Bingham Number Research Articles

Investigation of mixing viscoplastic fluid with a modified anchor impeller inside a cylindrical stirred vessel using Casson–Papanastasiou model

In process engineering as chemical and biotechnological industry, agitated vessels are commonly used for various applications; mechanical agitation and mixing are performed to enhance heat transfer and improve specific Physico-chemical characteristics inside a heated tank. The research subject of this work is a numerical investigation of the thermo-hydrodynamic behavior of viscoplastic fluid (Casson–Papanastasiou model) in a stirred tank, with introducing a new anchor impeller design by conducting some modifications to the standard anchor impeller shape. Four geometry cases have been presented for achieving the mixing process inside the stirred vessel, CAI; classical anchor impeller, AI1; anchor impeller with added horizontal arm blade, AI2 and AI3 anchor impeller with two and three added arm blades, respectively. The investigation is focused on the effect of inertia and plasticity on the thermo-hydrodynamic behavior (flow pattern, power consumption, and heat transfer) by varying the Reynolds number (Re = 1, 10, 100, 200), Bingham number (Bn = 1, 10, 50), in addition to the effect of geometry design in the overall stirred system parameters. The findings revealed an excellent enhancement of flow pattern and heat transfer in the stirred system relatively to the increase of inertia values. Also, an energy reduction has been remarked and the effect of anchor impeller shape. AI3 geometry design significantly improves the flow pattern and enhances heat transfer by an increased rate of 10.46% over the other cases.

SPH simulations of thixo-viscoplastic fluid flow past a cylinder

Thixotropic materials are complex fluids that display time-dependent viscosity and/or yield-stress response upon the application of a fixed deformation, while recovering their original structured-state when the deformation is discontinued. Thixotropic effects are presents in many different systems and applications, ranging from food products, such as ketchup, to metals, such as molten aluminum. In this work we present a first attempt to simulate the rheological properties of thixo-viscoplastic flows using a Smoothed Particle Hydrodynamic (SPH) method. The study set up is a 2D flow around a circular cylinder as well as a simple shear flow between parallel plates to validate our numerical results. SPH solutions are compared with simulations performed using the open-source Finite Volume Method solver RheoTool, based on OpenFOAM. The viscoplastic model used in this work is the Papanastasiou model combined with a recently developed microstructural one, in order to include thixotropy. In this thixo-viscoplastic framework, we analyze the flow properties in terms of yield-fronts, streamlines and structure-parameter fields at different Bingham and Thixotropy numbers, through microstructural thixotropic and yield-stress parameters variation. Obtained results show an important novelty: an asymmetry in the thixo-viscoplastic flow around the cylinder. • First attempt to simulate thixo-viscoplastic flows using SPH. • 2D flow around a circular cylinder as well as a simple shear flow are analyzed. • Papanastasiou model is combined with a microstructural one to include thixotropy. • Flow properties are analysed using different Bingham and Thixotropy numbers. • Main finding is an asymmetry in the thixo-viscoplastic flow around a cylinder.

On the Use of Viscous Micropumps for the Transport of Yield-Stress Liquids in Microfluidic Systems

• A fluid's yield stress is predicted to negatively affect the performance of viscous micropumps. • The drop in performance depends on the Bn number and is estimated to be around 60% for Bn = 10 and 20% for Bn = 2. • Viscous micropumps can be used for pumping weakly-viscoplastic liquids in microfluidic systems. The effect of a fluid's yield stress is numerically studied on the performance of a single-rotor, I-shaped viscous micropump. The device comprises a long cylindrical rotor which rotates in a two-dimensional rectilinear channel filled with a yield-stress fluid. A net flow is established for viscous fluids provided that the cylinder is installed in an off-center position across the channel. Having assumed that the viscoplastic fluid of interest obeys the Bingham-Papanastasiou model we have obtained converged results for Bingham numbers (Bn) up to 10. Numerical results obtained at low Reynolds numbers (typical of microfluidic systems) have revealed that a fluid's yield stress has a negative effect on the device performance. Specifically, the flow rate of the micropump is decreased while its driving torque is increased the larger Bn number. At the maximum Bn number tried in this work (i.e., Bn = 10) the drop in performance is predicted to be around 60%. For Bn < 1, however, it is predicted to be less than 20% suggesting that viscous micropumps can still be regarded as a viable option for the transport of weakly-viscoplastic liquids in microfluidic systems.

Effects of Thermal Diffusion and Diffusion Thermo on a Chemically Reacting MHD Peristaltic Transport of Bingham Plastic Nanofluid

In this work we investigated the solution of the peristaltic motion of Bingham plastic nanofluid through a vertical symmetric channel. The system is stressed by an external strong magnetic field to produce a hall currents. Moreover, we involved effects of Joule heating, radiation, chemical reaction and couple stresses. Further, we considered Soret and Dufour effects. This phenomenon is represented mathematically by a system of non-linear equations which describe the problem. We used the approximation of low Reynolds number and long-wavelength approximation to simplify the governing equations. Then, we used Homotopy perturbation method (HPM) to solve the equations. We sketched a graph for the influence of various parameters on velocity, temperature and nanoparticles concentration profiles and then discussed them physically. Finding revealed that an increase of Bingham number and the regularization parameter of Bingham fluid decreases the velocity and increases the temperature, but radiation parameter has an opposite relative to them. Also, we found that increasing of chemical reaction parameter reduces the nanoparticles concentration, whereas the reverse is the case when the reaction order is increased.

Flows of Dense Suspensions of Polymer Particles through Oblique Bifurcating Channels: Two Continua Approach.

A two-velocity mathematical model is proposed for dense suspension flows through channel bifurcations. Equations agree with thermodynamic laws and they are suitable for both heavy and light particles. The pulsatile mode of injection of particles is considered. In the 2D-case, we address the issue of partitioning particles and study how a loss of particles into the side branch depends on the bifurcation angle. A qualitative agreement with experiment data are established. We capture the Zweifach–Fung effect. We treat polymer particles as a phase enjoying the rheology of the Bingham viscoplastic material. We prove that the polymer particle distribution between two branches correlates with the averaged-in-time Bingham number in these branches.

Interaction between a complex fluid flow and a rotating cylinder

The flow of a thixotropic Bingham material past a rotating cylinder is studied under a wide range of Reynolds and Bingham numbers, thixotropic parameters, and rotational speeds. A microstructure transition of the material involving breakdown and recovery processes is modeled using a kinetic equation, and the Bingham- Papanastasiou model is employed to represent shear stress-strain rate relations. Results show that the material’s structural state at equilibrium depends greatly on the rotational speed and the thixotropic parameters. A layer around the cylinder resembling a Newtonian fluid is observed, in which the microstructure is almost completely broken, the yield stress is negligibly small, and the apparent viscosity approximates that of the Newtonian fluid. The thickness of this Newtonian-like layer varies with the rotational speed and the Reynolds number, but more significantly with the former than with the latter. In addition, the lift and moment coefficients increase with the rotational speed. These values are found to be close to those of the Newtonian fluid as well as of an equivalent non-thixotropic Bingham fluid. Many other aspects of the flow such as the flow pattern, the unyielded zones, and strain rate distribution are presented and discussed.

Poiseuille flow of a Bingham fluid in a channel with a superhydrophobic groovy wall

Plane Poiseuille flow of a Bingham fluid in a channel armed with a superhydrophobic (SH) lower wall is analysed via a semi-analytical model, accompanied by complementary direct numerical simulations (DNS). The SH surface represents a groovy structure with air trapped inside its cavities. Therefore, the fluid adjacent to the wall undergoes stick–slip conditions. The model is developed based on introducing infinitesimal wall-induced perturbations into the motion equations, followed by Fourier series expansions, and solving the resulting equations as a boundary value problem. The Navier slip law accounts for the slip at the liquid/air interface (assuming the Cassie state). The presented analysis is fairly comprehensive, covering the creeping and inertial regimes for thick channels (via the semi-analytical and DNS solutions). The main dimensionless numbers are the Reynolds ( $Re$ ), Bingham ( $Bi$ ) and slip ( $b$ ) numbers, as well as the groove periodicity length ( $\ell$ ) and the slip area fraction ( $\varphi$ ). By increasing $Bi$ , the perturbation and slip velocity fields grow. As $Re$ increases, the perturbation and slip velocity fields become asymmetric. For certain flow parameters, an unyielded plug zone may appear on the SH wall liquid/air interface, while its formation is accelerated by inertial effects. The results classify the regimes of creeping and inertial flows via predicting the onset of the unyielded plug zone formation at the SH wall.

Computational rheometry of yielding and viscoplastic flow in vane-and-cup rheometer fixtures

A planar two-dimensional computational analysis is presented to qualify traditional and fractal vane-in-cup geometries for accurate rheometry of simple viscoplastic fluids with and without slip. Numerical simulations based on an adaptive augmented Lagrangian scheme are used to study the two-dimensional flow field of yield-stress fluids within and around vane tools with N=3 to 24 arms for a wide range of Bingham numbers, B (i.e. the ratio of the yield stress over the characteristic viscous stress). This allows for accurate calculations of the velocity and stress fields around vanes with various geometries, as well as direct comparison to experimental observations of the output torque measured by a rheometer, revealing sources of variation and error. We describe the impact of the vane structure on the fluid velocity field, from few-arm cruciform vanes (N≤6) that significantly perturb the flow away from ideal azimuthal kinematics, to many-arm fractal vanes (N≥12) in which the internal structural features are successfully “cloaked” by a yield surface. This results in the shearing of an almost-circular ring of viscoplastic fluid that is indistinguishable from the annular ring of fluid deformed around a slip-free rotating cylindrical bob and leads to more accurate rheometric measurements of the material flow curve. Moreover, in direct comparison with data from previous literature, we show that slip conditions on the vane surface do not impact the velocity field or measured overall torque T, whereas slip conditions on the smooth outer wall have significant impact on data, even when using a vane geometry. Finally, we describe the impact of vane topography and Bingham number, B, on the measured torque and rheometric accuracy of vane-in-cup geometries for “simple” (inelastic) yield-stress fluids described by either the Bingham plastic or Herschel–Bulkley constitutive model.

Immersed buoyant viscoplastic injections

In this study, the injection of a viscoplastic (yield stress) fluid into a quiescent Newtonian medium is experimentally investigated, using a range of experimental methods, including high-speed camera imaging, laser-induced fluorescence (LIF), and ultrasound Doppler velocimetry (UDV) techniques. The case of our consideration is the continuous downward injection of a Carbopol solution, as a core fluid, through an eccentric inner pipe, placed inside a vertical closed-end pipe filled with a glycerin–water solution. The injected fluid is always heavier than the in-situ fluid and the fluids are miscible. As the injection continues, a filament of the core fluid immersed in the in-situ fluid is formed, for which it has been shown that there exist three distinct flow patterns, namely, the breakup, coiling, and bulging (buckling) regimes. Our focus in this study is to analyse the detailed flow features and sub-regimes at longer times, from an experimental perspective. To generalize our results, they are presented versus the governing dimensionless numbers of the flow, i.e. the buoyancy number, the Bingham number, the viscosity ratio, and the falling height. The breakup regime is analysed in terms of the yielding, necking and pinch-off process, the length of filaments, and their subsequent falling behaviours, the coiling regime in terms of the regular, free and irregular coiling behaviours, and the bulging (buckling) regime in terms of the appearance of a detached falling part and the maximum penetration depth. The results of this work can find applications, in particular, in the plug and abandonment (e.g. cementing processes) of oil and gas wells and 3D printing technologies. • Experimental analysis of immersed buoyant viscoplastic injection in Newtonian fluid. • Observed flow regimes classified into breakup, coiling, and bulging. • Analysis of yielding, necking/pinch-off, and length of filaments for breakup regime. • Coiling subregimes categorized as regular, free, and irregular coiling. • Analysis of maximum penetration depth and subregimes in bulging regime.

Non-dimensional analysis of an unsteady flow in a magnetorheological damper

Theoretical modeling is often applied to study magnetorheological dampers (MRDs) with dimensional and non-dimensional analyses. In contrast to dimensional models, non-dimensional analyses can eliminate the influence of dimensionality and reduce the number of redundant parameters to simplify theoretical modeling and provide more universal applicability. However, most previous non-dimensional analyses have been based on quasi-steady flows that cannot reflect the transient response of an MRD because of the key assumption that the fluid velocity changes instantaneously. This study presents an investigation of the transient response of an MRD using a non-dimensional analysis approach based on an unsteady model. We focus on the step response of the MRD with a step excitation of the piston speed, while the magnetic field is kept constant. For a comprehensive analysis, a set of dimensionless parameters are defined, including a non-dimensional coordinate, a non-dimensional time parameter, the Bingham number, a non-dimensional pre-yield thickness, a damping coefficient, and a hydraulic amplification ratio. The relationships between these dimensionless numbers are analyzed. An unusual “concave area” is found in the velocity profile instead of a simple rigid flow during the transient process under a magnetic field. However, when the non-dimensional time is 0.4, the delayed concave area disappears, and the rigid area fluid velocity reaches 98% of its stable value.

Fluid structure interaction study of non-Newtonian Casson fluid in a bifurcated channel having stenosis with elastic walls

Fluid–structure interaction (FSI) gained a huge attention of scientists and researchers due to its applications in biomedical and mechanical engineering. One of the most important applications of FSI is to study the elastic wall behavior of stenotic arteries. Blood is the suspension of various cells characterized by shear thinning, yield stress, and viscoelastic qualities that can be assessed by using non-Newtonian models. In this study we explored non-Newtonian, incompressible Casson fluid flow in a bifurcated artery with a stenosis. The two-dimensional Casson model is used to study the hemodynamics of the flow. The walls of the artery are supposed to be elastic and the stenosis region is constructed in both walls. Suitable scales are used to transform the nonlinear differential equations into a dimensionless form. The problem is formulated and discretized using Arbitrary Lagrangian–Eulerian (ALE) approach. The finite element method (FEM) technique is used to solve the system of equations, together with appropriate boundary conditions. The analysis is carried out for the Bingham number, Hartmann number, and Reynolds number. The graphical results of pressure field, velocity profile, and load on the walls are assessed and used to study the influence of hemodynamic effects on stenotic arteries, bifurcation region, and elastic walls. This study shows that there is an increase in wall shear stresses (WSS) with increasing values of Bingham number and Hartmann number. Also, for different values of the Bingham number, the load on the upper wall is computed against the Hartmann number. The result indicate that load at the walls increases as the values of Bingham number and Hartmann number increase.

Inferring the rheology of the crust from the uplift observed above the Altiplano-Puna Magma Body

SUMMARYGeophysical imaging techniques together with numerical models have shown that the surface uplift measured above the Altiplano-Puna Magma Body (APMB) can be explained by the presence and propagation of a diapir from the top of the APMB itself. In this work, we model deformation that characterizes the crustal region above and around APMB through the use of a viscoplastic rheology. That is, we assume that at large scale the crust that surround the magmatic mushy diapir behaves as a yield-stress fluid described by the Herschel–Bulkley (HB) model, whereby motion develops only when the local deviatoric stress is greater than a critical value, the yield stress. In this scenario, laboratory and numerical results show that there are two main critical conditions needed for the growth and subsequent rise of a diapir: (1) the ratio between the yield stress and viscous stresses, namely the Bingham number Bi, has to be less than 1, that is Bi ≤ 1 and (2) the ratio between buoyancy stresses and the yield stress, namely the inverse Yield number Yinv, has to be larger than a critical value $Y_{{\rm inv}_C}$. Using these two conditions allows us to estimate the bulk rheological properties of the heterogeneous crust above APMB as a function of the diapir’s size and density contrast with the crust. For the development of a 10–100 km wide diapir, 100–400 kg m–3 lighter than the surrounding crust, the crust yield stress should range between 0.5 and 15 MPa. Then, the regional uplift velocity measured at the surface implies a strain rate greater than ∼10−15–10−16 s−1 and a crust maximum bulk effective viscosity of ηc=1021 Pa.s.

Surface-tension-driven evolution of a viscoplastic liquid coating the interior of a cylindrical tube

One mechanism for airway closure in the lung is the surface-tension-driven instability of the mucus layer which lines the airway wall. We study the instability of an axisymmetric layer of viscoplastic Bingham liquid coating the interior of a rigid tube, which is a simple model for an airway that takes into account the yield stress of mucus. An evolution equation for the thickness of the liquid layer is derived using long-wave theory, from which we also derive a simpler thin-film evolution equation. In the thin-film case we show that two branches of marginally yielded static solutions of the evolution equation can be used to both predict the size of the initial perturbation required to trigger instability and quantify how increasing the capillary Bingham number (a parameter measuring yield stress relative to surface tension) reduces the final deformation of the layer. Using numerical solutions of the long-wave evolution equation, we quantify how the critical layer thickness required to form a liquid plug in the tube increases as the capillary Bingham number is increased. We discuss the significance of these findings for modelling airway closure in obstructive conditions such as cystic fibrosis, where the mucus layer is often thicker and has a higher yield stress.

Viscoplastic and pseudoplastic flows through fractal fractures

A new mathematical model is developed to examine the coupled effects of yield stress, shear-thinning or shear-thickening property and fractal characteristics of fractures on the flow of Herschel–Bulkley fluid through fractal fractures. The flowrate is integrated in a range of fractures that follow a fractal distribution. Increasing the fractal dimension increases the flowrate for both Herschel–Bulkley fluid and Newtonian fluid, but the increase for the Newtonian fluid flow is more significant. The influence of maximum and minimum trace length ratio on the flowrate ratio of the Herschel–Bulkley fluid over the Newtonian fluid is more significant than the fractal dimension. The scaling exponent has very similar effect as maximum and minimum trace length ratio on the flowrate ratio in the fractal fractures. The flowrate ratio drops very quickly with the increasing modified Bingham number. For both shear-thinning and shear-thickening fluids, increasing the pressure head gradient amplifies the effect of the power-law index.

Flow and Heat Transfer in Bingham-Papanastasiou Fluid over a Wedge

This investigation focuses on flow and corresponding heat transfer behavior of viscoplastic Bingham- Papanastasiou fluid over a static radiated wedge. The modeled boundary layer flow governing equations are simplified by using similarity transformations. Numerical results are obtained for the economized equations by employing Runge-Kutta and Newton’s method. The impact of the associated parameters concerned with the fluid properties, friction factor and local Nusselt number are determined. It is found that the thermal radiation parameter improves the heat transfer rate. Bingham number boosts the friction at the wall and reduces the heat transfer rate. Improvement in Bingham number reduces the free surface movement and velocity of the fluid increases.

Entropy optimized flow of hydromagnetic Reiner–Philippoff fluid over a stretching surface

Here, we investigate magnetohydrodynamic flow of an incompressible Reiner–Philippoff fluid over a stretched surface. The stretching property of the sheet induced flow. Joule heating and dissipation effects are considered in energy communication. The energy equation is developed through the first law of thermodynamics. Irreversibility analysis is constructed. Furthermore, the first-order chemical reaction is also accounted. Adequate transformation is used to get the ordinary differential system tackled through a local non-similar technique via the built-in Matlab function bvp4c. Prominent characteristics of flow parameters on the entropy rate, temperature, velocity, and concentration are studied. Thermal and solutal transport rates are studied. Reverse impacts for velocity and temperature are noted for the Reiner–Philippoff liquid parameter. Reduction in velocity is seen for the Bingham number. A larger Prandtl number reduces temperature distribution. Concentration is decreased for both the Lewis number and chemical reaction parameter. A reverse trend is observed for the entropy rate against Brinkman and Bingham numbers. A larger magnetic variable enhances entropy generation.

Wall properties and Joule heating effects on MHD peristaltic transport of Bingham non-Newtonian nanofluid

We analysed Soret and Dufour effects on peristaltic flow of magnetohydrodynamic (MHD) non-Newtonian nanofluid in a uniform symmetric channel with wall properties. Moreover, we involved effects of Joule heating, chemical reaction. Furthermore, we considered Brownian motion and thermophoresis. Then, we simplified the governing equations to a system of partial differential equations by applying low Reynolds number and long wavelength approximations and we solved them by using Homotopy Perturbation Method (HPM). We sketched the influence of various parameters on the stream function, velocity, temperature and nanoparticles concentration in graphs and we discussed them physically. Also, we obtained graphs for heat transfer coefficient, skin friction coefficient, Nusselt number and Sherwood number at the upper wall of the channel. Finding revealed that as the wall proprieties parameters E1 and E2 increased the velocity and temperature increased, but the stream function increased and decreased. While, Bingham number Bn had an opposite relative to the wall proprieties parameters E1 and E2.

Numerical simulations for heat transfer in peristalsis of Bingham fluid utilizing partial slip conditions

The buoyancy-assisted peristaltic motion of viscoplastic fluid in a vertical channel is revisited by accounting partial slip effect at the channel boundaries. Accompanying heat transfer problems due to fluid friction is addressed through the temperature jump assumption. The governing system formed under lubrication approximation is coupled owing to the existence of buoyant force. The system contains a parameter , measuring the fluid yield stress. Moreover, in contrast to the no-slip case, the boundary conditions are Robbin-type conditions and nonlinear in the axial velocity component. An easy-to-use yet reliable package NDSolve of the software MATHEMATICA is utilized to solve the problem for broad ranges of Bingham number and slip coefficients. Plots of stream function, velocity, and temperature are generated, which clearly depict that the yield stress combined with partial slip assumption considerably affects the flow phenomena. To illustrate heat transfer rates, the expression of heat transfer rate is analyzed across the channel walls under different controlling parameters.

Viscoplastic corner eddies

When viscous fluid in a corner is disturbed, eddies can form in the absence of inertia. Examples of flow configurations in which this motion occurs include flow through an abrupt contraction and over a cavity. Six decades ago, Moffatt (J. Fluid Mech., vol. 18, 1964, pp. 1–18) calculated the slow viscous flow of Newtonian fluids in sharp corners, detailing his eponymous ‘Moffatt eddies’. In this study we examine corner flows of viscoplastic materials, a class of non-Newtonian fluids which exhibit solid-like behaviour for stresses below a yield stress. Specifically, we consider a Bingham fluid, for which the material is perfectly rigid at stresses below the yield stress. While a static unyielded plug forms at the tip of the corner, eddies analogous to those found by Moffatt can also form. We examine these viscoplastic eddies numerically, by computing finite element solutions using the augmented-Lagrangian method, and analytically, by employing a viscoplastic boundary layer formulation and scaling arguments. We measure the depth of the static plug as a function of the Bingham number (dimensionless yield stress), show that the process of a new eddy forming as the Bingham number is decreased is driven by the pressure in the yielded fluid adjacent to the static plug, and provide a heuristic argument for the critical Bingham number at which this occurs.

Forced convective MHD flow of Reiner-Philippoff fluid induced by hybrid nanofluid past a nonlinear moving sheet with nonlinear heat sink/source

The utilization of hybrid nanofluids is important in industry because of their high effective thermal conductivity, resulting in high heat transfer rates. The present work examines the influence of nonlinear heat sink/source on forced convective flow induced by Reiner–Philippoff fluid conveying hybrid nanofluid through a moving sheet. Using similarity variables, the PDEs in the current model are transformed into ODEs. Furthermore, the bvp4c approach is used to deal with the dimensionless model. Graphs depict the effects of various physical factors discovered during the scrutiny of numerical simulation on the temperature, heat transfer rate, friction factor, and velocity field. It’s worth noting that increasing the heat source/sink parameters reduces the temperature. Furthermore, increasing the Bingham number parameter increases the temperature and decreases the velocity. A change in the volume fraction parameter augments the temperature and declines the velocity field. The numerical results’ reliability is determined by comparing the acquired results to the existing literature.