive two uncoupled problems corresponding to the two branches of the T-like shaped structure. We then analyze and give a physical justification of the limit problem; Dans ce travail nous étudions l'écoulement stationnaire d'un fluide incompressible de Bingham, dans un domaine mince en forme de T, sous l'action d'une force extérieure et avec des conditions de non glissement aux parois. Ce phénomène est décrit par des inéquations variationnelles non linéaires. En faisant tendre vers zéro le petit paramètre qui caractérise l'épaisseur du domaine mince, on obtient à la limite deux problèmes découplés, correspondant chacun à l'une des branches du T-domaine. Ensuite on analyse le problème limite et on en donne une interprétation physique.

© 2018 Elsevier B.V. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ Supplementary Raw Research Data, is open data under the CC BY license http://creativecommons.org/licenses/by/4.0/ This author accepted manuscript is made available following 24 month embargo from date of publication (October 2018) in accordance with the publisher’s archiving policy, The start-up flow of a Bingham plastic in a channel is considered and Safronchik’s solution [1] for the initial evolution of the yield surface and the core velocity is revisited. Stricter time bounds for the validity of the above solution are derived and the solution is extended to include the velocity profile in the evolving yielded zone. Comparisons are made with another approximate solution derived under the assumption that the velocity in the yielded zone is parabolic adjusting with the evolving yield surface. This approximation performs well for small values of the yield stress, or, equivalently, for large values of the imposed pressure gradient.

The linear instability characteristics of a core-annular pipe flow of a Newtonian fluid and a Bingham fluid separated by a mixed region is investigated. The main objective of this study is to investigate the effect of the yield stress of the annular fluid on the stability behaviour. The present linear stability analysis reveals that the corkscrew mode is the most dangerous mode for the parameters considered. The Bingham number seems to destabilise the flow via modifying the velocity profile. The mechanism of instability is found to be inviscid in nature. An energy budget analysis shows that the energy due to the gradient of viscosity perturbation in the radial direction, and the transfer of energy from the basic flow to the perturbation via the “Reynolds stress” term are responsible for the instability observed. The positive contributions of these terms towards instability increases with increasing the yield stress. The critical Reynolds number also decreases with increasing the Bingham number for the corkscrew mode, which indicates the destabilising influence of the yield stress.

This paper presents a lattice Boltzmann framework for the transient simulation of blood flow using biologically inspired geometries and pressure boundary conditions. The Kuang-Luo rheological model is used to represent blood as a homogeneous continuum. This model includes the two primary non-Newtonian characteristics of blood, namely viscoplasticity and pseudoplasticity. This paper makes two contributions. First, the numerical challenges associated with zero strain rates and infinite viscosity, as a consequence of the yield stress in the Kuang-Luo model, were addressed by regularising the constitutive equation so that the viscosity tends towards a finite value at low strain rates. A two-relaxation-time operator, which exhibits improved performance over the single-relaxation-time operator and lower computational overhead than the multiple-relaxation-time operator, is employed in the collision process. The recursive relationship between the local strain rate and relaxation rate was addressed by use of an implicit solver for these two quantities. The implemented model was benchmarked against analytic solutions for Poiseuille flow between parallel plates in two dimensions and in a cylindrical tube in three dimensions. More importantly, the transient performance of the implemented model was demonstrated by matching the predicted start-up flow of the Poiseuille flow of a Bingham fluid with the corresponding analytical solution. Second, the numerical developments were applied in the simulation of transient blood flow in complex configurations. The development and implementation of physically inspired pressure profiles highlighted the shortcomings of using a sinusoidal pressure profile in the prediction of velocity and stress distributions. Finally, the simulation of blood flow in a section of a carotid artery indicated a number of flow characteristics that will be of interest to future investigations of clinical problems.

A mathematical model is constructed for MHD flow of Bingham fluid and Jeffrey fluid in a horizontal porous space with heat transfer effect. The dimensionless governing equations for this problem are solved analytically and expressions for velocity and temperature field are presented. Further, impacts of physical parameters on flow and heat transfer charcteristics has been studied and discussed graphically. It is found that increasing viscosity ratio enhances velocity and temperature field whereas thermal conductivity ratio gives the reverse effect. Increasing magnetic field parameter tends to suppress fluid velocity significantly whilst mixed behavior shown in temperature distribution. Velocity of the fluid increased considerably with increasing material parameter whereas the reverse trend can be noticed in temperature distribution. Further, it is observed that pressure gradient influenced significantly on velocity and temperature field. Increasing magnetic parameter reduces mass flow rate whereas it shows opposite trend in shearstress distribution.

This paper presents a new family of semi-implicit hybrid finite volume/finite element schemes on edge-based staggered meshes for the numerical solution of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations in combination with the k−ε turbulence model. The rheology for calculating the laminar viscosity coefficient under consideration in this work is the one of a non-Newtonian Herschel–Bulkley (power-law) fluid with yield stress, which includes the Bingham fluid and classical Newtonian fluids as special cases. For the spatial discretization, we use edge-based staggered unstructured simplex meshes, as well as staggered non-uniform Cartesian grids. In order to get a simple and computationally efficient algorithm, we apply an operator splitting technique, where the hyperbolic convective terms of the RANS equations are discretized explicitly at the aid of a Godunov-type finite volume scheme, while the viscous parabolic terms, the elliptic pressure terms and the stiff algebraic source terms of the k−ε model are discretized implicitly. For the discretization of the elliptic pressure Poisson equation, we use classical conforming P1 and Q1 finite elements on triangles and rectangles, respectively. The implicit discretization of the viscous terms is mandatory for non-Newtonian fluids, since the apparent viscosity can tend to infinity for fluids with yield stress and certain power-law fluids. It is carried out with P1 finite elements on triangular simplex meshes and with finite volumes on rectangles. For Cartesian grids and more general orthogonal unstructured meshes, we can prove that our new scheme can preserve the positivity of k and ε. This is achieved via a special implicit discretization of the stiff algebraic relaxation source terms, using a suitable combination of the discrete evolution equations for the logarithms of k and ε. The method is applied to some classical academic benchmark problems for non-Newtonian and turbulent flows in two space dimensions, comparing the obtained numerical results with available exact or numerical reference solutions. In all cases, an excellent agreement is observed.

This paper investigates the remarkable effects for introducing the Ion-slip characteristics in the magnetohydrodynamic phenomena on the drift of Bingham fluid when flowing through the inner territory of two porous parallel plates in correspondence with the suction case. The lower plate (L-plate) is as steady while the upper one (U-plate) is oscillatory, which oscillates in its own plane at time t>0. A magnetic field, which is uniform, is affixed perpendicular to the plates. The U-plate temperature oscillates while the L-plate temperature is constant. Numerical performance is presented by the MATLAB R2015a simulation tool with the explicit Finite difference Method (FDM) algorithm. To ensure the preciseness and convergence of the solutions, careful attention has been given on the criteria of stability and convergence of the FDM schemes. As an outcome, the converged solution is obtained for Pr ≥ 0.066, βi ≥ 2, Ha ≤ 20, h ≤ 8, S ≥ −10 and Re ≤ −0.017 with the arbitrary choice of βe = 0.10 and Ec = 0.01. The mesh sensibility test gives the competent mesh space at (m,n)=(60,60). The time sensibility test ensures that the solutions at dimensionless time, τ=2.0 will be steady-state. The exactitude of the current study is obtained by comparing with the published results. Finally, the physical influences of several governing parameters, including Ionslip on the fluid property like velocity, local shear stress, temperature and Nusselt number are discussed and decorated graphically.

Recently, Hoang et al. (2021) discussed our paper Zou et al. (2020). In our paper, we made a statement that Dai and Bird (1981)'s solution for two-dimensional (2D) radial Bingham fluid flow between parallel plates violates mass balance. Hoang et al. pointed out that Dai and Bird (1981)'s solution does not violate the mass balance because Dai and Bird (1981)'s solution and our analysis are based on different assumptions, i.e. with consideration of the vertical velocity component in the continuity equation or not, which leads to two different approximation models. In this sense, the mass balance of Dai and Bird (1981)'s solution should not be checked using our solution as a reference. In this reply, we add remarks on the two approximation models and their implication for rock grouting analysis. The discussion by Hoang et al. and this reply are helpful to thoroughly eliminate the existing confusion regarding the two solutions in the rock grouting research community.

In the present paper, the non-iterative PISO algorithm and finite volume method are employed to solve the exact Bingham fluid flow. The OpenFOAM solver, i.e., icoFoam, is modified to achieve this goal. The accuracy of numerical procedures is obtained by reproducing the results of Chupin and Dubois (Comput Math Appl 72:1263–1286, 2016. doi:10.1016/j.camwa.2016.06.026). Then, the Bingham fluid flow in a cylindrical enclosure with the rotating top wall is numerically studied by the modified solver for the following ranges of conditions: Reynolds number, $$ 1 \le \text{Re} \le 1000 $$, Bingham number, $$ 0 \le {\text{Bn}} \le 1000 $$ and aspect ratio (AR) of 1, 1/2, 1/4, 1/8 and 1/16. The variation of yielded regions percentage with Re and Bn is plotted for AR = 0.5 and 1. It is found that Reynolds number change in the range $$ 1 \le {\text{Re}} \le 10{\text{Bn}} $$ has no significant effect on the size and shape of the unyielded regions for constant Bn and AR = 1. The effect of Re, Bn and AR on the primary ($$ Q_{p}^{*} $$) and secondary ($$ Q_{s}^{*} $$) volumetric flow rate is also discussed. A correlation is proposed in terms of Re and Bn to approximate the torque coefficient ($$ C_{\text{T}} $$) for $$ 1\le {\text{Re}} \le 1 0 0,\,\, 0\le {\text{Bn}} \le 1000\, $$ and different aspect ratios.

By using dimension reduction and homogenization techniques, we study the steady flow of an incompresible viscoplastic Bingham fluid in a thin porous medium. A main feature of our study is the dependence of the yield stress of the Bingham fluid on the small parameters describing the geometry of the thin porous medium under consideration. Three different problems are obtained in the limit when the small parameter $\varepsilon$ tends to zero, following the ratio between the height $\varepsilon$ of the porous medium and the relative dimension $a_\varepsilon$ of its periodically distributed pores. We conclude with the interpretation of these limit problems, which all preserve the nonlinear character of the flow., Comment: 21 pages, 1 figure

Dense granular materials are universal in nature and some common examples of this kind of materials in our daily lives are sand, rice, sugar etc. Research in this area is motivated by numerous applications encountered in industrial processes, such as hopper discharge, chute flow, moving beds, sandpipe flow, etc. and also in geophysics for the description and prediction of natural hazards like landslide and rock avalanches. Here, a continuum description of these granular flows is appropriate. Granular materials in liquid-like state show strong non-Newtonian behavior, which is typically described by phenomenological constitutive laws with local rheology. Very often, fluidization and flows of granular materials are localized in thin shear bands, whereas the granular material remains in solid-like state outside these bands. In this paper we establish a benchmark problem for granular materials, namely as Couette flow which is a very common example for granular materials in industrial application. We formulate the rheology for the continuum approach in the framework of the regularized version of a Bingham fluid and use advanced numerical methods regarding discretization as well as solution aspects, so that we can provide mesh independent results with two different software packages, Featflow and Openfoam, which can be used for validation and evaluation of the different methods and approaches. The authors thank the German Research Foundation DFG for funding under the grants TU 102/44-1 and SCHW 1168/6-1 within the pilot for transnational cooperation of the Dutch Technology Foundation STW and the DFG. This work was also supported by DAAD.

The theory of slow viscous flow around a slender body is generalized to the situation where the ambient fluid has a yield stress. The local flow around a cylinder that is moving along or perpendicular to its axis, and rotating, provides a first step in this theory. Unlike for a Newtonian fluid, the nonlinearity associated with the viscoplastic constitutive law precludes one from linearly superposing solutions corresponding to each independent component of motion, and instead demands a full numerical approach to the problem. This is accomplished for the case of a Bingham fluid, along with a consideration of some asymptotic limits in which analytical progress is possible. Since the yield stress of the fluid strongly localizes the flow around the body, the leading-order slender-body approximation is rendered significantly more accurate than the equivalent Newtonian problem. The theory is applied to the sedimentation of inclined cylinders, bent rods and helices, and compared with some experimental data. Finally, the theory is applied to the locomotion of a cylindrical filament driven by helical waves through a viscoplastic fluid.

The Folgar–Tucker fiber orientation model coupled with weakly compressible Smoothed Particle Hydrodynamics is used to simulate the process of casting of fiber reinforced concrete and to predict the spatial-temporal evolution of the probability density function of fiber orientation. The flowable concrete-fiber mix is modeled as a viscous Bingham-type fluid. Model predictions qualitatively agree with fiber orientations observed in an L-box test with fibers suspended in transparent gel. Important factors and assumptions regarding the fiber flow are reviewed and conclusions are drawn based on numerical experiments.

Buoyant miscible displacement flow of a Bingham fluid by a Newtonian fluid along a vertical plane-channel is studied, in the high Peclet number regime. The displacing fluid is lighter than the displaced fluid and the flow direction is upwards, hence density-unstable. The flow is effectively governed by 4 dimensionless parameters: the Newtonian Bingham number (BN), the viscosity ratio (m), the Reynolds number (Re), and modified Froude number (Fr). This is a simple model for micro-annulus formation in the primary cementing of oil and gas wells. The micro-annulus layers correspond to residual drilling mud that remains on the walls after displacement. Here we show that in density unstable situations static residual wall layers can exist for yield stresses below the minimum for density stable regimes. The layers are partially static and may also be thicker than the fully static layers encountered in density stable flows. Adverse buoyancy gradients (represented here by large χ = 2 R e F r 2 ), also seems to promote instability of the flow. These instabilities are various, ranging from Rayleigh–Taylor type frontal instabilities, through to Kelvin–Helmholz type inertial waves, and more viscous-controlled inverse bamboo and mushroom morphologies. We present an overview of flow regimes within our parameter space and approximate onset criteria. It is unclear whether or not the instabilities observed result in improved fluid displacement efficiency.

We consider the steady axisymmetric Poiseuille flow of a Bingham plastic under the assumption that both the plastic viscosity and the yield stress vary linearly with pressure. An analytical solution is derived for the case where the growth coefficients of both rheological parameters are equal, which is indeed a reasonable assumption for certain oil-drilling fluids allowing the existence of a separable solution with a cylindrical unyielded core. The conditions for the occurrence of flow and the effects of the growth coefficient on the radius of the unyielded plug, the velocity profiles and the pressure distributions are discussed.

The Bingham fluid model represents viscoplastic materials that display yielding, that is, behave as a solid body at low stresses, but flow as a Newtonian fluid at high stresses. In any Bingham flow, there may be regions of solid material separated from regions of Newtonian flow by so-called yield boundaries. Such materials arise in a range of industrial applications. Here, we consider the helical flow of a Bingham fluid between infinitely long coaxial cylinders, where the flow arises from the imposition of a steady rotation of the inner cylinder (annular Coutte flow) on a steady axial pressure driven flow (Poiseuille flow), where the ratio of the rotational flow compared to the axial flow is small. We apply a perturbation procedure to obtain approximate analytic expressions for the fluid velocity field and such related quantities as the stress and viscosity profiles in the flow. In particular, we examine the location of yield boundaries in the flow and how these vary with the rotation speed of the inner cylinder and other flow parameters. These analytic results are shown to agree very well with the results of numerical computations.

We study a generalization of the Navier–Stokes–Fourier system for an incompressible fluid where the deviatoric part of the Cauchy stress tensor is related to the symmetric part of the velocity gradient via a maximal monotone 2-graph that is continuously parametrized by the temperature. As such, the considered fluid may go through transitions between three of the following regimes: it can flow as a Bingham fluid for a specific value of the temperature, while it can behave as the Navier–Stokes fluid for another value of the temperature or, for yet another temperature, it can respond as the Euler fluid until a certain activation initiates the response of the Navier–Stokes fluid. At the same time, we regard a generalized threshold slip on the boundary that also may go through various regimes continuously with the temperature. All material coefficients like the kinematic viscosity, friction or activation coefficients are assumed to be temperature-dependent. We establish the large-data and long-time existence of weak solutions, applying the L ∞ L ∞ -truncation technique to approximate the velocity field.

The Bingham fluid model has been successfully used in modeling a large class of non-Newtonian fluids. In this paper, the authors extend to the case of Bingham fluids the results previously obtained by Chipot and Mardare, who studied the asymptotics of the Stokes flow in a cylindrical domain that becomes unbounded in one direction, and prove the convergence of the solution to the Bingham problem in a finite periodic domain, to the solution of the Bingham problem in the infinite periodic domain, as the length of the finite domain goes to infinity. As a consequence of this convergence, the existence of a solution to a Bingham problem in the infinite periodic domain is obtained, and the uniqueness of the velocity field for this problem is also shown. Finally, they show that the error in approximating the velocity field in the infinite domain with the velocity in a periodic domain of length 2l has a polynomial decay in l, unlike in the Stokes case (see [Chipot, M. and Mardare, S., Asymptotic behaviour of the Stokes problem in cylinders becoming unbounded in one direction, Journal de Mathematiques Pures et Appliquees, 90(2), 2008, 133–159]) where it has an exponential decay. This is in itself an important result for the numerical simulations of non-Newtonian flows in long tubes.

In this study, applicability of peristaltic waves as an efficient means for bio-particle manipulation is numerically investigated using the lattice Boltzmann method (LBM). The analysis is carried out in a flow configuration which comprises a two-dimensional, closed-ended, vertical cavity filled with a Bingham fluid and equipped with two flexible side walls and two rigid upper and lower plates. It is shown that by judicially adjusting the working parameters of the peristaltic waves (e.g., wavelength, amplitude, speed, and/or phase lag) it is possible to control the dynamics/trajectory of a large, circular bio-particle immersed in the fluid. It is also shown that a suspending circular particle moves faster if the surrounding fluid possess a yield-stress. Based on our obtained numerical results, for a given cross-sectional area, circular particles move faster than elliptical particles when excited peristaltically.

The evolution equations for the particle distribution functions are written in a divergence form applicable in three dimensions. From this set, it is shown that the continuity equation and the equations of motion are satisfied in Cartesian, cylindrical and spherical coordinates for all fluids when additional source terms are added to the equations of evolution in the latter two coordinate systems. If the body forces are present, a new set of source functions is required in each coordinate system and these are described as well. Next, the energy equation is derived by using a separate set of particle distribution functions. Modifications of the relevant equations to be applicable to incompressible fluids is described. The incorporation of boundary conditions and the description of the numerical scheme for the simulation of the flows employing the new approach is given. Validation results obtained through the modelling of a mixed convection flow of a Bingham fluid in a lid-driven square cavity, and the steady flow of a Bingham fluid in a pipe of square cross-section are presented. Next, using the cylindrical coordinate version of the evolution equations, numerical modelling of the steady flow of a Bingham fluid and the Herschel–Bulkley fluid in a pipe of circular cross-section have been performed and compared with the simulation results using the augmented Lagrangian method as well as the analytical solutions for the velocity field and the flow rate. Finally, some comments on the theoretical differences between the present approach and the existing formulations regarding Lattice Boltzmann Equations are offered.

The unclassified tailings paste (UTP) is a typical non-Newtonian fluid with a mass fraction over 70%, and the UTP disposal technology is of great significance to solve the disaster of tailings ponds and underground voids. The widely used Buckingham equation for predicting the pressure gradient of UTP in pipe flow does not consider the effect of wall slip drag reduction and the prediction accuracy is relatively low. To this end, a theoretical derivation based on the apparent wall slip equations established by previous scholars, sedimentation tests, loop tests and rheological tests were carried out. When the UTP is a Bingham fluid, the proposed new pressure gradient model which considers the wall slip effect is composed of the Buckingham equation and a dimensionless wall slip correction coefficient. The correction coefficient decreases with increases in either slip layer thickness and plastic viscosity of UTP, and increases with increases in either pipe diameter and slip layer viscosity. In this study, the dimensionless correction coefficient is 0.83∼0.85. When using the new pressure gradient model, the ratio of solids volume fraction to the maximum packing volume fraction of UTP (the filling degree) is at least 0.76. Compared with the measured pressure gradient from the loop tests in this study and the literature, the average error of the new pressure gradient model is 11.3%, whereas the average error of the Buckingham equation is 29.8%. The advantages of the new pressure gradient model are that the required parameters are easy to obtain, the model is easy to solve, and the prediction accuracy is better than the Buckingham equation. When the UTP is a Herschel-Bulkley fluid, the reliability of the pipe pressure gradient model which considers the wall slip effect still needs to be verified.

The plane Poiseuille flow of a Bingham plastic with pressure-dependent material parameters is analysed. Both the plastic viscosity and the yield stress are assumed to vary linearly with pressure and analytical solutions are derived for the two-dimensional pressure and the one-dimensional velocity. The effects of the plastic-viscosity and yield-stress growth parameters on the thickness of the unyielded plug and the conditions for the occurrence of flow are discussed.

We study the peristaltic transport of a Bingham fluid in a channel with small aspect ratio whose walls behave as a periodic traveling wave. The governing equations in the unyielded phase are obtained writing the integral formulation for the momentum balance. As shown in Fusi et al. (2015), this approach allows to overcome the so-called “lubrication paradox” which may arise in the thin film approximation. We consider the case in which the inlet flux is prescribed and the one in which the flow is driven by a given pressure drop. In both cases the solution of the problem is determined solving a nonlinear integral equation for the yield surface. We perform some numerical simulations to illustrate the behavior of the yield surface, assuming that the traveling wave describing the peristaltic motion has a sinusoidal shape.

We develop an asymptotic solution for the axisymmetric squeeze flow of a viscoplastic medium. The standard lubrication-style expansions of the problem predict a plug speed which varies slowly in the principal flow direction. This variation implies that the plug region cannot be truly unyielded. Our solution shows that this region is a pseudo-plug region in which the leading order equation predicts a plug, but really it is weakly yielded at a higher order. We follow the asymptotic technique suggested earlier by Balmforth and Craster (1999) and Frigaard and Ryan (2004). This method involves no relaxation of the exact Bingham model. The analytical expression for the squeeze force is in good agreement with previous numerical results. In the second part of the paper, the axisymmetric squeeze flow of a Bingham fluid with slip yield boundary condition at the wall is considered. We provide an asymptotic solution for this type of flow. Depending on the ratio of two dimensionless parameters partial slip (stick-slip) or full slip at the wall (slip) are possible.

The effect of modifying yield stress on turbulent pipe flow of generalised Newtonian fluids at a friction Reynolds number of 323 is investigated using direct numerical simulations. Simulations are carried out for Bingham and Herschel–Bulkley fluids with the yield stress varying from 0% to 20% of the mean wall shear stress. Results show that the effect of increasing yield stress is mostly similar to shear thinning in power-law fluids. The turbulent viscous stress which arises due to viscosity fluctuations is negative for a yield stress fluid and is higher in magnitude for higher yield stress. An analysis of the turbulent kinetic energy budget showed that the effect of yield stress is mainly significant near the wall for y + ≲ 60 which was also seen for shear-thinning power-law fluids at similar Re τ . Additional shear thinning enhances the yield stress effect. The main difference between shear thinning and yield stress is that the effect of yield stress is maximum outside the viscous sublayer whereas shear thinning has a more significant effect inside the viscous sublayer.

We study numerically isoviscous displacement flows of a Bingham fluid by a Newtonian fluid in a horizontal plane channel. We consider a heavy-light displacement flow with a laminar imposed flow. We analyze the flow from various perspectives, using the three dimensionless parameters that largely describe the flow: the Bingham number (Bn), the densimetric Froude number (Fr) and the Reynolds number (Re). In particular, we are able to categorize three classes for the main flow regimes, viz., center-type and slump-type, no-back-flow and temporary-back-flow, and stable and unstable displacements, for which we also quantify their transition boundaries versus the aforementioned dimensionless numbers. We show that the appearance of unstable flows can eventually lead to a number of different flow patterns, such as sinusoidal shaped wave and periodic-detachment forms. In addition, we study in depth an interesting feature exclusively associated to viscoplastic fluid displacements, i.e., the formation of static residual layers on the upper/lower walls of the channel. We also study the evolution of the key displacement flow features, i.e., the trailing and leading fronts, and classify certain secondary flow features related to the leading front, such as plug-like, inertial tip, semi-detached and fully-detached fronts. Finally, we quantify the velocity of the leading front at long times.

Buoyant miscible displacement flow of a Bingham fluid by a Newtonian fluid along a vertical plane-channel is studied, in the high Peclet number regime. The displacing fluid is denser than the displaced fluid and the flow direction is density-stable (upwards). The flow is effectively governed by 4 dimensionless parameters: the Newtonian Bingham number (BN), the viscosity ratio (m), the Reynolds number (Re), and modified Froude number (Fr). This is a simple model for micro-annulus formation in the primary cementing of oil and gas wells. We show that the residual layer thickness is largely determined by two parameters: (BN/m, χ*/m), where χ * = 2 R e F r 2 . Residual wall layers may be either static or mobile, and mobile layers either evolve to become static or are washed from the channel at long times. We show that the different behaviours of the residual layers are linked to different characteristic behaviours of the displacement front, and we show how the latter behaviours can be predicted using a lubrication/thin-film approximation.

Elasto-viscoplastic fluid flow in tubes of arbitrary cross-section is explored. A constitutive equation superposing elasticity on the viscoplastic behavior through a linear combination of the non-linear simplified Phan–Thien–Tanner constitutive model of viscoelasticity and the non-linear Bingham constitutive equation of viscoplasticity is developed. The equation of motion is solved analytically for the longitudinal field for steady flow in tubes of non-circular cross-section. The evolution of the plug and stagnant zones, hallmarks of viscoplastic behavior, is studied when elasticity is present, and the rate of flow is determined in terms of the Weissenberg and Bingham numbers. Elasticity tends to enhance the rate of flow for given viscoplastic conditions, and stagnant and plug zone configurations are substantially altered.

Flow of Bingham plastics through straight, long tubes is studied by means of a versatile analytical method that allows extending the study to a large range of tube geometries. The equation of motion is solved for general non-circular cross-sections obtained via a continuous and one-to-one mapping called the shape factor method. In particular the velocity field and associated plug and stagnant zones in tubes with equilateral triangular and square cross-section are explored. Shear stress normal to equal velocity lines, energy dissipation distribution and rate of flow are determined. Shear-thinning and shear-thickening effects on the flow, which cannot be accounted for with the Bingham model, are investigated using the Hershey-Bulkley constitutive formulation an extension of the Bingham model. The existence and the extent of undeformed regions in the flow field in a tube with equilateral triangular cross-section are predicted in the presence of shear-thinning and shear-thickening as a specific example. The mathematical flexibility of the analytical method allows the formulation of general results related to viscoplastic fluid flow with implications related to the design and optimization of physical systems for viscoplastic material transport and processing.

We develop a multiscale approach to describe the behavior of a suspension of solid magnetizable particles in a viscous non-conducting fluid in the presence of an externally applied magnetic field. By upscaling the quasi-static Maxwell equations coupled with the Stokes' equations we are able to capture the magnetorheological effect. The model we obtain generalizes the one introduced by Neuringer & Rosensweig for quasistatic phenomena. We derive the macroscopic constitutive properties explicitly in terms of the solutions of local problems. The effective coefficients have a nonlinear dependence on the volume fraction when chain structures are present. The velocity profiles computed for some simple flows, exhibit an apparent yield stress and the flow profile resembles a Bingham fluid flow., 25 pages

The aim of this paper is to calibrate a Eulerian-Eulerian multiphase CFD model for the discharge process of an off-road self-loading concrete mixer drum, by using the viscosity model applied to fresh concrete. Two different viscosity models for the simulated concrete were studied: the Newtonian fluid model and the Bingham fluid model. The final parameters applied for calibration were defined by means of comparisons between numerical simulations, literature search on concrete rheology, experimental tests of drum discharge process and experimental Abrams' cone tests applied to concrete. The results show that concrete, simulated as a Bingham fluid, matches with the experimental results for the discharge process. On the other hand, a good fit between numerical and experimental results is not reached with concrete simulated as a Newtonian fluid. The calibration of the material model is essential in further numerical analyses for new mixer drums and concrete machinery production design.

International audience; We consider the motion of a rigid body in a viscoplastic material. This material is modeled by the 3D Bingham equations, and the Newton laws govern the displacement of the rigid body. Our main result is the existence of a weak solution for the corresponding system. The weak formulation is an inequality (due to the plasticity of the fluid), and it involves a free boundary (due to the motion of the rigid body). We approximate it by regularizing the convex terms in the Bingham fluid and by using a penalty method to take into account the presence of the rigid body.

We present an assessment of viscosity regularization (VR) in predicting transient flows of viscoplastic fluids (VPF). Although VR has been the most popular approach for computational studies of VPF in the last decade, the understanding of its effects on the predicted flow behaviour has remained largely qualitative. To reveal the effects of VR, we present a systematic quantitative comparison of the solutions of the regularized and exact (i.e. not regularized) problems. We consider free convection of a Bingham fluid in a two-dimensional square cavity with differentially heated sidewalls. We use bi-viscosity as a representative VR model and use augmented Lagrangian (AL) method to solve the exact problem. We show that common methods of assessing the solution error can hugely overestimate the accuracy of the results of VR. In particular, complete loss of accuracy is observed when flow conditions approach the motionless state, rendering the lower computational cost and algorithmic simplicity of VR irrelevant. Furthermore, we quantify the accuracy of bi-viscosity regularization in predicting the unyielded regions. This analysis disproves by counterexample that using yield stress contours produces convergent numerical estimates of the unyielded regions.

In the present paper we analyze the dimensionless numbers that concern the flow of viscoplastic materials. The Bingham material is used to conduct the main discussion but the ideas are generalized to more complex viscoplastic models at the end of the article. Although one can explore the space of solutions with a set of dimensionless numbers where only one of them takes into account the yield stress, like the Bingham number for example, we recommend that the characteristic stress should be defined as the extra-stress intensity evaluated at a characteristic (maximum) deformation rate. Such a definition includes the yield stress in every dimensionless number that is related to viscous effects like the Reynolds number, the viscosity ratio, and the Rayleigh number. This procedure was shown to be more effective on collapsing data into master curves and to provide a fairer comparison with the Newtonian case. This happens because a more representative viscous effect is taken into account, concentrating the plastic effects into a single parameter. The plastic number, the ratio of the yield stress to the maximum stress of the domain, is shown to better capture plastic effects than the usual Bingham number. The analysis of problems where a characteristic stress, but not a characteristic velocity, is provided, indicates that a more representative characteristic velocity should be defined with respect to the driving potential for the motion, i.e., the difference between the characteristic and yield stresses. This method is in contrast to the majority of the literature, where for Bingham materials, the dimensionless numbers are maintained in the same form as the original Newtonian ones, replacing the Newtonian viscosity by the viscous parameter of the Bingham model.

Shear-induced particle migration in monodisperse and bi-disperse suspension of spherical particles in a Bingham fluid is considered. Previous models of particle migration in Newtonian suspensions are extended to account for the existence of un-yielded regions in the visco-plastic domain when forced to flow in a tube and in a concentric Couette device. In suspension with a monodisperse particle phase, it is shown that particle concentration is continuously augmented in low shear rate regions. The yield boundary is monotonically shifting, affecting the velocity profiles, and the power to maintain the flow is monotonically reduced. When the suspension size distribution is bi-modal, the migration, eventually, results in a separation of the species. Larger particles migrate to the low shear rate zones and the smaller phase is pushed away from there. The velocity profiles, yield boundary and power do not change monotonically, and several stages in this dynamics can be identified.

Starting from the basic thermodynamic principles, we derive equations of micropolar Bingham fluids allowing for the variable concentration of polar particles. On the one hand, such fluids exhibit couple stresses, a non-symmetrical stress tensor, microrotations, and microinertia. On the other hand, the fluids support a yield stress. It was proved earlier that a micropolar fluid model can reproduce the Segre–Silberberg effect. Here, we study how the yield stress impacts this effect.