Viscoplastic Fluid Research Articles

Effect of Ultrasound on Microstructure and Properties of Aluminum–Copper Friction Stir Lap Welding

In this paper, the influence mechanism of ultrasound on plastic flow and microstructure features of the aluminum–copper friction stir lap welding (Al/Cu-FSLW) process is systematically investigated by adjusting the welding speed and improving the shear rheology in the plastic stirring zone. Through adjusting the ultrasonic vibration and welding speed, the directional control of mechanical properties is realized. It is found that increasing the welding speed properly is beneficial to enhance the mechanical shear between the tool and the workpiece, thus forming more staggered layered structures at the copper side and improving the tensile strength of the weld. The acoustic softening enhances the viscoplastic fluid mixing and strengthens the mechanical interlock of the Al/Cu lap interface. As the welding speeds increase or ultrasonic vibration is applied, the thickness of Al/Cu intermetallic compound (IMC) decreases, and the tensile strength and elongation of the Al/Cu joints are enhanced. Compared with adjusting the welding speed, the ultrasonic vibration can further refine the copper particles which are stirred into the plastic zone, and the thinning effect of ultrasound on IMC layers is better than that of increasing welding speed. At the welding speed of 60 mm/min, the IMC layer thickness is reduced by 42% under ultrasonic effect. In three welding speed conditions, the UV reduced the absolute value of the effective heat of formation (EHF) for Al2Cu and Al4Cu9 and suppressed the formation of AlCu phase. Meanwhile, only when the welding speed is increased from 60 mm/min to 100 mm/min can the formation of AlCu be suppressed. Under the ultrasonic optimization, the stable improvement of welding efficiency is ensured.

Stokes layers in complex fluids

Stokes’s second problem is reconsidered for three models of complex fluids: an elasto-viscoplastic fluid, a thixotropic viscoplastic fluid and a discontinuously shear-thickening fluid. In each case, the Stokes-layer dynamics is interrogated with a view to examining the signatures of the detailed rheology. Significant deformations are possible below the yield stress for elasto-viscoplastic fluids as a result of the excitation of elastic waves, particularly near resonances. Thixotropic fluids with viscosity bifurcations layer internally, but surface-speed signatures mostly appear similar to those for simple yield-stress fluids. Stokes-layer oscillations of discontinuous shear thickening fluids can prompt abrupt increases in viscosity, introducing sudden jumps in surface speed. Pre-existing experimental results for layers of kaolin slurries in a motorized, oscillating tray are reconsidered and compared with the results for elasto-viscoplastic and thixotropic fluids.

RANS predictions of turbulent non-isothermal viscoplastic fluid in pipe with sudden expansion

A transition of Newtonian turbulent fluid to viscoplastic non-Newtonian fluid by cooling in a pipe with a sudden expansion is numerically studied. A recirculation region with negative velocities appears for fluid velocity profiles corresponding to the zone of flow recirculation. A small corner eddy disappears for a non-Newtonian fluid. Significant anisotropy between axial and radial components of Reynolds stresses is numerically shown. The heat transfer distributions along the pipe surface for turbulent non- and Newtonian fluids are qualitatively similar. The peak of heat transfer is shifted upstream in the Schwedoff-Bingham fluid in comparison with the Newtonian one. Authors’ numerical predictions are compared with numerical simulations by other authors for turbulent Schwedoff-Bingham fluids.

Lattice Boltzmann simulations of unsteady Bingham fluid flows

Transient flows of viscoplastic fluids have very peculiar characteristics. The startup and cessation flows of viscoplastic materials have been subject to many theoretical and numerical investigations. The most challenging aspect of numerical solutions of viscoplastic fluids is the viscosity singularity during the transition from yielded to unyielded material. Hence, the proper representation of yield surfaces is the most critical aspect of numerical methods in viscoplastic fluid flow. In the present work, we use a lattice Boltzmann scheme to solve an ideal Bingham fluid’s startup and cessation flows. This numerical scheme advantage is that can represent infinite viscosity without noticeable numerical instabilities, producing yield surfaces with more accuracy and quality. Theoretical solutions for the startup flow are available in the literature. However, it is unclear which is more accurate and what their validity ranges are. Nonetheless, these solutions served as a reference for the present simulations. The overall aspect of the numerical solutions agreed with the theoretical models. The cessation flow of the Bingham fluid was also simulated. Unlike a Newtonian fluid, this type of flow is known to have a finite period until cessation. The simulations correctly reproduced this behavior. The transient yield surfaces matched very well with augmented Lagrangian solutions.

Immiscible non-Newtonian displacement flows in stationary and axially rotating pipes

We examine immiscible displacement flows in stationary and rotating pipes, at a fixed inclination angle in a density-unstable configuration, using a viscoplastic fluid to displace a less viscous Newtonian fluid. We employ non-intrusive experimental methods, such as camera imaging, planar laser-induced fluorescence (PLIF), and ultrasound Doppler velocimetry (UDV). We analyze the impact of key dimensionless numbers, including the imposed Reynolds numbers (Re, Re*), rotational Reynolds number (Rer), capillary number (Ca), and viscosity ratio (M), on flow patterns, regime classifications, regime transition boundaries, interfacial instabilities, and displacement efficiency. Our experiments demonstrate distinct immiscible displacement flow patterns in stationary and rotating pipes. In stationary pipes, heavier fluids slump underneath lighter ones, resulting in lift-head and wavy interface stratified flows, driven by gravity. Decreasing M slows the interface evolution and reduces its front velocity, while increasing Re* shortens the thin layer of the interface tail. In rotating pipes, the interplay between viscous, rotational, and capillary forces generates swirling slug flows with stable, elongated, and chaotic sub-regimes. Progressively, decreasing M leads to swirling dispersed droplet flow, swirling fragmented flow, and, eventually, swirling bulk flow. The interface dynamics, such as wave formations and velocity profiles, is influenced by rotational forces and inertial effects, with Fourier analysis showing the dependence of the interfacial front velocity's dominant frequency on Re and Rer. Finally, UDV measurements reveal the existence/absence of countercurrent flows in stationary/rotating pipes, while PLIF results provide further insight into droplet formation and concentration field behavior at the pipe center plane.

3D-Numerical simulation of free convection inside a cubical cavity filled with non-Newtonian-Bingham fluid

This work focuses on a numerical study of the natural convection of a non-Newtonian viscoplastic fluid within a cubic enclosure. The viscoplastic behavior is described by the Bingham model. The considered three-dimensional convective flow is confined within a cavity, subjected to a horizontal temperature gradient, where the vertical walls have two imposed temperatures while the rest of the walls are adiabatic. The Navier-Stokes equations, along with the mass and energy conservation equations, are numerically solved. Fluid flow and heat transfer characteristics are systematically studied over a wide range of Rayleigh numbers Ra (103 - 106) and Bingham number Bn (0 - 20). Finally, comparisons were made with previous results obtained in two dimensions in order to analyze the existence of a three-dimensional effect on the flow of the Bingham fluid. The results show that the Nusselt number decreases with the increase of the Bingham number, and for the large values ​​of the latter the heat transfer is done by conduction. It is also noteworthy that the critical Bn of the 2D model is higher than that of the 3D model, which confirms the existence of the three-dimensional effect. This is attributed to the presence of a wall along the Z axis which hinders and limits the flow of fluid within the enclosure.

Yield-stress effects on spontaneous imbibition in paper-based kits

The classic Richards equation is a good model for predicting imbibition of viscous fluids in porous materials such as dry soils or filter papers. It cannot, in principle, be used for physiological fluids such as blood simply because such fluids often exhibit a variety of non-Newtonian behavior such as a yield stress. In the present work, we have theoretically extended the classic Richards equation to viscoplastic fluids obeying the Bingham model using the concept of the effective viscosity together with the bundle-of-tube model. The new imbibition model could partly resolve the discrepancy reported in the literature between the predictions of the classic Richards equation for the stain growth of sessile blood droplets in a typical filter paper. A better fit, however, requires considering other non-Newtonian effects of the blood such as its viscoelasticity. Using the Bingham-modified Richards equation, it is demonstrated that yield stress in a test fluid has a retarding effect on the imbibition phenomenon, so that such fluids may not necessarily reach the test line of a paper-based diagnostic kit. But yield stress is predicted to extend the duration of the quasi-steady regime on the test line of diagnostic kits, which is a desirable effect. The results suggest that inducing (or elevating) the level of yield stress in a test liquid such as blood can be used as a passive means to control imbibition characteristics in paper-based systems.

Variable viscosity characteristics of two-layered isothermal calendering phenomena with non-Newtonian fluids

Two-layer calendering using non-Newtonian fluids is used frequently in different industries to improve the working and the quality of the products. In this article, two-layer calendering of incompressible non-Newtonian fluids, that is, couple stress fluid and viscoplastic fluid in the upper and lower layer respectively is investigated. For the simplified governing equations, Lubrication approximation theory is applied. By the use of no-slip condition and non-dimensional variables, the exact solution for pressure gradient, pressure, and velocity profiles for both layers of fluid is calculated. The parameters are important from an engineering point of view like roll separating force, power delivered by rollers, torque acting by each roller, and adiabatic temperature are also calculated analytically. Additionally, the results of a couple stress parameter, viscoplastic Casson parameter, and viscosity ratios on all the parameters are also discussed and investigated in detail. Moreover, all the effects investigated are also verified with the literature results of single-layer calendering of non-Newtonian fluids by using large numeric values for the couple stress parameter and viscoplastic Casson parameter.

Horizontal buoyant jets into viscoplastic ambient fluids

This study investigates the horizontal injection of a heavy Newtonian fluid into a lighter viscoplastic ambient fluid, in a large reservoir. The flow dynamics is experimentally captured via camera imaging, laser-induced fluorescence, and particle image velocimetry techniques. The flow parameters include various density differences, injection velocities, and ambient fluid viscoplastic properties. Our analysis identifies two key dimensionless numbers, the Froude number (Fr) and the effective viscosity ratio (m), which includes the rheology of the viscoplastic fluid. Our study also examines the effects of these dimensionless numbers on critical jet characteristics, such as bifurcation length, transition length, deviation length, and jet trajectory, and provides correlations using Fr and m, to predict these characteristic lengths. A regime classification based on the bifurcation phenomenon is also presented in the Fr−m plane.

H(div)-conforming and discontinuous Galerkin approach for Herschel–Bulkley flow with density-dependent viscosity and yield stress

This paper presents a comprehensive study on Herschel–Bulkley flow, where the flow parameters are dependent on the density. The Herschel–Bulkley model is a generalized power-law model used to simulate viscoplastic fluids defined by a plasticity threshold. We consider the case where the plasticity threshold and the viscosity depend on the shear rate and fluid density. To analyze this model, we use a Huber regularization of the stress and propose an H(div)-conforming and discontinuous Galerkin (DG) numerical approximation for the coupled equations governing the flow. We discuss the stability and existence of discrete solutions and propose a semismooth Newton linearization for the numerical solution of the discretized system. Our numerical scheme is validated through several experiments that explore the behavior of Herschel–Bulkley flow under different conditions. The results demonstrate the robustness of our numerical method.

Stability of fully developed pipe flow of a shear-thinning fluid that approximates the response of viscoplastic fluids

The stability of steady, fully developed flow in a long cylindrical pipe for a shear-thinning fluid (which approximates a class of viscoplastic materials) is studied using linear stability analysis. The eigenvalues of the frequency of the perturbation of the steady-state solution are obtained using the shooting method. The eigenvalues are negative in the Reynolds number range studied and asymptotically tend to zero as the Reynolds number increases. This shows the pipe flow is stable in the Reynolds number range studied. A qualitatively similar trend is shown by the eigenvalues of a Navier–Stokes fluid of equivalent viscosity. However, the eigenvalues are much lesser than those of the shear-thinning fluid, and this shows that the flow of the Navier–Stokes fluid can be expected to be stable over a much larger Reynolds number range than the shear-thinning fluid.

Swimming in viscoplastic fluids

Abstract Locomotion at small scales in the absence of inertia is a classical and enduring research topic. Here, recent developments in the theory of such locomotion through a viscoplastic ambient fluid are reviewed and explored. The specific focus here applies to motion of cylindrical filamentary bodies that are long and thin, for which an asymptotic slender-body theory can be exploited. Details of this theory are summarised and then applied to describe different swimming waveforms: undulation, peristalsis, and helical motion. It is shown that, in general, strong force anisotropy close to the limit of axial cylindrical motion has a significant effect on locomotion in viscoplastic media, allowing for highly efficient motion in which the swimmer is able to ‘cut’ through the material following very closely the path of its own axis. Some qualitative comparison with experiments is presented, and future extensions and research directions are reviewed. Graphical abstract Deformation fields around cylinders moving at different angles to their axis through a yield stress fluid, showing (a) a low yield stress and (b) a high yield stress

Derivation and numerical resolution of 2D shallow water equations for multi-regime flows of Herschel–Bulkley fluids

This paper presents mathematical modelling and simulation of thin free-surface flows of viscoplastic fluids with a Herschel–Bulkley rheology over complex topographies with basal perturbations. Using the asymptotic expansion method, depth-averaged models (lubrication and shallow water type models) are derived for 3D (three-dimensional) multi-regime flows on non-flat inclined topographies with varying basal slipperiness. Starting from the Navier–Stokes equations, two flow regimes corresponding to different balances between shear and pressure forces are presented. Flow models corresponding to these regimes are calculated as perturbations of the zeroth-order solutions. The classical reference models in the literature are recovered by considering their respective cases on a flat-inclined surface. In the second regime case, a pressure term is non-negligible. Mathematically, it leads to a corrective term to the classical regime equations. Flow solutions of the two regimes are compared; the difference appears in particular in the vicinity of sharp changes of slopes. Nonetheless, both regime models are compared with experiments and are found to be in good agreement. Furthermore, numerical examples are shown to illustrate the robustness of the present shallow water models to simulate viscoplastic flows in 3D and over an inclined topography with local perturbations in basal elevation and basal slipperiness. The derived models are adequate for direct (engineering and geophysical) applications to real-world flow problems presenting Herschel–Bulkley rheology like lava and mud flows.

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.

Viscosity-model-independent generalized Reynolds number for laminar pipe flow of shear-thinning and viscoplastic fluids

Understanding the flow dynamics of non-Newtonian fluids is crucial in various engineering, industrial, and biomedical applications. However, the existing generalized Reynolds number formulations for non-Newtonian fluids have limited applicability due to their dependencies on their specific viscosity models. In this study, we propose a new viscosity-model-independent generalized Reynolds number formulation for laminar pipe flow. The proposed method is based on the direct adaptation of the measurement principles of rotational viscometers for wall shear rate estimation. We assess the accuracy of this proposed formulation for power-law and Carreau-Yasuda viscosity models through robust friction factor experiments. The experimental results demonstrate the applicability and effectiveness of the proposed viscosity-model-independent Reynolds number, as the measured friction factor data align closely with our Reynolds number predictions. Furthermore, we compare the accuracy of our Reynolds number formulation against established generalized Reynolds formulations for pure shear-thinning (Carreau-Yasuda) and viscoplastic (Herschel-Bulkley-extended) models. The results of the comparative analysis confirm the reliability and robustness of this generalized Reynolds number in characterizing and interpreting flow behavior in systems with visco-inelastic non-Newtonian fluids. This unified generalized Reynolds number formulation presents new and significant opportunities for precise pipe flow characterization and interpretation as it is applicable to any visco-inelastic (time-independent) viscosity model without requiring additional derivations.

On the efficient non-linear solver for hydraulic fracturing and well cementing simulations based on Anderson acceleration

The aim of this study is to create a fast and stable iterative technique for numerical solution of a quasi-linear elliptic pressure equation. We developed a modified version of the Anderson acceleration (AA) algorithm to fixed-point (FP) iteration method. It computes the approximation to the solutions at each iteration based on the history of vectors in extended space, which includes the vector of unknowns, the discrete form of the operator, and the equation's right-hand side. Several constraints are applied to AA algorithm, including a limitation of the time step variation during the iteration process, which allows switching to the base FP iterations to maintain convergence. Compared to the base FP algorithm, the improved version of the AA algorithm enables a reliable and rapid convergence of the iterative solution for the quasi-linear elliptic pressure equation describing the flow of particle-laden yield-stress fluids in a narrow channel during hydraulic fracturing, a key technology for stimulating hydrocarbon-bearing reservoirs. In particular, the proposed AA algorithm allows for faster computations and resolution of unyielding zones in hydraulic fractures that cannot be calculated using the FP algorithm. The quasi-linear elliptic pressure equation under consideration describes various physical processes, such as the displacement of fluids with viscoplastic rheology in a narrow cylindrical annulus during well cementing, the displacement of cross-linked gel in a proppant pack filling hydraulic fractures during the early stage of well production (fracture flowback), and multiphase filtration in a rock formation. We estimate computational complexity of the developed algorithm as compared to Jacobian-based algorithms and show that the performance of the former one is higher in modelling of flows of viscoplastic fluids. We believe that the developed algorithm is a useful numerical tool that can be implemented in commercial simulators to obtain fast and converged solutions to the non-linear problems described above.

Natural convection of viscoplastic fluids in a triangular enclosure

This study extends the analysis of natural convection in a viscoplastic fluid to flow within a triangular enclosure. The finite-element approach provided a numerical solution to the continuity, momentum, and energy equations. The governing parameters for this problem are the Rayleigh number, (Ra=104−106), yield number, (Y=0−Ymax), aspect ratio, (HL=0.5−2.5), and slope angle, (∅=0−π2). The influence of these parameters on the heat and mass transfer, morphology of yielded/unyielded regions, and fluid flow were thoroughly examined. The results show that two opposing factors influence the flow behavior and heat transmission within the triangular enclosure. The proximity of the walls restricts the convective movement, leading to reduced heat transfer. However, the proximity of hot and cold sources increases the temperature gradient and heat transfer. The unique influence of the viscoplastic material properties, particularly the yield stress, further distinguishes the heat transfer in this triangular enclosure from other geometries. The results indicate that an increase in the Rayleigh number mitigates the effects of yield stress to some extent. However, the accumulation of unyielded material at the triangular apex hinders convection flow. Furthermore, the viscoplastic fluid flow and heat transfer changed significantly with changes in triangle height. In particular, the maximum yield stress increased by more than 100 % as the aspect ratio increased from 0.5 to 2.5. A change in the slope angle causes a continuous transition from subcritical to supercritical bifurcation, significantly affecting the morphology of the yielded and unyielded areas, and the maximum (critical) yield number. Finally, correlations were developed to predict the Nusselt number and maximum yield stress in all cases.

Diagnostics of velocity-flow characteristics in laminar flows

Understanding and evaluating various types of fluid movement is important for the analysis and design of systems and devices related to hydraulics and hydrodynamics, including hydrocarbon production and transport systems. Knowledge of stationary and non-stationary types of motion makes it possible to predict and control the behavior of a fluid under various conditions. The article discussed the issues of diagnosing stationary and non-stationary and gravitational flows. It has been established that the time of transition to a stationary mode in pipelines mainly depends on the geometric parameters of the pipeline and the rheological properties of the pumped systems. It is known that many issues related to the management of technological processes and the improvement of efficiency in oil and gas production are closely associated with the movement of homogeneous and heterogeneous fluids with various rheological and physico-chemical properties in pipelines during extraction, collection, and transportation, which have different forms and hydraulic characteristics. In the laminar flow regime, a methodology has been developed to determine the transition time to a steady-state operation mode, based on the flow characteristics of viscous and visco-plastic fluids, taking into account inertia forces. It has been determined that, unlike viscous fluids, the time to reach a steady-state operation mode for visco-plastic flows also varies depending on the formation of the flow core, which is determined by the initial yield stress. In the article, a mathematical expression is provided for determining the flow rate for visco-plastic fluids in laminar flow, taking into account the pressure losses due to inertia forces for viscous fluids. The results obtained are explained to eliminate the difference and increase accuracy when determining the flow rate using the Poiseuille formula for visco-plastic laminar flows.

Turbulent impingement jet cleaning of thick viscoplastic layers

An experimental study is conducted on the use of a normally impinging turbulent water jet (with the Reynolds number of Re≈11800), for cleaning thick layers of a Newtonian fluid and two viscoplastic fluids (i.e., transparent Carbopol solutions). The layer thickness is larger than the jet radius. Non-intrusive techniques are used to track the geometrical features of the cleaning process in real time. The effects of layer thickness and fluid yield stress on removal behavior, including cleaning radius, cavity radius, and angle, are investigated. A yield stress promotes the initial formation of a blister rather than a cavity, and the rate of removal decreases with increasing layer thickness and yield stress. A relation is presented for the growth of the cavity radius, which fits our experimental observations well. A comparative analysis of submerged and impinging jets reveals, for the first time, the role of air entrainment in the process, with bubble characteristics such as trajectory, size distribution (diameter), and velocity being determined by the yield stress.

General hydrodynamic features of elastoviscoplastic fluid flows through randomised porous media

A numerical study of yield-stress fluids flowing in porous media is presented. The porous media is randomly constructed by non-overlapping mono-dispersed circular obstacles. Two class of rheological models are investigated: elastoviscoplastic fluids (i.e. Saramito model) and viscoplastic fluids (i.e. Bingham model). A wide range of practical Weissenberg and Bingham numbers is studied at three different levels of porosities of the media. The emphasis is on revealing some physical transport mechanisms of yield-stress fluids in porous media when the elastic behaviour of this kind of fluids is incorporated. Thus, computations of elastoviscoplastic fluids are performed and are compared with the viscoplastic fluid flow properties. At a constant Weissenberg number, the pressure drop increases both with the Bingham number and the solid volume fraction of obstacles. However, the effect of elasticity is less trivial. At low Bingham numbers, the pressure drop of an elastoviscoplastic fluid increases compared to a viscoplastic fluid, while at high Bingham numbers we observe drag reduction by elasticity. At the yield limit (i.e. infinitely large Bingham numbers), elasticity of the fluid systematically promotes yielding: elastic stresses help the fluid to overcome the yield stress resistance at smaller pressure gradients. We observe that elastic effects increase with both Weissenberg and Bingham numbers. In both cases, elastic effects finally make the elastoviscoplastic flow unsteady, which consequently can result in chaos and turbulence.Graphical abstract