Analysis of Bingham fluid radial flow in smooth fractures

Solutions for radial flow of a Bingham fluid are analyzed in this paper.It aims to eliminate confusions in the literature concerning the plug flow region in different solutions for analysis and design of grouting in rock fractures.The analyses based on the force balance equation reveal that the plug flow region in Bingham radial flow is independent of the fracture radius,and is not a growth function adapted from the solution of one-dimensional(1D)slit flow according to‘similarity’.Based on the shear stress distribution,we analytically proposed that a non-uniform plug flow region cannot exist.The Bingham fluid(grout)penetration and flowrate evolution as functions of grouting time are given using the correct expression for the plug flow region.The radius-independent plug flow region and the presented flowrate evolution equation are also verified numerically.For radial flow,the relative penetration length is equal to the relative width of plug flow region,which is the same as that for 1D channel flow.Discrepancies in analytical solutions for grout penetration and flowrate evolution were also illustrated.The clarification of the plug flow region and evaluation of discrepancies in analytical solutions presented in this work could simplify modeling and design of grouting in rock engineering applications.

Discussion on “Analysis of Bingham fluid radial flow in smooth fractures”[J Rock Mech Geotech Eng 12(2020) 1112e1118]

Recently,Zou et al.(2020 a)published a theoretical analysis on the radial flow of a Bingham fluid,where they argued that the classical analysis by Dai and Bird(1981)violates the mass conservation.The present discussion aims to clarify this conflict between those two studies.It is noted that Zou et al.(2020 a)presumed the gap-wise mass flux is negligible in the mass conservation equation,while Dai and Bird(1981)did not require so in their model,and this is found to be the origin of the conflict.In fact,Dai and Bird(1981)’s model is shown to not violate the mass conservation.Therefore,those two models should be viewed as separate models derived from different perspectives.Details of the major difference between the two models are discussed.

Motion Law of a Sphere in Bingham Fluid with Thixotropy

The thixotropy properties and the motion law of a sphere in the Bingham fluid have been studied. Through observation of the settling motion of a single sphere in the Bingham fluid on the X-ray screen, it has been discovered that the mud in estuaries and along sea bay, and the hyperconcentrated flow all behave as the Bingham fl fluid with thixotropy properties as the large sediment concentration. Through derivation, the theoretical relationship between the yield stress and non-settling maximum sphere supported by the stress for the Bingham fluid has been developed, the equations for calculating the increasing yield stress and the non-settling maximum sphere diameter with the duration at rest of the slurry have been obtained. In consideration of the effect of thixotropy on fluid motion, the Navier-Stokes equation group for the Bingham thixotropy fluid has been developed. Through further study of the flow boundary condition of settling motion of ii single sphere in the Bingham thixotropy fluid, and the solving of the Navier-Stokes equation group, under the small Reynolds number, the theoretical equation of the drag force of the Bingham thixotropy fluid flowing around a sphere has been deduced. The theoretical relationship between drag coefficient and Reynolds number has been derived. By use of the experimental data of rheological test of various slurries measured with viscometer and those of single sphere motion observed on the X-ray screeen, the above equations have been verified. The equations are in good agreement with the experimental data for various slurries.

SECOND-ORDER MOMENT MODEL FOR DENSE TWO-PHASE TURBULENT FLOW OF BINGHAM FLUID WITH PARTICLES

The USM-θ model of Bingham fluid for dense two-phase turbulent flow was developed, which combines the second-order moment model for two-phase turbulence with the particle kinetic theory for the inter-particle collision. In this model, phases interaction and the extra term of Bingham fluid yield stress are taken into account. An algorithm for USM-θ model in dense two-phase flow was proposed, in which the influence of particle volume fraction is accounted for. This model was used to simulate turbulent flow of Bingham fluid single-phase and dense liquid-particle two-phase in pipe. It is shown USM-θ model has better prediction result than the five-equation model, in which the particle-particle collision is modeled by the particle kinetic theory, while the turbulence of both phase is simulated by the two-equation turbulence model. The USM-θ model was then used to simulate the dense two-phase turbulent up flow of Bingham fluid with particles. With the increasing of the yield stress, the velocities of Bingham and particle decrease near the pipe centre. Comparing the two-phase flow of Bingham-particle with that of liquid-particle, it is found the source term of yield stress has significant effect on flow.

Reply to Discussion on “Analysis of Bingham fluid radial flow in smooth fractures”

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(2 D)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.

The Analysis of Stability of Bingham Fluid Flowing Down an Inclined Plane

In this paper, the stability problem of Bingham fluids flowing down an inclinedplane is studied with respect to two dimensional disturbances. The crilical Reynolodsnumber is given in ihe case of long waves, and the effect of yield stress on stability isanalysed.

A Direct Effective Viscosity Approach for Modeling and Simulating Bingham Fluids with the Cumulant Lattice Boltzmann Method

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.

THE ANALYSIS OF STABILITY OF BINGHAM FLUID FLOWIGN DOWN AN INCLINED PLANE

In this paper the stability problem of Bingham flowing down an inclinedplane is studied with respect to two dimensional disturbances, The critical Reynolodsnumber is given in the. case. of long wayes and the effect of yield stress on stability isanalysed.

AN IMPLICIT SOLUTION OF BI-PENALTY APPROXIMATION WITH ORTHOGONALITY PROJECTION FOR THE NUMERICAL SIMULATION OF BINGHAM FLUID FLOW

An implicit algorithm of Bi-penalty approximation with orthogonality projection for the numerical simulation of Bingham fluid flow problems is proposed in this paper. A Newton fluid flow with two kinds of artificial viscosity subjected to the inequality constraint is introduced to approximate the Bingham fluid flow. This approach can effectively simulate the Bingham fluid flow with floating rigid cores or fixing rigid cores.

Heat transfer and Helmholtz-Smoluchowski velocity in Bingham fluid flow

A mathematical study is developed for the electro-osmotic flow of a nonNewtonian 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.

粘时变Bingham流体在砂性地层中的渗透扩散机理研究与应用

由于砂性地层存在大量的孔隙管道,管道孔径也并不相等,浆液的流变性除了与水灰比有关之外,还与外加剂、时间等有关,不能仅以初始粘度作为判断流变性的唯一指标,为使注浆材料扩散机理更加符合砂性地层实际扩散情况,对粘度时变性的宾汉姆流体在不均匀管组中的柱形扩散机理进行深入研究,并将此扩散机理应用至掺入工业废渣的新型注浆材料研究中,获得新型注浆材料在砂性地层中的理论扩散范围,为该注浆材料在实际工程中的应用提供参考。

Asymptotic Analysis of a Bingham Fluid in a Thin Domain with Fourier and Tresca Boundary Conditions

In this paper we prove first the existence and uniqueness results for the weak solution,to the stationary equations for Bingham fluid in a three dimensional bounded domain with Fourier and Tresca boundary condition;then we study the asymptotic analysis when one dimension of the fluid domain tend to zero.The strong convergence of the velocity is proved,a specific Reynolds limit equation and the limit of Tresca free boundary conditions are obtained.

Magnetic Bingham fluid-assisted deterministic polishing for super-smooth surfaces

The characteristics of magnetorheological fin- ishing (MRF) technique such as the behavior of magnetic particles and the magnetic field distribution have been inves- tigated. Based on the established theoretical model, material removal function and removal rate experiments involving a K9 glass mirror are designed and carried out. Further experiments are carried out to improve the surface roughness of the workpiece. The final surface roughness with an initial value of 17.58 nm reached 0.4351 nm rms after 35 min of deterministic MRF, and the AFM measurements on microstructures of the polished surface are also improved by 0.445 nm rms without obvious defects.

Numerical analysis on intermittent flow in a longitudinal section of Bingham fluid along a slope

The transition process of intermittent flow in a longitudinal section of Bingham fluid from initial distribu-tion to fully developed state was numerically investigated in this paper. The influences of slope ,J dimensionless runoff Q* and viscosity m*0on the dimensionless surge speed U* were also presented in a wide range of parameters. By one typical example, the intermittent flow possessed wave char-acteristics and showed a supercritical flow conformation for a fully developed flow. The distributions of gravity and bed drag along the flow path and the velocity distribution of flow field were also analyzed.

A Bingham-Plastic Model for Fluid Mud Transport Under Waves and Currents

Simplified equations of fluid mud motion, which is described as Bingham-Plastic model under waves and currents, are presented by order analysis. The simplified equations are non-linear ordinary differential equations which are solved by hybrid numerical-analytical technique. As the computational cost is very low, the effects of wave current parameters and fluid mud properties on the transportation velocity of the fluid mud are studied systematically. It is found that the fluid mud can move toward one direction even if the shear stress acting on the fluid mud bed is much smaller than the fluid mud yield stress under the condition of wave and current coexistence. Experiments of the fluid mud motion under current with fluctuation water surface are carried out. The fluid mud transportation velocity predicted by the presented mathematical model can roughly match that measured in experiments.

蓝宝石光学曲面柱形宽缎带磁流变抛光仿真分析及实验研究

目的针对目前光滑无损伤光学曲面蓝宝石加工成本高、效率低的问题,对加工过程中磁流变抛光缎带进行流体仿真,进而优化抛光轮表面结构。方法设计并提出3种表面结构柱形宽缎带磁流变抛光轮,介绍了磁流变抛光轮加工的基本原理,建立了磁流变抛光垫Bingham流体特性加工仿真模型,分析了3种抛光轮表面结构对工件表面磁通密度模、流场流速、流场压力分布的影响。同时对3种抛光轮的抛光效果进行了实验探究,探究了抛光轮表面结构对材料去除率和抛光后表面粗糙度的影响规律。结果仿真结果表明,抛光轮表面槽型结构具有能增强磁通密度模、增大流体流速和流体压力的特性。实验结果表明,螺旋槽抛光轮的抛光效果最好,在螺旋抛光轮作用下,材料去除率为0.22mg/h,抛光后蓝宝石表面粗糙度为1.08nm。最终抛光轮近壁区总压力和速度的乘积结果与抛光轮实验去除率结果具有较好的一致性。结论槽型结构可以提高抛光液在抛光轮表面的固着效果,影响工件表面流场运动状态,增强工件表面受到抛光垫的作用力。相较于光滑和横条槽抛光轮,螺旋槽抛光轮的抛光效率最高,表面粗糙度最低,可有效提高抛光效果。

考虑多孔介质渗透率变化的宾汉流体渗透注浆扩散模型

渗透注浆是一种广泛应用于岩土多孔介质加固的方法,但已有基于浆液流变方程和平衡方程的研究,侧重分析浆液本身性质如黏度时变性、滤过效应、自身重力等对渗透注浆扩散范围的影响,尚未考虑浆液在孔隙内的流动阻力对多孔介质渗透率的影响,导致现有理论解析结果与试验测试结果相差较大。因此,基于黏性流体的运动方程、连续方程和平衡方程,引入浆液在孔隙内的流动阻力对多孔介质渗透率的影响,推导获得多孔介质渗透率随介质孔隙特性、注浆施工参数和注浆扩散半径的函数解析表达式,据此建立考虑多孔介质渗透率变化的浆液扩散控制方程。与已有理论结果对比表明,考虑浆液流动阻力影响的理论结果更接近试验结果,可以验证该理论方法的有效性。参数分析结果表明:考虑浆液在孔隙内的流动阻力时,岩土多孔介质渗透率随注浆时间的增加呈现先迅速降低后缓慢上升趋势,随注浆压力的增大呈现逐渐下降趋势;对于工程中常见的符合宾汉流体性质的水泥浆液材料,当注浆时间小于5 min时,多孔介质渗透率变化是影响浆液扩散的主控因素,超过7 min后,浆液黏度时变性成为主控因素。研究成果能为岩土多孔介质渗透注浆工程实际中注浆参数选取和质量控制起到良好的理论指导意义。

磁流变阻尼器的分数阶Bingham模型研究

为解决磁流变阻尼器的Bingham模型在拟合阻尼力-位移关系时"具有较高精度,但是无法准确描述阻尼力-速度的滞后"的问题,基于传统Bingham模型进行了改进,应用分数阶微分形式的Bingham模型,更加准确地拟合了速度-阻尼力的滞回特性。在改进的模型中以分数阶导数代替了传统模型中位移的一阶导数,既能够描述磁流变液屈服后的粘性剪切流动,又包含磁流变液体在低剪切速率下未屈服时的弹性变形。对一种磁流变阻尼器进行了实验,比较传统Bingham模型和改进分数阶模型阻尼力-位移和阻尼力-速度拟合的精度,分析分数阶微分项阶数与控制电流及阻尼力-速度滞后特性的关系。实验结果表明,分数阶微分形式的Bingham模型拟合磁流变阻尼器力-位移和力-速度关系,其精度均较传统Bingham模型有明显提高。

Bingham体的连续流和阵性流研究中无量纲参数讨论

本文首先对轴对称管流、二维管流和明渠连续流研究中无量纲参数的选择进行评述 ,通过对这些有分析解的流动的讨论 ,说明采用无量纲表达式的优点、无量纲方法的多样性 ,以及适当选择基本量的重要性。

盾构克泥效浆液成分与渗透扩散研究

盾构隧道近接既有结构施工中常采用克泥效工法辅助控制地层位移。克泥效浆液的流变和渗透扩散性能对盾壳外环向间隙的填充效果具有重要影响。基于克泥效浆液X射线衍射试验(X-ray diffraction, XRD)和黏度试验,研究浆液微观成分及流变性能。基于多孔介质多相流控制方程,建立砂土中克泥效浆液扩散数值模型,分析浆液配比、地层渗透系数和注浆压力对扩散性能的影响。结合宾汉姆流体砂土渗流公式,验证数值模型的可靠性。结果表明:克泥效浆液成分主要为石英、斜长石、伊利石、正长石,且随A液水粉比增加,斜长石含量增加,伊利石含量减小。克泥效浆液的塑性黏度与屈服强度随A液水粉比的减小而增大,宾汉姆模型对浆液流变曲线的拟合优度大于0.95。克泥效浆液在环向间隙中的填充扩散阶段约为25s,而后再向地层中发生渗透扩散,渗透扩散速度远小于填充扩散。随A液水粉比、地层渗透系数、注浆压力增加,克泥效在地层渗透距离均会增加。