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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.

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).

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.

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.

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.

In this paper, experimental experiments were conducted using advanced computerized rheometer. The lean solid slurry (10 to 40 percent by weight) requires an immense amount of water and requires high-energy pumping power. The rheological properties are highly dependent on shear strength, solid density and particle size distribution. The plastic viscosity and yield stress enhanced with a solid concentration. At higher concentrations of solids, the experimental results fitted well with the Bingham plastic model. The results reveal that the plastic viscosity and yield stress of the slurry samples are increased due to finer particles. Bingham plastic viscosity depends primarily on solid concentration and particle size.

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.

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.

This study examined the magnetorheological (MR) behavior of magnetite-based MR fluids according to the magnetite concentration, magnetic field strength, and different viscosity of silicone oil as the carrier fluid. To this end, an experimental design was chosen to decrease the experimental error and provide a better rheological interpretation. The flow behavior parameters increased remarkably with increasing medium viscosity and nanoparticles (NPs) concentration, whereas the magnetic field strength had a milder effect, exhibiting a saturation value above which its effect became almost negligible. Moreover, both the Bingham plastic and power law models were found to fit their flow behaviors well. Almost all the critical rheological parameters of the aforementioned models experienced remarkable improvement after increasing the magnetic field strength, NPs concentration and carried fluid viscosity. In addition, mathematical correlations were derived to model each of the rheological parameters such as plastic viscosity and yield stress (for Bingham fluid model) and the consistency and power-law indices (for power-law model).

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.

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...

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.

Considering a mathematical model for the bilateral, frictionless contact between two Bingham fluids, establish a variational formulation for the problem and prove estimates on the velocity and pressure which are independent of the small parameter. The passage to the limit on \(\varepsilon \) permits us to obtain the existence and uniqueness of the velocity. A specific Reynolds equation associated with variational inequalities is obtained.

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.

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.

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 study aims to analyze mixed convection in a square cavity with two moving vertical walls by finite volume method. The cavity filled with Non-Newtonian fluid of Bingham model is heated from below and cooled by the other walls. This study has been conducted for certain parameters of Reynolds number (Re=1- 100), Richardson number (Ri=1-20), Prandtl number (Pr=1-500), and Bingham number has been studied from 0 to 10. The results indicate that the increase in yield stress drops the heat transfer and the flow become flatter, while increasing Reynolds number augments it. The convective transport is dominant when increasing Richardson number which leads to enhanced heat transfer in the cavity for both Newtonian and Non-Newtonian fluid. A correlation of Nusselt number is given in function of different parameters.

In this paper, we describe our study of the mixed convection of a Boussinesquian Bingham fluid in a vertical channel in the absence and presence of an external uniform magnetic field normal to the walls. The velocity, the induced magnetic field, and the temperature are analytically obtained. A detailed analysis is conducted to determine the plug regions in relation to the values of the Bingham number, the buoyancy parameter, and the Hartmann number. In particular, the velocity decreases as the Bingham number increases. Detailed considerations are drawn for the occurrence of the reverse flow phenomenon. Moreover, a selected set of diagrams illustrating the influence of various parameters involved in the problem is presented and discussed.

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.

The present analysis emphasizes the effects of variable properties on Bingham fluid under MHD peristaltic transport. Due to the impact of mechanical forces on the applied magnetic field on the conducting fluid, the fluid stream gets altered. These principle targets drug transport and control of blood flow during surgeries; hence the impact of MHD flow with convective and porous boundary conditions is considered. Further, the implications of homogeneous and heterogeneous reactions are analyzed by considering wall properties. The governing equations are turned dimensionless by appropriate similarity transformations. The series solution is obtained for temperature, velocity, and concentration by perturbation method with lubrication approach. The graphical representation of the pertinent parameters on the physiological flow quantities is depicted by applying for MATLAB 2019b program. The obtained results reveal that the rise in the magnetic parameter diminishes the velocity and temperature profiles. Further, the impact of variable viscosity slightly improves the magnitude of the trapped bolus. The homogenous and heterogeneous reaction parameters have a converse effect on the concentration distribution. Moreover, the present investigation finds its applications to perceive the complex rheological functioning of blood flow through narrow arteries.

This study presents the numerical solution of velocity and temperature fields based on mass conservation, momentum and energy balances for the time-dependent Couette-Poiseuille flow of Bingham materials through channels. The channel flow of Bingham fluid concerns the flow of cement paste in the building industry and the mudflow in the drilling industry. The specific aim is to introduce the magnetohydrodynamic (MHD) phenomena specified by both Ion-slip and Hall currents into the non-isothermal channel flow in a theoretical approach. The Bingham constitutive equation is formulated by the generalized Newtonian fluid technique and solved by employing the explicit Finite Difference Method (FDM) using the MATLAB R2015a and Compaq Visual FORTRAN 6.6a both. For the exactness of numerical performance estimations, the criteria for stabilization and the convergence factor are analyzed. The velocity and temperature profiles are discussed individually at the moving and stationary walls of the channel. It is observed that magnetohydrodynamic phenomena accelerate the flow, and the temperature distributions reach the steady-state situation earlier than velocity distributions. Furthermore, the dominance of MHD parameters on the velocity distributions, shear stress, temperature distributions, and Nusselt number are discussed.

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.

It is of great significance for the improvement of grouting technology and engineering practice to master the flow law of grout between parallel plates. However, the traditional calculation model ignores the influence of the inertia term and only considers the stable flow of slurry, so there is a big error in some cases. It is difficult to solve the motion equation of a Bingham fluid considering the inertial force term directly. Combined with the relationship between the steady-state flow equation of a Bingham fluid and a Newtonian fluid, the approximate unsteady-state flow equation of a Bingham fluid suitable for describing slurry flow is constructed. In addition, according to the unsteady flow equation, the relationship between the time and distance of slurry flow in parallel plate fractures can be obtained, and the simplified conditions of the Bingham fluid unsteady flow model are given. Finally, the accuracy of the flow equations and the simplified conditions are verified by experiments and numerical calculations.

Laminar Bingham fluid flow in the entrance region of a circular pipe is investigated using a momentum integral model. The fully developed flow is uniform in the core region, while the velocity changes in the annular part of the cross section of the pipe. The inlet-filled region concept is adopted. In the inlet region, the boundary layer thickness increases until the size of the plug flow area reaches the fully developed flow size. The model converges to the fully developed solution in the filled region. The model provides the velocity, pressure drop, and skin friction coefficient profiles. The pressure drop results are in good agreement with published experimental data. The flow results asymptotically converge to the fully developed values. In addition, the results are consistent with published Newtonian fluid flow experimental data and theoretical results for the boundary layer thickness, pressure drop, and centerline velocity for small values of the Bingham number.

The study of the entrance region flow is, nowaday, assuming considerable technical importance due to its immediate applications in various designs of chemical and biomedical equipments in which the flows of non-Newtonian fluids are encountered. The flow characteristic of non-Newtonian fluid that is independent of time, and which obeys Bingham’s stress–strain relations, has been investigated at the entrance region of the annular space between two rotating coaxial cylinders. This analysis has been carried out for both the cases when each rotates with different speed in the same direction and as well as in the opposite direction with constant angular velocity. Discussions have been presented for a steady, laminar, isothermal flow condition of the Bingham fluid. The continuity and momentum equations are solved iteratively with finite difference method by using the Prandtl’s boundary layer assumptions. The velocity components and pressure distributions have been obtained numerically for various values of Bingham number and aspect ratio.

The effects of inertia on the squeeze flow of a Bingham fluid between two approaching parallel plates with a constant squeeze velocity is investigated using matched asymptotic expansions. This analysis is an extension to the prior study of Muravleva (2015), who has investigated the planar squeeze flow of a Bingham fluid in the absence of inertia. In the present study, the expressions for the shear stress field, velocity, pressure field and squeeze force are derived. The combined effects of the fluid inertia and yield stress on the pressure field and squeeze force are investigated. We found that the pressure and eventually squeeze force increases with increase in Reynolds number. The squeeze force decreases with an increase in the value of the gap aspect ratio.

20 pages, 12 ref., 9 fig.; A lubrication equation is obtained for a simplified shear-thinning fluid. The simplified rheology consists of a piecewise linear stress tensor, resulting in a two-viscosity model. This can be interpreted as a modified Bingham fluid, which can be recovered in a specific limit. The lubrication equation is obtained in two steps. First two scalings are performed on the incompressible Navier-Stokes equations, namely the long-wave scaling and the slow motion scaling. Second, the resulting equations are averaged along the vertical direction. Numerical illustations are provided, bringing to light the different possible behaviours.

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

The rheological behavior of many lubricants used in hydrodynamic bearings can reasonably be modeled using the Bingham plastic material model. This behavior is characterized by a strong discontinuity, from a pure solid state to a viscous fluid state depending on the local shear stress. In literature three methods have been presented to model a Bingham plastic lubricated film. A full CFD and thus expensive, numerical simulation has been used. A general Reynolds equation based simulation has been used, however with a less accurate numerical regularization of the material discontinuity. Or a general Reynolds equation based simulation has been used, but with a severe reduction of the geometric complexity. In this paper, an ’exact’ thin film lubrication simulation for a Bingham plastic fluid is presented. The model is said to be exact in the sense that it requires no additional approximations to those used in the derivation of the general Reynolds equation, and requires no numerical regularization of the Bingham plastic fluid model and can still be used on any lubricating film geometry. Simulations on both infinite and finite journal bearings shows that the results of this new method are in good accordance with literature, demonstrating the validity of the method.

We study the stability of laminar Bingham&ndash, Poiseuille flows in a sheet of fluid (open channel) down an incline with constant slope angle &beta, &isin, (0,&pi, /2). This problem has geophysical applications to the evolution of landslides. In this article, we apply to this problem recent results of Falsaperla et al. for laminar Couette and Poiseuille flows of Newtonian fluids in inclined channels. The stability of the basic motion of the generalised Navier&ndash, Stokes system for a Bingham fluid in a horizontal channel against linear perturbations has been studied. In this article, we study the flows of a Bingham fluid when the channel is oblique and we prove a stabilizing effect of the Bingham parameter B. We also study the stability of the linear system with an energy method (Lyapunov functions) and prove that the streamwise perturbations are always stable, while the spanwise perturbations are energy-stable if the Reynolds number Re is less than the critical Reynolds number Rc obtained solving a generalised Orr equation of a maximum variational problem.

We consider two unsteady free convection flows of a Bingham fluid when it saturates a porous medium contained within a vertical circular cylinder. The cylinder is initially at a uniform temperature, and such flows are then induced by suddenly applying either a new constant temperature or a nonzero heat flux to the exterior surface. As time progresses, heat conducts inwards and this may or may not overcome the yield threshold for flow. For the constant temperature case, flow begins immediately should the parameter, Rb, which is a nondimensional yield parameter, be sufficiently large. The ultimate fate, though, is full immobility as the cylinder eventually tends towards a new constant temperature. For the constant heat flux case, the fluid remains immobile but will begin to flow eventually should Rb be sufficiently large. The two cases have different critical values for Rb.

In order to investigate hydrodynamic characteristics of Bingham porous-flow mechanics in a heavy oil reservoir, this paper originally proposes a physical model of Bingham fluid flow for a horizontal well in a heavy oil reservoir. A corresponding mathematical model is established and solved. A series of typical pressure and pressure derivative curves are drawn by the use of the semi-analytical solution of the mathematical model. Hydrodynamic flow characteristics of Bingham fluid are thoroughly analyzed. Through the recognition of flow regimes from the typical curves, it is found that the flow process of Bingham fluid mainly includes the early radial flow regime, linear flow regime and late pseudo-radial flow regime. The curves controlled by different boundary conditions are especially studied, and it is found that the late curve shape, especially the shape of pressure derivative curve, differs completely for the various property of boundary. The curve sensitivity to different model parameters, such as dimensionless threshold pressure gradient and horizontal wellbore length, is deeply discussed. The findings are as follow: except the pure wellbore storage regime, the threshold pressure gradient influences the hydrodynamic flow behavior in all the regimes; the curve of Bingham fluid governed by the threshold pressure gradient upward deviates from that of Newtonian fluid; and the deviation of the curves increases with the increase of the threshold pressure gradient and the elapsed time. The research results of this paper sufficiently demonstrate the essential property of non-Newtonian porous-flow mechanics.