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The exploration addresses the effect of variable viscosity and thermal conductivity on the peristaltic mechanism of Bingham fluid. A two-dimensional non-uniform porous channel is considered for the fluid flow, which is assumed to be inclined. The impact of heat, slip conditions, wall properties, homogeneous and heterogeneous reactions are examined. The resulting nonlinear differential equations are solved by employing the perturbation method. The solutions acquired are analyzed and sketched through graphs that show that the variable viscosity renders a critical role in regulating the velocity of the fluid in the channel's central part. The stream function has been analyzed to observe the trapping phenomenon. Further, the obtained results find its application in understanding the flow of blood in micro arteries.

An artery narrowing referred to as atherosclerosis or stenosis causes a reduction in the diameter of the artery. When blood flow through an artery consists of stenosis, the issue of solute dispersion is more challenging to solve. A mathematical model is developed to examine the unsteady solute dispersion in an overlapping stenosed artery portraying blood as Bingham fluid model. The governing of the momentum equation and the constitutive equation is solved analytically. The generalized dispersion model is imposed to solve the convective-diffusion equation and to describe the entire dispersion process. The dispersion function at steady-state decreases at the center of an artery as the stenosis height increase. A reverse behavior is shown at an unsteady-state. As the plug core radius, time and stenosis height increase, the dispersion function decreases at the center of an artery. There is a high amount of red blood cells at the center of the artery but no influences near the wall. Hence, this model is useful in transporting the drug or nutrients to the targeted stenosed region in the treatment of diseases and in understanding many physiological processes.

A crucial part in a two-dimensional discrete fracture network (DFN) model of two-phase flow in a naturally fractured rock is the updating algorithm for the fluid interface positions inside the fractures. In this study, a single fracture situated in an impermeable rock matrix is considered. Displacements of a Newtonian fluid by another Newtonian fluid, by a power-law fluid, or by a Bingham fluid are analysed. It is shown that a significant improvement in accuracy and computational efficiency in all three types of displacement can be achieved by updating the interface position with predictor–corrector scheme instead of forward Euler scheme currently used in such simulators. In order to achieve the same accuracy in the breakthrough time as with the forward Euler method, predictor–corrector can be run with a timestep that is 3 to 50 times larger. The gain in accuracy (or, equivalently, increase of the adjustable timestep) achieved with predictor–corrector is shown to outweigh the extra computational cost of this scheme. It is also shown that further increase in accuracy, e.g. by using the Runge–Kutta 4 scheme, does not pay off. In order to have control over the error in the interface positions, a viable strategy is to choose the timestep equal to a given fraction of the shortest single-fracture breakthrough time in the network. The value of this fraction can be chosen based on the required accuracy in the breakthrough time. In the case of Newtonian/Newtonian displacement, the optimal ratio of timestep to the single-fracture breakthrough time is found to depend only on the mobility ratio and the prescribed accuracy in the numerically computed breakthrough time. In case of a single interface in the fracture, numerical experimentation yields simple equations for recommended timestep in Newtonian/Newtonian displacement based on the fluid mobility ratio and the maximum allowed error in the breakthrough time.

In this paper aims, we found the fluid concentration after calculating the velocity and temperature of the fluid with a variable viscosity that depends on the fluid moving through an inclined porous channel. We examined the influences of certain parameters that are active on fluid velocity by analyzing the graphs obtained after we reached the momentum equation solution, and used the MATHEMATICA program for plot the velocity and temperature of the fluid for two types of flow (Poiseuille and Couette).

Recently, Zou et al. (2020a) 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. (2020a) 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.

In this study, we performed direct simulations of particle suspensions in a simple shear flow with a viscoplastic model (Bingham fluid) using a finite-element/fictitious-domain method, as a model system for cuttings transport with a drilling mud in the oil and gas exploration. Bulk rheology and particle interaction were investigated via several example problems including single-particle, two-interacting particle, and many-particle problems for various solid fractions in both Bingham and Newtonian fluids. We report that, at low shear rate, huge reductions occur in both the relative bulk viscosity and angular velocity of particle suspension in a Bingham fluid compared to that in a Newtonian fluid. This indicates that, although particle incorporation increases bulk viscosity in a Bingham fluid, the amount of increase appears much smaller than that with the Newtonian medium, and particle motions were restricted to some extend at a low shear rate with a yield stress fluid. Moreover, the suppression in viscosity increase with particles disappears at a high shear rate, just like Newtonian particle suspension. This phenomenon has been interpreted by the suppressed particle angular rotation in a Bingham fluid. In addition, flow fields with low shear regions, where the viscosity of a Bingham fluid is high and the fluid mobility decreased significantly, on the left and right sides of a particle also confirm this interpretation. The two-interacting particle problem reveals delayed particle interaction with increased time period in particle trajectories (drafting, kissing, tumbling and separation) in a Bingham fluid. Similar behaviors with the suppression in viscosity increase can be also observed in the many-particle problem.

Avalanche is a natural disaster in the snow-covered mountainous area in winter, which may cause great disasters to human life and property. It is also a danger for skiers and climbers. This paper presents a new physically based algorithm to simulate the dynamic avalanches under position-based dynamics framework. To realistically simulate avalanches’ dynamic characteristics, we introduce the Bingham plastic model from geodynamics to model snow flow motion in avalanches. The interaction between snow flow in the avalanche and the surrounding objects is simulated by a level set-based two-way fluid–solid coupling model. We also propose static and kinetic friction mixed model to determine the accumulated transition of the avalanche. To create an avalanche scene with more realistic details, we employ an aerodynamics-based snow drag force model to generate snow fog effect. Finally, by choosing different criterion shear rate and friction parameters, different kinds of wet and dry avalanche scenes are realistically rendered. Compared with the real photographs of avalanches, our simulated results are quite satisfactory.

In nature, many fluid-like materials exhibit a yield stress below which they behave like a solid. The Bingham model aims to describe such materials. This paper draws some mathematical considerations on the flow of a Bingham fluid in a vertical channel. The situation due to the presence of an external magnetic field and natural convection is analyzed: the external magnetic field, which is orthogonal to the walls of the channel, generates the Lorentz forces that influence the motion through the Hartmann number. The behavior of the velocity, the induced magnetic field and the thickness of the plug regions are discussed and presented graphically. We find that the velocity is a decreasing function of the Bingham and Hartmann numbers. In particular, the presence of the external magnetic field increases the thickness of the plug region. The modulus of the induced magnetic field is not monotone when the Hartmann number changes, but it is a decreasing function of the Bingham number.

The paper examined a theoretical investigation of the stimulus of mass and heat transfer on the channel's peristaltic utilization of the MHD Bingham liquid. The research study is motivated to explore blood circulation in the little vessels by taking the slip, variable thermal conductivity, and thickness of the wall features into account. The leading constitutive equations are established based on low Reynolds number and approximations for long wavelengths. The solution for the resulting nonlinear energy and momentum equations is obtained using a semi-analytic method, while the exact solution for the concentration field is obtained. Using the MATLAB software, the influences of different constraints on the interest of physiological quantities are represented graphically. One of the considerable outcomes of the current model exposes that the existence of variable fluid properties boosts the rate as well as temperature level areas. The rheological and flow properties of various biological fluids can be derived from this model as a particular case. Moreover, the formation of stuck bolus diminishes for larger values of the magnetic and velocity slip constraints.

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.

Accurate pressure drop estimation is important for drill string and bit nozzles design and optimized fluid circulations as well as identifying the drilling problems such as bit nozzle(s) washout or plugging. In this study, the Bingham Plastic model has been modified by applying a coefficient to its turbulent pressure loss calculations. This coefficient encompasses the effects of the drill pipe tool joints and other effects in estimation of pressure losses. The range of the coefficient was determined in field applications for different hole sizes and mud types. The results showed that applying a correction coefficient of 1.08–1.12 to turbulent pressure loss equations (depending on borehole size and mud type) improves the pressure loss estimation. By applying this coefficient, the estimated pressure losses are increased to compensate the under-estimation of the Bingham Plastic model. This is considered a significant contribution to accurate calculation of borehole hydraulics and in-time detection and identification of borehole problems and reduction of invisible lost time. The findings also showed that this enhanced effect is independent of the mud type. The use of this coefficient removes the necessity of using rather complex mud rheological models such as the Herschel–Bulkley model.

Present analysis is conducted for Bodewadt flow along a permeable plane surface immersed in a Bingham fluid, which represents a class of non-Newtonian fluids exhibiting yield stress. The resulting heat transfer phenomenon is analyzed in the existence of viscous dissipation. After implementing boundary layer approximations, von-Karman substitutions are employed resulting in a coupled non-linear system. For the numerical solution, a reliable numerical routine bvp4c based on collocation method, is adopted. Flow model contains a parameter Bn , measuring the relative importance of fluid yield stress. Boundary layer is markedly risen as a consequence of fluid yield stress. Viscous dissipation also remarkably influences the resulting heat transfer process. Subtle fluid dynamics quantities including normalized skin friction, disk cooling rate and volumetric flow rate are scrutinized under the influence of fluid yield stress. The results predict a deceleration in entrained flow whenever fluid exhibits yield stress.

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.

The superimposition of mechanical oscillations can lead to a remarkable modification in the friction factor (Cf) with a fixed flowrate for viscous non-Newtonian fluids in a pipe but cannot affect the flow of Newtonian types. Therefore, less driving pressure is required for fluids with a shear-thinning nature, while more for shear-thickening types. A validated computational fluid dynamics (CFD) model is employed to compare the efficacy of three modes of oscillation: longitudinal, transverse, and rotational within the laminar flow region. Rheologies considered here are power-law fluids and Bingham plastic types. At a given Reynolds number (Re), Cf is found to be a function of rheological parameters (i.e., flow behavior index, n and Bingham number, Bn) and the vibration intensity, λ. Longitudinal oscillation is the most effective, with transverse oscillation also producing comparable results. Rotational oscillation causes the least effect on the viscosity and therefore, the friction factor is even nearly unchanged while λ ≤ 20 for power-law fluids, and λ ≤ 5 for Bingham plastic rheologies. Expressions for Cf are proposed, giving a quantitative estimation on the effect of mechanical vibration on the driving pressure.

Accurate determination of hydraulic parameters such as pressure losses, equivalent circulation density (ECD), etc. plays profound roles in drilling, cementing and other well operations. Hydraulics characterization requires that all factors are considered as the neglect of any could become potential sources of errors that would be detrimental to the overall well operation. Drilling Hydraulics has been extensively treated in the literature. However, these works almost entirely rely on the assumption that the drill string lies perfectly at the center of the annulus—the so-called “concentric annulus”. In reality, concentricity is almost never achieved even when centralizers are used. This is because of high well inclination angles and different string geometries. Thus, eccentricity exists in practical oil and gas wells especially horizontal and extended reach wells (ERWs) and must be accounted for. The prevalence of drillstring (DS) eccentricity in the annulus calls for a re-evaluation of existing hydraulic models. This study evaluates the effect of drilling fluid rheology types and DS eccentricity on the entire drilling hydraulics. Three non-Newtonian fluid models were analyzed, viz: Herschel Bulkley, power law and Bingham plastic models. From the results, it was observed that while power law and Bingham plastic models gave the upper and lower hydraulic values, Herschel Bulkley fluid model gave annular pressure loss (APL) and ECD values that fall between the upper and lower values and provide a better fit to the hydraulic data than power law and Bingham plastic fluids. Furthermore, analysis of annular eccentricity reveals that APLs and ECD decrease with an increase in DS eccentricity. Pressure loss reduction of more than 50% was predicted for the fully eccentric case for Herschel Bulkley fluids. Thus, DS eccentricity must be fully considered during well planning and hydraulics designs.

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.

This framework presents heat transfer analysis for swirling flow of viscoplastic fluid bounded by a permeable rotating disk. Problem formulation is made through constitutive relations of Bingham fluid model. Viscous dissipation effects are pre-served in the mathematical model. Entropy production analysis is made which is yet to be explored for the von-Karman flow of non-Newtonian fluids. Having found the similarity equations, these have been dealt numerically for broad parameter values. The solutions are remarkably influenced by wall suction parameter and Bingham number which measures the fluid yield stress. Akin to earlier numerical results, thermal boundary-layer suppresses upon increasing wall suction velocity. Thermal penetration depth is much enhanced when fluid yield stress becomes large. Higher heat transfer rate can be accomplished by employing higher suction velocity at the disk. However, deterioration in heat transfer is anticipated as fluid yield stress enlarges. Current numerical results are in perfect line with those of an existing article in limiting sense.

We study the onset of convection within a rectangular porous cavity which is saturated with a Bingham fluid and subjected to a uniform internal heat generation. When such a cavity is saturated by a Newtonian fluid, convection takes place at all nonzero values of the Darcy–Rayleigh number, Ra. In such cases convection takes the form of two contrarotating cells with flow down the cold sidewalls when Ra first increases from zero. However, when the enclosure is saturated by a Bingham fluid, then we find that the cavity remains stagnant until the Darcy–Rayleigh number is sufficiently large that buoyancy overcomes the yield threshold. Numerical solutions are obtained using a second-order accurate finite-difference methodology where convergence is accelerated using line relaxation. The presence of the yield surfaces, which mark the boundaries of stagnant regions, is modelled by means of a regularization of the yield threshold. It is found that the critical value of Ra above which convection arises depends roughly linearly on the value of Rb, which may be described as a convective porous Bingham number. When the cavity has a sufficiently large aspect ratio, the fluid admits more than one stable steady-state solution.

Understanding flow of non-Newtonian fluids in porous media is critical to successful operation of several important processes, including polymer flooding, filtration, and food processing. In some cases, such as when a non-Newtonian fluid can be represented by the power-law model, simulation of its flow in a porous medium is a straightforward extension of that of Newtonian fluids. In other cases, such as flow of Bingham fluids, there is a minimum external threshold pressure below which there would be no macroscopic flow in the porous medium. Computing the threshold pressure is a difficult problem, however. We present a new algorithm for determining the threshold pressure for flow of a Bingham fluid through a porous medium, modeled by a pore-network (PN) model. The algorithm, the ant colony optimization (ACO), is described in detail and together with the PN model is used to determine the minimum pressure for flow of Bingham fluids in a heterogeneous porous medium, the Mt. Simon sandstone, whose PN and morphological properties were extracted from the sandstone’s image. To assess the accuracy and computational efficiency of the ACO algorithm, we also carry out the same computations with two previous methods, namely invasion percolation with memory (IPM) and the path of minimum pressure (PMP) algorithms. The IPM does not guarantee identification of the optimal flow path with the minimum threshold pressure, while the PMP algorithm provides an approximate, albeit accurate, solution of the problem. We also compare the computational complexity of the three methods. For large PNs, both the IPM and PMP are much less efficient than the ACO algorithm. Finally, we study the effect of the morphology of the pore space on the minimum threshold pressure.

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.

The current paper is focused on analyzing the flow of viscoplastic fluid in a cavity that is driven by the two walls. The Lattice Boltzmann method (LBM) is used to solve the discrete Boltzmann equation. To represent the stress-strain rate relationship of viscoplastic fluids, the Bingham Papanastasiou constitutive model is considered. Cavity flow filled with Bingham fluids is considered for validating the present LBM code. After successful validation of the code, the analysis is extended for three dissimilar wall motions-simultaneous and opposed movement of the parallel facing walls, and the simultaneous motion of non-facing walls. The flow dynamics of Bingham fluid is influenced by Reynolds and Bingham numbers which can be studied using velocity and streamline plots. Subsequently, the yielded and un-yielded zones in a cavity have been effectively tracked using the limiting condition of yield stress. Further, the effect of wall motion on the variation of those zones inside a cavity has been studied. Finally, the drag coefficient for considered wall motions is presented.

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.

In the present work, the sloshing phenomenon in a rectangular tank with non-Newtonian Bingham fluid is modeled. The numerical method used is the Incompressible Smoothed particle Hydrodynamics (ISPH...

Bio-mimicking the natural structure is one of the important aspects which have been developed with various inventions in the fields of engineering. There has been a subsequent effort to mimic the branched structure that can be seen in the leaves of trees and also in the branched structure of veins and arteries in the human body. This is done on the Lifting plate Hele-Shaw Cell (LHSC) setup. In LHSC, Bingham plastic fluid is squeezed between two parallel plates and then one of the plates is lifted up. This leads to some indefinite branched pattern commonly known as viscous fingers or micro fractals. Viscous fingering can be obtained using Newtonian fluid, Bingham plastic fluid. In this paper Bingham plastic fluid is used to study. Saffman-Taylor instability explains the formation of these viscous fingers. The main part of this paper is to control the formation of the viscous fingers and to get it in the desired shape. This can be done by providing some anisotropies on the surface of the plate in the form of pits or holes. The experiments are conducted under wide range of lifting velocities in Hele-Shaw cell. The geometrical dimensions of the viscous fingers are measured and their variation with the lifting velocity is plotted.

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.

On the basis of the laws of conservation and the principles of thermodynamics, a mathematical model of the flow of a two-phase granular fluid is proposed. One of the phases is the viscoplastic granular Bingham fluid; the other phase is a viscous Newtonian fluid. The equations for flows in the Hele–Shaw cell are analyzed asymptotically, i.e., when the flat-channel width is much less than its length. The correlations between the phase flow rates and the pressure gradient leading to equations of filtration for a two-phase granular viscoplastic fluid are constructed. The criterion is found for the initiation of motion of a granular phase in a porous medium. It is established that, depending on the shear-yield stress, such a phase does not flow if either the pressure gradient or the channel thickness is small. The phase flow rates are analyzed numerically at various input parameters such as the phase viscosities, phase resistivities, ultimate shear stress, etc. The factors slowing down the penetrating motion of the solid phase into the porous medium are revealed.

A thin fluid film formed by withdrawing a vertical plate is being studied theoretically. It is the case of a viscoplastic fluid described by the Bingham model. The boundary-value problem is formulated by the lubrication theory. The analysis presented herein required to solve numerically a system of ordinary differential equations at appropriate boundary conditions. For the viscoplastic fluid, we have predicted a film thickness over a wide range of a capillary number (10−4 to 10), with a particular emphasis on finding the free surface location at various conditions: (a) the unyielded region is much larger than the yielded one; (b) without the yielded region; and (c) the unyielded region is smaller but is compared with the yielded one. For the case of Bingham fluid and the limiting case of Newtonian fluid, the agreement was found between our theory and the experiment for the considered capillary number range.A thin fluid film formed by withdrawing a vertical plate is being studied theoretically. It is the case of a viscoplastic fluid described by the Bingham model. The boundary-value problem is formulated by the lubrication theory. The analysis presented herein required to solve numerically a system of ordinary differential equations at appropriate boundary conditions. For the viscoplastic fluid, we have predicted a film thickness over a wide range of a capillary number (10−4 to 10), with a particular emphasis on finding the free surface location at various conditions: (a) the unyielded region is much larger than the yielded one; (b) without the yielded region; and (c) the unyielded region is smaller but is compared with the yielded one. For the case of Bingham fluid and the limiting case of Newtonian fluid, the agreement was found between our theory and the experiment for the considered capillary number range.

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.

This work analyses the entrance region flow of Bingham nanofluids in cylindrical concentric annuli. In this discussion, water is used as the base fluid which is embedded with the silver(Ag) and copper(Cu) nanoparticles coalescing with Bingham fluid. The investigation has been carried out by rotating the inner cylinder, while the outer cylinder is assumed to be at rest. A finite-difference analysis is used to obtain the axial, radial, tangential velocity components and the pressure along the radial direction. With the Prandtl's boundary layer assumptions, the continuity and momentum equations are solved iteratively using a finite difference method. Computational results are obtained for various non-Newtonian flow parameters, different volume fraction parameters and geometrical considerations. This work’s main interest is to study the development of velocity profiles and pressure drop in the entrance region of the annuli. The present results are compared with the results available in the literature for various particular cases and it is found to be in good agreement.

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.

In this paper, we study the peristaltic transport of incompressible Bingham plastic fluid in a curved channel. The formulation of the problem is presented through, the regular perturbation technique for small values of is used to find the final expression of stream function. The numerical solution of pressure rise per wave length is obtained through numerical integration because its analytical solution is impossible. Also the trapping phenomenon is analyzed. The effect of the variation of the physical parameters of the problem are discussed and illustrated graphically.

During well drilling operations, unexpected gas influx may take place when the bottom hole pressure falls below the pore pressure. The conventional procedure to control the influx consists of shutting-in the well and waiting for pressure stabilization. The pressures measured at the surface after the well closure and the pit gain are used to estimate the pore pressure, the gas density, and the kick magnitude. However, the drilling fluid compressibility and rheology can undermine pore pressure transmission to the surface, which can affect the pit gain and the well closing pressure. The current work presents a transient compressible isothermal mathematical model to predict pit gain and pressure transmission along the well during gas kicks. The model is based on the balance equations of mass and momentum that are solved by the method of characteristics. Three types of drilling fluids are analyzed: Newtonian, Bingham Plastic and thixotropic. The Darcy's law is used to represent the gas influx through the rock formation. The results indicate that the usual pressure balance used to estimate the pore pressure may lead to mistaken values and also that the drilling fluid compressibility and rheology have considerable effect on the pit gain and on the gas volume that invades the well. In all cases investigated, the gas volume within the well is at least 50% higher than the pit gain. Therefore, drilling fluid compressibility cannot be neglected on the estimation of the formation fluid volume within the well.

In this paper, thermosolutal natural convection in an open porous cavity filled with Bingham fluids has been simulated numerically. Fluid flow, heat and mass transfer, and yielded/unyielded parts have been conducted for certain pertinent parameters of Rayleigh number (Ra = 104-106), Darcy number (Da = 10-2-10-6), porosity (e = 0.1-0.9), Lewis number (Le = 2.5-100), the Buoyancy ratio (N = 0.1-20), and Prandtl number (Pr = 1-100). Moreover, the Bingham number (Bn) is studied in a wide range of different studied parameters. Results indicate that the heat and mass transfer increases and the unyielded section diminishes as Rayleigh number rises. For specific Rayleigh and Darcy numbers, the increase in the Bingham number decreases the heat and mass transfer. However, the decreases in heat and mass transfer due to Bingham number becomes stronger as Darcy number drops from Da = 10-2 to 10-6. The growth of the Bingham number expands the unyielded sections in the cavity generally in different Darcy numbers and porosities. The increases in Darcy number changes the size and shape of the unyielded section while heat and mass transfer drop gradually. For fixed Rayleigh, Darcy and Bingham numbers, the unyielded region is decreased by the augmentation of the porosity. In addition, heat and mass transfer augments gradually as the porosity increases. The enhancement of buoyancy ratio increases heat and mass transfer for the studied parameter. The rise of Lewis number has a marginal effect on heat transfer, but the mass transfer rises considerably.

Modal and non-modal stability analyses are performed for Poiseuille flow of a Bingham fluid overlying an anisotropic and inhomogeneous porous layer saturated with the same fluid. In the case of modal analysis, the resultant Orr–Sommerfeld type eigenvalue problem is formulated and solved via the Chebyshev collocation method, using QZ decomposition. It is found that no unstable eigenvalues are present for the problem, indicating that the flow is linearly stable. Therefore, non-modal analysis is attempted in order to observe the short-time response. For non-modal analysis, the initial value problem is solved, and the response of the system to initial conditions is assessed. The aim is to evaluate the effects on the flow stability of porous layer parameters in terms of depth ratio (ratio of the fluid layer thickness $d$ to the porous layer thickness $d_{m}$), Bingham number, Darcy number and slip coefficient. The effects of anisotropy and inhomogeneity of the porous layer on flow transition are also investigated. In addition, the shapes of the optimal perturbations are constructed. The mechanism of transient growth is explored to comprehend the complex interplay of various factors that lead to intermediate amplifications. The present analysis is perhaps the first attempt at analysing flow stability of viscoplastic fluids over a porous medium, and would possibly lead to better and efficient designing of flow environments involving such flow.

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.

To describe the macroscopic rheological behavior of thixotropic elasto-visco-plastic (TEVP) materials, phenomena that take place in their microstructure must be accounted for. To this end, we couple the tensorial constitutive model by Saramito for EVP materials with thixotropy, extending the ideas of isotropic hardening, and with kinematic hardening (KH), to account for back stresses. We use a scalar variable that describes the level of structure at any instance and a modified Armstrong–Frederick KH equation, thus providing rules governing the dynamics of the apparent yield stress. The material viscosity, yield stress, and back stress modulus feature a nonlinear dependence on the structural parameter, enabling the model to make accurate predictions with a single structural parameter. To avoid unphysical stress evolution in both shear and extensional flows, we propose a modified back stress constitutive equation that keeps the components of the stress tensor bounded. The predictions of the new model are compared to experimental data and predictions of previously proposed TEVP models in simple rheometric flows, including steady and step-shear tests, flow reversal, intermittent step tests, small amplitude oscillatory shear (SAOS) and large amplitude oscillatory shear. In most cases, the proposed model reproduces more accurately these experimental data than the other models, highlighting its predictive capabilities. Moreover, SAOS illustrates that introducing viscoplasticity via the Saramito model necessarily reduces G″ to zero in the linear strain regime. This calls for model adjustments in the solid state. Finally, we examined the proposed model in uniaxial elongation and concluded that it is important to include this flow in the rheological characterization and modeling of such systems.To describe the macroscopic rheological behavior of thixotropic elasto-visco-plastic (TEVP) materials, phenomena that take place in their microstructure must be accounted for. To this end, we couple the tensorial constitutive model by Saramito for EVP materials with thixotropy, extending the ideas of isotropic hardening, and with kinematic hardening (KH), to account for back stresses. We use a scalar variable that describes the level of structure at any instance and a modified Armstrong–Frederick KH equation, thus providing rules governing the dynamics of the apparent yield stress. The material viscosity, yield stress, and back stress modulus feature a nonlinear dependence on the structural parameter, enabling the model to make accurate predictions with a single structural parameter. To avoid unphysical stress evolution in both shear and extensional flows, we propose a modified back stress constitutive equation that keeps the components of the stress tensor bounded. The predictions of the new model are co...

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...

The influence of a sudden change in the external magnetic field on the flow of an electrically conducting Bingham fluid in a two-dimensional channel is considered. It is demonstrated that a method of independent descriptions of the flows in plastic and viscous regions can be used for studying magnetohydrodynamic flows of the Bingham fluid. An exact equation is derived for the position of the plastic flow region boundary as a function of time and magnetic field induction. It is shown that the corresponding Cauchy problem has a unique asymptotically stable solution. Results of numerical integration for some values of parameters are presented; these result confirm the qualitative conclusions.

The aim of this experimental study is to determine the jet breakup of non-Newtonian, Bingham plastic slurries in air, notably to obtain an experimentally verified formula for the breakup length. The ash-to-water concentration varies between appr. 9% and 57% resulting in Ohnesorge-numbers between 0.0215 and 0.0476 . The Bingham plastic fluid discharge from a circular orifice and the jet exit Weber-numbers were varied between 983 and 46866 . The Hedstrom-numbers are between 172 and 235, the Reynolds-numbers were in the range of 906–7169 and the tested Froude-numbers are between 7.06 and 42.13 at standard temperature and ambient pressure. Most of our experiments fall into the second wind-induced regime. A linearly increasing correlation was found for the dimensionless breakup length as a function of We , the Reynolds-, and the Froude-number. The rheology and the particle size distribution of the (fly and bottom) ash were measured and are reported, respectively.

The present study investigates the effects of slip and heat transfer on peristaltic mechanism of Bingham fluid in an inclined tube. The sinusoidal, multi-sinusoidal, triangular, square and trapezoidal wave forms are considered. The analysis has been carried out under the assumptions of long wavelength and small Reynold's number approximations. The closed-form solutions are obtained for velocity, plug flow velocity, pressure gradient, streamlines, and temperature. The numerical integration is employed to investigate the effects of pressure rise and frictional force. The influence of relevant parameters on physiological quantities of interest is analyzed and discussed through graphs. The study reveals that velocity and thermal slip have a decreasing effect on velocity and temperature. Further, it is noticed that the volume of trapped bolus increases for increasing values of velocity slip parameter.

Effects of the orientation and apex angle on the settling velocity of conical particles in Bingham plastic fluids are numerically investigated over a Reynolds number range of 1 ≤ Re ≤ 100, Bingham number range of 0 ≤ Bn ≤ 100, and cone angle range of 20° ≤ α ≤ 150°. Governing equations (of continuity and momentum) are solved numerically using the finite element method to obtain velocity and pressure fields that are postprocessed to obtain values of the drag coefficient. Furthermore, the effect of cone orientation on the flow field is visualized and explored in terms of streamline contours and the morphology of yielded/unyielded regions. Finally, the obtained values of drag are correlated via a simple predictive expression in terms of the modified Reynolds number. The effect of the cone angle on drag is found to be moderate because the drag force acting on the lateral surface is a small fraction of the overall drag.