Unyielded Regions Research Articles

A revisit of the development of viscoplastic flow in pipes and channels

This study revisits the development of viscoplastic flow in pipes and channels, focusing on the flow of a Bingham plastic. Using finite element simulations and the Papanastasiou regularisation, results are obtained across a range of Reynolds and Bingham numbers. The novel contributions of this work include: (a) investigating a definition of the development length based on wall shear stress, a critical parameter in numerous applications; (b) considering alternative definitions of the Reynolds number in an effort to collapse the development length curves onto a single master curve, independent of the Bingham number; (c) examining the patterns of yielded and unyielded regions within the flow domain; and (d) assessing the impact of the regularisation parameter on the accuracy of the results. The findings enhance the existing literature, providing a more comprehensive understanding of this classic flow problem.

Insight into the electroosmotic vortex modulated reaction characteristics of viscoplastic fluids

Using positively charged patches embedded in the walls of a microreactor, we generated electroosmotic vortices to analyze chemical reactions involving the flow of viscoplastic species. Reactant species A and B undergo a reaction to produce species C, which possesses physical properties suitable for biomedical applications. We developed a modeling framework, extensively validated with the available experimental results as well, to solve relevant transport equations considering pertinent boundary conditions. By varying parameters, such as the Bingham number, diffusive Peclet number, relative concentration of species B, flow-behavior index, and Damkohler number within physically justified ranges, we examined the flow field, species concentration, average product concentration, and generated species flow rate. Our findings indicate that the liquid yield stress and shear-thinning nature strongly influence vortex strength and the structure of yielded and unyielded regions. Notably, electroosmotic vortices enhance product species concentration compared to cases without vortices across the chosen range of diffusive Peclet numbers, providing convective mixing strength for reactants. For lower Bingham number values, product concentration trends increase then decrease with increasing Peclet numbers, whereas for higher Bingham numbers, it exhibits a monotonic decrease. Additionally, lower Bingham numbers lead to increased average product concentration as flow-behavior index decreases, while higher Bingham numbers show the opposite trend. Furthermore, average product concentration increases up to critical Damkohler number values for smaller Bingham numbers but becomes insensitive to Damkohler number changes with greater Bingham numbers. These insights of our analysis pave the way for designing innovative, highly effective microreactors largely used for biochemical and biomedical applications.

Sedimentation of a spheroidal particle in an elastoviscoplastic fluid

The sedimentation dynamics of a prolate spheroidal particle in an unbounded elastoviscoplastic (EVP) fluid is studied by direct finite element simulations under inertialess flow conditions. The Saramito–Giesekus constitutive equation is employed to model the suspending liquid. The arbitrary Lagrangian–Eulerian formulation is used to handle the particle motion. The sedimentation, lift, and angular velocities of spheroids with aspect ratio between 1 and 8 are computed as the initial orientation, Bingham, and Weissenberg numbers are varied. Similar to the purely viscoelastic case, a spheroid in an EVP fluid rotates up to align its major axis with the applied force. As the Bingham number increases, the settling rate monotonically reduces, while the angular velocity first increases and then decreases. The initial orientation has a relevant effect on the particle stoppage because of the different drag experienced by the spheroid as its orientation is varied. The yielded and unyielded regions around the spheroid reveal that, for particle oriented transversely to the force, the yielded envelope shrinks near the tips due to the fast spatial decay of the stresses, and unyielded regions appear along the surface of the particle, similar to the solid caps observed at the front and back of a sphere. Fluid plasticity enhances the negative wake phenomenon that is observed at Weissenberg numbers significantly lower than the purely viscoelastic case. The results of the drag correction coefficient for particles aligned with longest axis along the force are presented.

Elasto-visco-plastic flows in benchmark geometries: I. 4 to 1 planar contraction

We present predictions for the flow of elastoviscoplastic (EVP) fluids in the 4 to 1 planar contraction geometry. The Saramito-Herschel-Bulkley fluid model is solved via the finite-volume method with the OpenFOAM software. Both the constitutive model and the solution method require using transient simulations. In this benchmark geometry, whereas viscoelastic fluids may exhibit two vortices, referred to as lip and corner vortices, we find that EVP materials are unyielded in the concave corners. They are also unyielded along the mid-plane of both channels, but not around the contraction area where all stress components are larger. The unyielded areas using this EVP model are qualitatively similar to those using the standard viscoplastic models, when the Bingham or the Weissenberg numbers are lower than critical values, and then, a steady state is reached. When these two dimensionless numbers increase while they remain below the respective critical values, which are interdependent, (a) the unyielded regions expand and shift in the flow direction, and (b) the maximum velocity increases at the entrance of the contraction. Increasing material elasticity collaborates with increasing the yield stress, which expands the unyielded areas, because it deforms the material more prior to yielding compared to stiffer materials. Above the critical Weissenberg number, transient variations appear for longer times in all variables, including the yield surface, instead of a monotonic approach to the steady state. They may lead to oscillations which are damped or of constant amplitude or approach a flow with rather smooth path lines but complex stress field without a plane of symmetry, under creeping conditions. These patterns arise near the entrance of the narrow channel, where the curvature of the path lines is highest and its coupling with the increased elasticity triggers a purely elastic instability. Similarly, a critical value of the yield stress exists above which such phenomena are predicted.

Interplay between wall slip and shear banding in a thixotropic yield stress fluid.

We study the local dynamics of a thixotropic yield stress fluid that shows a pronounced non-monotonic flow curve. This mechanically unstable behavior is generally not observable from standard rheometry tests, resulting in a stress plateau that stems from the coexistence of a flowing band with an unyielded region below a critical shear rate c. Combining ultrasound velocimetry with standard rheometry, we discover an original shear-banding scenario in the decreasing branch of the flow curve of model paraffin gels, in which the velocity profile of the flowing band is set by the applied shear rate  instead of c. As a consequence, the material slips at the walls with a velocity that shows a non-trivial dependence on the applied shear rate. To capture our observations, we propose a differential version of the so-called lever rule, describing the extent of the flowing band and the evolution of wall slip with shear rate. This phenomenological model holds down to very low shear rates, at which the dimension of the flowing band becomes comparable to the size of the individual wax particles that constitute the gel microstructure, leading to cooperative effects. Our approach provides a framework where constraints imposed in the classical shear-banding scenario can be relaxed, with wall slip acting as an additional degree of freedom.

Low yield stress measurements with a microfluidic rheometer.

Yield stress, τy, is a key rheological property of complex materials such as gels, dense suspensions, and dense emulsions. While there is a range of established techniques to measure τy in the order of tens to thousands of pascals, the measurement of low τy, specifically below 1 Pa, remains underexplored. In this article, we present the measurement of low apparent τy using a Hele-Shaw microfluidic extensional flow device (MEFD). Using the MEFD, we observe a gradient in shear stress, τ, such that τ is lower near the center or stagnation point, and higher away from the stagnation point. For a yield stress fluid, we observe that, below a certain flow rate, τ exceeds τy only in the outer region, leading to stagnation or unyielding of the fluid in the inner region. We use scaling analysis based on a Hele-Shaw linear extensional flow to deduce τy by measuring the size of the unyielded region, S. We validate this scaling relationship using Carbopol solutions with concentrations ranging between 0.015 to 0.3%, measuring τy as low as ∼10 mPa to ∼1 Pa, and comparing it with τy measured using a standard rheometer. While the experimental lower limit of our technique is 5 mPa, modifying the geometry or improving the image analysis can reduce this limit to the order of 10-4 Pa. The MEFD facilitates rapid measurement of τy, allowing for its real-time assessment. We further report τy of human blood samples between 30 to 80 mPa with their hematocrit ranging between 14 to 63%. Additionally, we determine τy for a mucus simulant (∼0.7 Pa), and lactic drink (∼7 mPa) to demonstrate the versatility of the MEFD technique.

Predicting yielding in the pressure-driven flow of a Bingham fluid in a serpentine channel

The use of foams in applications such as enhanced oil recovery and varicose vein sclerotherapy has led to renewed interest in their flow as yield stress fluids. In each application one seeks to determine the ability of a foam to displace another fluid from a tortuous channel. We simulate the pressure-driven flow of a Bingham fluid through a serpentine two-dimensional channel to predict the effects of varying the channel geometry on the size and location of the unyielded regions in which the fluid is either stationary or moving as a plug. We discuss the implications for the effective displacement of blood in varicose vein sclerotherapy.

Yield stress prediction from 3D reconstruction of fresh concrete slump

Relating the empirical slump test to yield stress facilitates the capture of fresh state concrete performance in fundamental quantitative terms whilst avoiding the requirement for expensive rheological equipment. This paper proposes a novel method to predict yield stress from slump by directly measuring the height of the unyielded region, via a 3D reconstruction approach. The efficacy of the proposed method is assessed through an experimental series of 21 tests. The results indicate a better correlation between the height of the unyielded region and yield stress compared to total slump height and yield stress. The proposed 3D reconstruction methodology predicts yield stress with significantly increased accuracy, indicated by a mean prediction error an order of magnitude lower than the average performance of existing models. The results of this study suggest that, for the first time, a valuable fundamental rheological property can be confidently derived from a standard concrete slump test.

The analytical solution for the flow of a Papanastasiou fluid in ducts with variable geometry

We solve the isothermal, steady and creeping flow of the regularized Papanastasiou model in ducts, such as symmetric channels and axisymmetric pipes, with varying rigid walls. The classic lubrication approximation is invoked to simplify the governing equations and to produce an exact analytical solution for the flow field with the aid of the Lambert W function. For specific suitable values of the parameters of the Papanastasiou model, the solution reduces to the corresponding exact analytical solutions for the ideal yield-stress Bingham and Newtonian fluids. The analytical solutions derived here are used to calculate the average pressure drop required to maintain the constant volumetric flowrate through the duct. Results are provided for symmetric undulating or linearly varying channels with the emphasis put on the effects of the Bingham and regularization numbers, and the amplitude of the undulation or the wall variation, respectively. For Bn→0 the yield surface approaches the axis of symmetry, σ(z)→0, i.e., no unyielded region appears in the flow. In this case, the last term in the formula for v is well-defined at y=0 provided that the numerator is zero. This implies that p′=c/h4 where c is constant. With the aid of the total mass balance, we find that c=−16, and thus p′→−16/h4 as Bn→0. Thus, Eqs. (34) and (37) reduce to the corresponding formulas for a Newtonian fluid given in Eq. (25). Noting that f(y,z)>0, and substituting the solution for γ˙ into inequality (52) gives the trivial W(f(y,z))≥0 which holds for any value of y and z because of the properties of the Lambert function. Eq. (53) is used in conjunction with Eq. (8) to find the unknown pressure gradient, p′(z). First, we integrate by parts Eq. (8) and apply the no-slip condition at the upper wall to find ∫0h(z)y2γ˙dy=2. Then, by substituting Eq. (53) into the latter, and performing the integration with respect to y from the axis of symmetry (y=0) to the wall (y=h(z)), gives: Using properties of the Lambert function and denoting with w=w(z) the value of W function at f(y=h(z),z), i.e.: we find the integral I(z) in Eq. (54) in terms of W and w. Then, we set Φ(z)≡−p′(z)h(z)/2>0, rearrange the resulting equation and upon simplification, we derive the strongly non-linear algebraic equation for Φ: The axial velocity component, u=u(y,z), can be found by integrating Eq. (53) with respect to y between y and h(z) and applying the no-slip condition at the wall:

Heat transfer in the Jeffery-Hamel flow of a yield-stress fluid

Heat transfer in the Jeffery-Hamel flow of a yield-stress fluid that obeys the Bingham-Papanastasiou model is investigated in the present study. Among various rheological models proposed to formulate the flow of yield-stress fluids, the Bingham model is usually used to address the effect of yield-stress. A fluid that obeys the Bingham model does not show any shear-thinning or shear-thickening behaviors; this makes the interpretation of the results much easier. Addressing the unyielded regions (regions at which the yield-stress material behaves like a solid) in a yield-stress fluid flow is not a trivial task, and it could be impossible, especially in complex geometries. Unfortunately, the Bingham model itself cannot be used to formulate complex fluid flow problems due to the mathematical difficulties (i.e., addressing the unyielded regions); hence it is usually regularized to avoid these mathematical challenges. One of these regularized models is the Bingham-Papanastasiou model. Using an exponential term enables the Bingham-Papanastasiou model to address the unyielded regions. However, this exponential term results in highly non-linear equations, which are difficult to be handled mathematically. In the present study, the effect of the yield-stress (Bingham number) on the hydrodynamics and heat transfer in the Jeffery-Hamel flow is investigated by using the Bingham-Papanastasiou model. The momentum transfer in the Jeffery-Hamel flow of a Bingham-Papanastasiou fluid is governed by a highly non-linear ordinary differential equation (ODE) which is a boundary value problem. Assuming that the fluid's properties are independent of the temperature and pressure, this highly non-linear ODE is solved numerically. The heat transfer in the flow, with a constant wall temperature boundary condition, is investigated, assuming the flow to be an equilibrium viscous dissipation flow (the fluid's mixed mean temperature variation is negligible along the channel). The resultant linear energy equation is also solved numerically to obtain the temperature distribution across the channel. The results obtained in this work show that the yield-stress increases the wall shear stress, Nusselt number, and Stanton number. In fact, the effect of the yield-stress on the Nusselt number is comparable to the effects of Prandtl and Eckert numbers. It is also found that the yield-stress can prevent the flow separation in a divergent channel. Considering the advantages of using the yield-stress fluids as the working fluids in the Jeffery-Hamel flow, they can be considered as an alternative to nanoparticles which are usually used to enhance the fluid's thermal conductivity and also the Nusselt number.

Numerical study of steady and periodic mixed convection of Bingham fluid in a narrow annulus with the exact Bingham model

This article numerically investigates the mixed convection of Bingham fluid in a narrow annulus. The temperature of the inner fixed cylinder ( θ i ) is higher than that of the outer one ( θ o ), which rotates at a constant angular velocity. An OpenFOAM solver, icoFoam, which works based on the finite volume method and the PISO algorithm, is modified to perform the simulations. The modified solver can solve unsteady heat transfer problems for Bingham fluids without using regularization methods. After validating the solver, it solves the problem in different conditions to show the effect of relevant parameters, such as Reynolds number, Bingham number, and aspect ratio, on the flow field and thermal performance of the enclosure for constant amounts of Grashof and Prandtl numbers. The results are presented as streamlines and isotherms, the size and shape of the unyielded regions, the local and mean Nusselt number, and the torque coefficient. The results demonstrate that the flow is steady at Re = 10 for all Bingham numbers, and a decrease in the Reynolds number may result in an unsteady flow depending on the value of the Bingham number. The flow’s time history reveals that at a fixed Reynolds number, the flow shifts from periodic to steady-state by enhancing the Bingham number so that the flow becomes entirely steady for a given Bingham number. An interesting phenomenon is observed, the formation, movement, and disappearance of the plug zones (unyielded regions) for the periodic flow at Re = 1 and Bn = 5 .

Viscoplastic flow between hinged plates

The incompressible motion of viscoplastic fluid between two semi-infinite rigid plates, hinged at their ends and rotating towards one another at constant angular velocity, generates self-similar flow fields because there is no externally imposed length scale in the absence of inertia. The magnitude of the strain rate scales with the angular velocity of the plates and the dimensionless deviatoric stresses are functions only of the polar angle and a dimensionless measure of the yield stress; they are independent of the radial distance from the corner. These flows feature unyielded regions adjacent to the boundaries for sufficiently large angles between the plates. Moreover, when the dimensionless yield stress is large, there are viscoplastic boundary layers that are attached to the boundary or the plug, the asymptotic structures of which are constructed.

Double-diffusive natural convection of Casson fluids in an enclosure

This study investigates the double-diffusive natural convection of the non-Newtonian Casson fluid in a square cavity based on the original viscoplastic stress model without simplification. Therefore, yield stress plays an essential role in understanding fluid behavior. The finite element approach provided a numerical solution to continuity, momentum, energy, and species governing equations. The governing parameters for this problem are Rayleigh number, Ra, yield number, Y, buoyancy ratio number, Nr, and Lewis number, Le. The influence of these parameters on heat and mass transfer, the morphology of yielded/unyielded regions, and fluid flow are thoroughly examined.The results show that unyielded regions increase at high Rayleigh numbers, despite the increase in buoyancy force and consequently increased heat and mass transfer. On the other hand, as the buoyancy ratio drops, the flow's strength and heat and mass transmission diminish, leading to an increase in plug regions. Accordingly, the mechanisms influencing the growth of unyielded regions are complex and follow different patterns. However, the plug regions always grow with increasing Y. The results indicate that increasing the Lewis number (mass transfer) reduces the effect of the buoyancy ratio on flow, heat transfer, and the unyielded regions in every case. Quantitative analysis of the results indicates that, while buoyancy ratio affects heat and mass transfer almost equally, the Lewis number increases mass transfer up to three times the heat transfer. Meanwhile, changing the buoyancy ratio can increase the maximum yield stress to 400%, while changing the Lewis number has a maximum effect of 20%.

Numerical modeling of an elasto-viscoplastic fluid around a plate perpendicular to the flow direction

• A new EVP constitutive equation coupling the bingham and maxwell models is presented. • Computations have been performed using the FEMLIP method. • Elasticity and plasticity have an influence on flow morphology. • The fore-aft asymmetry is due to elasticity. • Elastic effects decrease as plastic effects become stronger. This numerical study investigates the creeping flow of an elasto-viscoplastic fluid around a plate perpendicular to the flow direction. The analysis is performed with the Finite Element Method with Lagrangian Integration Points (FEMLIP). This numerical method is able to simulate large deformations problems and to describe yielded and unyielded regions by precisely accounting for elastic stresses. An elasto-viscoplastic model coupling the Bingham and the Maxwell constitutive equations is used. The influences of plasticity and elasticity on kinematic and stress fields are discussed. The influence of elasticity on the fore-aft asymmetry of the flow morphology is shown and quantified. The size and shape of the yielded and unyielded regions around the obstacle are also quantified. Finally, far and local stress fields as well as the drag coefficient are quantified as a function of the plastic and elastic contributions.

Experimental study and numerical modeling of mixing by air injection in yield stress fluids using the OpenFOAM software

AbstractMixing by gas injection is an operation used in industrial processes such as wastewater treatment, metallurgy, or methanization in which pressurized gas is injected into a fluid in order to reduce concentrations and temperatures gradients. This study demonstrates how the CFD toolbox OpenFOAM can be used to simulate such flows. Experimental measurements and observations have been performed on a pilot‐scale reactor where pressurized air is injected in a yield stress fluid. The volume of fluid method and an adaptive mesh with refinement at the interface have been used to track the gas inclusions. The numerical model accuracy has been assessed by comparing experimental and numerical results related to the bubble's frequency, dimensions, and rising velocities as well as the fluid recirculation, yielded, and unyielded regions in the tank. The influence of injection parameters such as the injection flow rate and the fluid rheological parameters has been quantified.

Effect of finite Weissenberg number on turbulent channel flows of an elastoviscoplastic fluid

Direct numerical simulations are carried out to study the effect of finite Weissenberg number up to $Wi=16$ on laminar and turbulent channel flows of an elastoviscoplastic (EVP) fluid, at a fixed bulk Reynolds number of $2800$ . The incompressible flow equations are coupled with the evolution equation for the EVP stress tensor by a modified Saramito model that extends both the Bingham viscoplastic and the finite extensible nonlinear elastic-Peterlin (FENE-P) viscoelastic models. In turbulent flow, we find that drag decreases with both the Bingham and Weissenberg numbers, until the flow laminarises at high enough elastic and yield stresses. Hence, a higher drag reduction is achieved than in the viscoelastic flow at the same Weissenberg number. The drag reduction persists at Bingham numbers up to 20, in contrast to viscoplastic flow, where the drag increases in the laminar regime compared with a Newtonian flow. Moreover, elasticity affects the laminarisation of an EVP flow in a non-monotonic fashion, delaying it at lower and promoting it at higher Weissenberg numbers. A hibernation phenomenon is observed in the EVP flow, leading to large changes in the unyielded regions. Finally, plasticity is observed to affect both low- and high-speed streaks equally, attenuating the turbulent dissipation and the fragmentation of turbulent structures.

A PIV investigation of laminar and turbulent viscoplastic fluid flow on axisymmetric abrupt contraction

The current work is devoted to analyzing viscoplastic fluid flow in a circular axisymmetric abrupt contraction with ratio . The velocity fields and static pressure along a hydraulic loop were measured employing Particle Image Velocimetry technique and a pressure transducers system, respectively. Four aqueous solutions of Carbopol 940 with different rheological parameters were used as viscoplastic fluids. Laminar and turbulent flow conditions are evaluated and compared with the generalized Hagen–Poiseuille law and the Malin friction factor correlations supported by turbulent fluctuations measurements. The formation of a plug core region is observed around the centerline of the pipe. The relationships between the velocity profiles and the shear stress distribution with the plug core were investigated supported via shear rate calculations. The results show that the yield stress highly influences the shape of the velocity profiles and the appearance of unyielded regions at the contraction edges. The shear stress maps show a layered distribution across the pipe diameter with higher values located at the contraction entrance. Instead, a constant shear stress distribution is found in the unyielded regions. The size of these unyielded regions increases with the yield stress but is reduced at regions with high shear rates due to increased inertial forces. Pressure drop measurements contrasted with the shear stress and velocity gradients maps illustrated that the pressure losses along the pipe are affected by the yield stress and the Reynolds number. However, a non-dependence of the pressure loss coefficient at the contraction plane with the rheological parameters is evidenced, as presented in earlier experimental results. This differs from many numerical studies since the contraction’s pressure loss coefficient for viscoplastic fluids is directly affected by the increase of inertial forces at this region.

Viscoplastic flow in a pipe of complex cross section with stick–slip at the wall

The Poiseuille flow of a Bingham fluid with the yield slip condition at the wall is examined. The governing equations are solved employing the accelerated augmented Lagrangian method for both the viscoplastic model and the yield slip equation. We solve numerically the flow in pipes with different cross sections: equilateral and isosceles triangles, L-shape, and hourglass-shape. The combined effects of viscoplasticity and slip on the flow have been investigated. The evolution of the unyielded regions has been studied. The computations cover the full range of the two main dimensionless numbers and exhibit complex flow patterns: all the different flow regimes are identified in ducts of various cross sections.

Peristaltic-Ciliary Flow of A Casson Fluid through An Inclined Tube

The paper is concerned with the peristaltic-ciliary transport of a viscoplastic fluid (Casson fluid) through an inclined cylindrical tube. The peristalsis-cilia induced motion is analysed in the moving frame of reference under the lubrication approximations. Solutions to the flow characteristics petering to yielded and unyielded regions are obtained. The effects of various physical parameters on the axial velocity, the pumping characteristics, the pressure rise, and the frictional force over one wavelength, along with the trapping phenomenon are presented through graphs. Further, the peristaltic flow and peristaltic-ciliary flow results are compared. It is noticed that the axial velocity and the size of trapping bolus in the unplug flow region decrease with an increase in the yield stress. In addition, the axial velocity and the axial pressure gradient in the peristaltic-ciliary pumping are higher than those in the peristaltic pumping.

A new mathematical technique for analysis of internal viscoplastic flows through rectangular ducts

The flow of Bingham materials inside rectangular channels is characterized by a plug region located in the central region of the conduit and four dead zones in the vicinity of the corners of the geometry (i.e., the unyielded regions) that grow as the Bingham number is increased. In this paper, a semianalytical technique is presented for solving the flow of Bingham yield-stress materials inside straight rectangular ducts. To capture the topological shapes of the yielded and unyielded regions, a “squircle” mapping approach is proposed and the interfaces between the yielded and unyielded regions are captured based on minimizing the variational formulation for the velocity principle. The nonlinear governing equations are consequently solved using the Ritz method. To find the optimized solution, a genetic algorithm method is implemented, applying bounded constraints and a sufficient number of populations. The presented results are benchmarked against numerical works available in literature, revealing that the presented solution can accurately capture the positions of both the plug and dead regions. The critical Bingham number at which the plug region meets the dead regions is reported for different aspect ratios. The velocity profiles and the friction factor of flow with different Bingham numbers and aspect ratios are investigated in detail. Following this, in the limiting case in which the Bingham number is sufficiently small, an alternative, simpler analytical approach using elliptical mapping is also proposed. The authors believe that the proposed method could be employed as a useful tool to obtain fast, accurate approximate analytical solutions for similar classes of problem.