Bingham Number Research Articles

The isolated effect of yield stress in viscoplastic turbulent flow

Turbulent flows of viscoplastic fluids are present in several industrial and natural applications. The effects of yield stress on this problem have always been studied as a part of a larger physical context, because real viscoplastic materials have many properties that cannot be easily isolated. Direct numerical simulations have recently emerged as a viable tool for investigating non-Newtonian fluid flow in turbulent regimes. In the present work, we solve the turbulent flow of an ideal Bingham fluid, focusing on the isolated effect of yield stress. A numerical scheme for viscoplastic flows was implemented based on the lattice Boltzmann method. An outstanding characteristic of this scheme is the possibility of representing infinite viscosity by setting the relaxation frequency to zero, enabling the representation of the Bingham constitutive equation without artifacts, and producing a more accurate representation of the yield surfaces. In the turbulent channel flow simulations, the friction Reynolds number was fixed at 180, while the Bingham number varied from 0 (Newtonian) to 0.15. It is shown that unyielded portions of material travel along with the flow near the centerline. These unyielded spots do not disappear quickly, but rather have a significant lifetime. Another interesting outcome is that the yield stress increases the turbulence anisotropy, by lowering the spanwise and normal velocity fluctuations, while the streamwise component becomes higher. Reynolds stresses and budgets of turbulent kinetic energy have been analyzed regarding the increased bulk velocities that were found by increasing the yield stress.

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.

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.

Cascaded lattice Boltzmann simulation of Newtonian and non-Newtonian mixture nanofluids with variable thermophysical properties in a cavity with vertical heat radiator

The central moments-based cascaded lattice Boltzmann method (CLBM) for then Newtonian and non-Newtonian Buongiorno’s model mixture nanofluids (CuO, ZnO, Al2O3-water) has been implemented and applied in a square chamber with a vertical heat radiator using accelerated graphics processing unit (GPU) computing through compute unified device architecture (CUDA) C/C++ platform. Due to the higher numerical stability, the CLBM is a superior numerical tool to the raw moments-based MRT-LBM (multiple-relaxation-time lattice Boltzmann method). Three different models for the viscosity and thermal conductivity of the nanofluids: (i) the Binkmann model for the constant viscosity and the Maxwell model for the constant thermal conductivity (ii) Binkmann and Maxwell model with temperature dependent Brownian motion and (iii) Corcione model with non-Newtonian fluid where the temperature and strain rate determine the nanofluid effective thermal conductivity and viscosity, have been used. The enclosure’s upper and bottom walls are thermally adiabatic, but the left and right walls are uniformly cold. A vertical heater is immersed in the middle position of the cavity. The benchmark results for non-Newtonian, Newtonian, and nanofluids for the various computational domains are used to validate the current code adequately. The Bingham number (Bn), the Rayleigh number (Ra), and The volume fraction of the nanoparticles (ϕ) are the three key parameters that are varied in this investigation to demonstrate the effects of natural convection on the isotherms, streamlines, isolines of nanoparticle volume fractionation, yielded and unyeilded zone, and average Nusselt number (Nu¯). The Brownian motion effects of the nanoparticles augmented the average rate of heat transfer and the use of the Bingham nanofluids reduced the heat transfer enhancement. For the CuO-water nanofluid, the augmentation of the rate of heat transfer is 15.42% from ϕ=0 to 4% while Ra=106 and the corresponding heat transfer enhancement for the ZnO-water nanofluid is 11.11%. For the Bingham fluid, the rate of heat transfer increases 7% from Ra=105 to Ra=106 while ϕ=2% and Bn=0.3. The findings can be applied to optimize automotive radiator systems, which are crucial for maintaining engine temperature.

Laminar planar jets of elastoviscoplastic fluids

We perform numerical simulations of planar jets of elastoviscoplastic (EVP) fluid (Saramito (2007) model) at low Reynolds number. Three different configurations are considered: (i) EVP jet in EVP ambient fluid, (ii) EVP jet in Newtonian ambient fluid (miscible), and (iii) EVP jet in Newtonian ambient fluid (immiscible). We investigate the effect of the Bingham number, i.e. of the dimensionless yield stress of the EVP fluid, on the jet dynamics, and find a good agreement with the scaling for laminar, Newtonian jets for the centerline velocity uc/u0∝(x/h)−1/3 and for the jet thickness δmom/h∝(x/h)2/3 at small Bingham number. This is lost once substantial regions of the fluid become unyielded, where we find that the spreading rate of the jet and the decay rate of the centerline velocity increase with the Bingham number, due to the regions of unyielded fluid inducing a blockage effect on the jet. The most striking difference among the three configurations we considered is the extent and position of the regions of unyielded fluid: large portions of ambient and jet fluids (in particular, away from the inlet) are unyielded for the EVP jet in EVP ambient fluid, whereas for the other two configurations, the regions of unyielded fluid are limited to the jet, as expected. We derive a power law scaling for the centerline yield variable and confirm it with the results from our numerical simulations. The yield variable determines the transition from the viscoelastic Oldroyd-B fluid (yielded) to the viscoelastic Kelvin–Voigt material (unyielded).

Hydrodynamic coupling of a cilia–mucus system in Herschel–Bulkley flows

The yield stress and shear thinning properties of mucus are identified as critical for ciliary coordination and mucus transport in human airways. We use here numerical simulations to explore the hydrodynamic coupling of cilia and mucus with these two properties using the Herschel–Bulkley model, in a lattice Boltzmann solver for the fluid flow. Three mucus flow regimes, i.e. a poorly organized regime, a swirly regime, and a fully unidirectional regime, are observed and analysed by parametric studies. We systematically investigate the effects of ciliary density, interaction length, Bingham number and flow index on the mucus flow regime formation. The underlying mechanism of the regime formation is analysed in detail by examining the variation of two physical quantities (polarization and integral length) and the evolution of the flow velocity, viscosity and shear-rate fields. Mucus viscosity is found to be the dominant parameter influencing the regime formation when enhancing the yield stress and shear thinning properties. The present model is able to reproduce the solid body rotation observed in experiments (Loiseau et al., Nat. Phys., vol. 16, 2020, pp. 1158–1164). A more precise prediction can be achieved by incorporating non-Newtonian properties into the modelling of mucus as proposed by Gsell et al. (Sci. Rep., vol. 10, 2020, 8405).

Empirical modeling and sensitivity analysis of pressure rise per wavelength and frictional forces for the peristaltic flow of Bingham plastic fluids: application of response surface methodology

The efficiency of mixed convection peristaltic flow can be investigated through pressure rise per wavelength (ΔPλ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\Delta P_{{{\\uplambda }}} )$$\\end{document} and frictional forces (Fλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${F}_{\\uplambda }$$\\end{document}). The main aim of the present study is to discover the sensitivity analysis of non-Newtonian fluids using the Bingham plastic fluid model. In order to achieve this objective, we have empirically modeled the pressure rise per wavelength (ΔPλ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\Delta {P}_{\\uplambda })$$\\end{document} and frictional forces (Fλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${F}_{\\uplambda }$$\\end{document}) as a function varying with leading parameters of problem. The flow problem is governed by three coupled nonlinear partial differential equations. They are reduced to nonlinear coupled ordinary differential equations by using the long wavelength and low Reynolds number approximations. They are solved numerically using MATLAB built-in routine bvp4c to analyze the sensitivity of pressure rise per wavelength (ΔPλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta {P}_{\\uplambda }$$\\end{document}) and frictional forces (Fλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${F}_{\\uplambda }$$\\end{document}). We first derive the empirical model among each of responses ΔPλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta {P}_{\\uplambda }$$\\end{document} and Fλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${F}_{\\uplambda }$$\\end{document} and physical parameters which govern the flow using response surface methodology. The goodness of fit of empirical model is decided on the basis of coefficient of determination (R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}^{2}$$\\end{document}) obtained from the analysis of variance (ANOVA). The coefficients of determination (R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}^{2}$$\\end{document}) are 99.78% both for ΔPλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta {P}_{\\uplambda }$$\\end{document} andFλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${F}_{\\uplambda }$$\\end{document}. The higher values of R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}^{2}$$\\end{document} determine the goodness of fit of empirical model. No correlation has been developed to optimize ΔPλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta {P}_{\\uplambda }$$\\end{document} and Fλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${F}_{\\uplambda }$$\\end{document} in peristaltic flow for Bingham plastic fluids using RSM. The results of sensitivity analysis revealed that ΔPλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta {P}_{\\uplambda }$$\\end{document} and Fλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${F}_{\\uplambda }$$\\end{document} are most sensitive to flow rate (q) at all levels such as low (− 1), medium (0) and high (+ 1). The sensitivity of ΔPλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta {P}_{\\uplambda }$$\\end{document} to Bingham number (Bn) shows a distinct behavior with varying levels of flow rate (q). At low level (− 1) of flow rate (q), the sensitivity is positive, and at high level (+ 1) of flow rate (q), the sensitivity becomes negative. Conversely, the sensitivity of Fλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${F}_{\\uplambda }$$\\end{document} to Bingham number (Bn) at low to high level of flow rate (q).

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.

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

A review on free miscible buoyant jets

Buoyant jets have been a topic of interest due to their presence in various natural and industrial phenomena, as well as in our daily lives. Covering extensive literature, our focus in this review article is on free, miscible buoyant jets with small-density differences (under the Boussinesq approximation) within uniform and stagnant environments. We examine the latest research on Newtonian jet flows and expand our scope to their viscoplastic counterparts, reflecting a burgeoning trend in contemporary literature. Our analysis covers negatively buoyant, pure, and positively buoyant jets, via delving into their dynamics. This includes regime classifications and key jet characteristics, such as laminar length, penetration length, mixing behavior, velocity profiles, and turbulence parameters. In particular, the effects of the key governing dimensionless parameters, including the Reynolds number, the Froude number, the Bingham number, the viscosity ratio, and the injection angle, on these characteristics are explored.

Modeling of periodical shearing flow in a fibrous space with applications in shear-induced brain injury

This paper presents a theoretical model examining the interaction between a fibrous network and viscous fluid flow driven by an oscillating boundary. The aim is to understand how oscillating impacts are transmitted from the skull, through the arachnoid trabeculae network filled with cerebrospinal fluid, as observed in shaken baby syndrome. The model uses an effective medium approach to determine the fluid velocity field while each fiber is treated as a soft string undergoing deformation. Results indicate that the frequency of oscillation, fiber stiffness, and porous structure resistance significantly influence the oscillating shearing flow, as indicated by the Womersley (Wo), Brinkman (α), and Bingham (Bm) numbers. Application of the model to shaken baby syndrome suggests that oscillations in the cerebrospinal fluid and arachnoid trabeculae can significantly surpass those on the skull, leading to intense shear stress penetration to the brain. This model is the first study to integrate the dynamic response of string-like fibrous networks in fluid flows with oscillating boundaries and offers a quantitative framework for predicting the transmission of shearing forces from the skull to the brain matter.

Reiner-Philippoff hybrid ferrofluid flow past a permeable stretching/shrinking sheet with radiation effect: Dual solutions with stability analysis

The fundamental goal of this study is to scrutinize the flow and the heat transfer performance of Reiner-Philippoff hybrid ferrofluid (magnetic nanofluid) magnetite-cobalt ferrite/water, magnetite/water, and cobalt ferrite/water with radiation effect past a permeable stretching/shrinking sheet. Different variations of Bingham number, Reiner-Philippoff parameter, and radiation effect are examined to estimate the thermal progress across the fluid at the boundary layer when the sheet is shrunk and stretched. The governing fluid flow model is remodeled into a nonlinear set of ordinary differential equations via transformation of similarity. The new transformed system is numerically solved aided by MATLAB's bvp4c solver. Within a certain set of physical parameters, two distinct solutions can be generated. Hybrid ferrofluid has been shown to have an improved heat transfer performance when the sheet is shrunk compared to mono ferrofluids. The laminar state of the flow can be maintained efficiently when a hybrid ferrofluid is considered. The radiation parameter can be used to enhance the thermal progress except when the sheet is shrunk. The first solution has been shown to be stable by stability analysis.

Natural convection and heat transmission of two-phase Bingham Fe3O4-CoFe2O4-water hybrid nanofluid flow in an enclosure with a corrugated heated cylinder

The Galerkin weighted residual finite element method (GFEM) has been used to numerically investigate the natural convection flow of two-phase Bingham hybrid nanofluid in a square cavity with a corrugated cylinder at the center. The base fluid here is water. The hybrid nanofluid is made of Fe3O4-CoFe2O4 nanoparticles. The flow inside the enclosure is considered laminar, non-Newtonian, and incompressible. The corrugated cylinder is taken heated. The top and bottom walls of the outer square are considered adiabatic, and the left and right walls are considered cold. Several non-dimensional parameters, including the Rayleigh number (Ra = 104, 105, 106), the Bingham number (Bn = 0, 0.5, 1), the volume fraction (ϕ = 0.00, 0.04), the Brownian parameter (Nb = 0.1, 0.2, 0.3), thermophoresis parameter (Nt = 0.1, 0.2, 0.3), buoyancy ratio (Nr = 0.1, 0.2, 0.3), corrugation (N = 5, 6, 7), the Prandtl number (Pr = 6.2), and the Lewis number (Le = 1000) have been numerically simulated. The main focus of this work is to witness streamlines, isotherm, and nanoparticles’ volume fraction distribution and understand heat transmission with the help of average Nusselt number and local Nusselt number for several parameters for a two-phase Bingham hybrid nanofluid. The results show that an upsurge in Ra increases the extent of the upper two vortices and decreases the extent of the lower two vortices. All four vortices are seen to increase in dimension for an increase in Bn. The same observation is found for an increase in ϕ. The lower vortices are growing, and the top ones are nearly eliminated for base fluid, while those are visible for nanofluid when corrugation is increased. A further expansion completely destroyed the higher ones and increased the size of the bottom ones. The average Nusselt number ( Nu¯ ) increases when Ra is incremented. Nu¯ rises with a rise in ϕ. An increment in ϕ to 0.04 results in a 4.43% gain in Nu¯ . An increase in Bn is causing Nu¯ to drop by 18.02%. For Ra = 105 and Bn = 2, ϕ = 0.02 has the maximum heat transfer enhancement between 0.00 ≤ ϕ ≤ 0.04. In the future, we plan to use mixed convection in a three-dimensional domain along with the impact of magnetohydrodynamics (MHD).

Numerical investigation on the deformation and breakup of an elastoviscoplastic droplet in simple shear flow

In this paper, direct numerical simulations (DNSs) are performed to investigate the deformation and breakup of an elastoviscoplastic (EVP) droplet in a Newtonian matrix under simple shear flow. The two-phase interface is captured by the volume-of-fluid (VOF) method with adaptive mesh refinement technique. The Saramito model (Bingham model coupled exponential Phan-Thien–Tanner viscoelastic model) is used to characterize the rheological behavior of the droplet. The droplet deformation and conformational state are studied with different Capillary numbers Ca, Weissenberg numbers Wi, and Bingham numbers Bi, which represent the surface tension, elasticity, and yield stress of the droplet, respectively. Our results show that droplet deformation occurs at low Ca, while breakup occurs at high Ca. The droplet non-monotonically deforms with increasing Wi and Bi, while is elongated for higher Ca. In addition, three breakup modes (mid-point pinching, transitional breakup, and homogeneous breakup) are reported for EVP droplets, in which transitional breakup disappears due to the influence of high elasticity. The conformational state of the droplet intuitively demonstrates the change of breakup from horizontal shear to vertical breakup. In spite of the fact that the surface tension always inhibits the deformation of droplets, the present work indicates that Bi has little effect on the deformation with high Wi and high Ca, while the influence is obvious at low Wi and Ca. The observed elastic and plastic effects on droplet deformation and breakup are believed to have significant impacts, as yield stress fluids are widely encountered in industrial applications.

Numerical Modeling of Non-Isothermal Laminar Flow and Heat Transfer of Paraffinic Oil with Yield Stress in a Pipe

This paper presents the results of a study on the non-isothermal laminar flow and heat transfer of oil with Newtonian and viscoplastic rheologies. Heat exchange with the surrounding environment leads to the formation of a near-wall zone of viscoplastic fluid. As the flow proceeds, the transformation of a Newtonian fluid to a viscoplastic state occurs. The rheology of the Shvedoff–Bingham fluid as a function of temperature is represented by the effective molecular viscosity apparatus. A numerical solution to the system of equations of motion and heat transfer was obtained using the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm. The calculated data are obtained at Reynolds number Re from 523 to 1046, Bingham number Bn from 8.51 to 411.16, and Prandl number Pr = 45. The calculations’ novelty lies in the appearance of a “stagnation zone” in the near-wall zone and the pipe cross-section narrowing. The near-wall “stagnation zone” is along the pipe’s radius from r/R = 0.475 to r/R = 1 at Re = 523, Bn = 411.16, Pr = 45, u1 = 0.10 m/s, t1 = 25 °C, and tw = 0 °C. The influence of the heat of phase transition of paraffinic oil on the development of flow and heat transfer characteristics along the pipe length is demonstrated.

Entropy production in the flow of Bingham-Papanastasiou fluid having temperature and shear rate-dependent viscosity

This article comprehensively examines entropy production in a peristaltic flow of Bingham-Papanastasiou fluid in an asymmetric wavy wall channel. The viscosity of the fluid is selected to depend on the temperature in addition to the shear rate. Furthermore, viscous dissipation is considered. The equations governing momentum and energy are adjusted to incorporate the existence of a nonlinear correlation between stress and strain rate. The energy dissipation caused by viscosity in the energy equation exhibits variation between Newtonian and non-Newtonian fluids due to their distinct rheological behaviors and response to shear forces. Newton’s law of viscosity states that the viscous dissipation term for Newtonian fluids follows a simple linear relationship between shear stress and shear rate. In contrast, non-Newtonian fluids exhibit more complex rheological behaviors, with the correlation between shear stress and shear rate described by various constitutive equations, depending on the specific type of non-Newtonian behavior. Consequently, the energy equation for non-Newtonian fluid flow problems becomes more intricate and nonlinear compared to the energy equation for Newtonian fluids, resulting in a more challenging analysis. Modeling of the equations is traditionally obtained using all steps involved in analyzing peristaltic flow. Using MATLAB's bvp5c solver, the obtained coupled set of differential equations is numerically solved. The research explores how the Bingham number and stress growth exponent parameters affect different dimensionless variables, including velocity, streamlines, temperature, axial pressure gradient, entropy production, and the Bejan number.

A comprehensive model for viscoplastic flows in channels with a patterned wall: longitudinal, transverse and oblique flows

We develop a comprehensive model for the creeping Poiseuille Bingham flow in channels equipped with a patterned wall, i.e. decorated with grooves or stripes that may represent a superhydrophobic (SH) or a chemically patterned (CP) surface, respectively, with longitudinal, transverse and oblique groove (stripe) orientations with respect to the applied pressure gradient. We rely on the Navier slip law to model the boundary condition on the slippery grooves. We develop semi-analytical, explicit-form and complementary computational fluid dynamics models, with solutions that have reasonable agreement. In contrast to its Newtonian analogue, a distinct solution for the oblique configuration, with an a priori unknown transform matrix, must be developed due to the viscoplastic nonlinear rheology. Our focus is to systematically analyse the effects of the Bingham number ( $B$ ), slip number ( $b$ ), groove periodicity length ( $\ell$ ), slip area fraction ( $\varphi$ ) and groove orientation angle ( $\theta$ ), on the slip velocities, effective slip length ( $\chi$ ), slip angle difference ( $\theta -s$ ), mixing index ( $I_M$ ), flow anisotropy and flow regimes. In particular, we demonstrate that, as $B$ increases, the maximum values of the shear component of $\chi$ , $\theta -s$ and $I_M$ occur progressively at smaller values of $\theta$ , compared with their Newtonian counterparts.

Heat transfer analysis for magnetohydrodynamic peristalsis of Reiner–Philippoff fluid: Application of an artificial neural network

Present communication explores a novel application of the computational intelligence technique, namely, the Levenberg–Marquardt scheme under a Backpropagated Neural Network (LM-BNN) to solve the mathematical model for the magnetohydrodynamic peristaltic transport of Reiner–Philippoff (R–Ph) pseudoplastic fluid considering the influences of Ohmic heating, mixed convection, and viscous dissipation through a symmetric channel. The R–Ph fluid model is used in this investigation to elucidate the non-Newtonian behavior of the fluid under consideration. The Reiner–Philippoff fluid model delineates the intricate relationship between stress and deformation rate within the fluid. There are a few studies available on the peristaltic transport of the Reiner–Philippoff fluid that do not incorporate Joule heating, mixed convection, and magnetic field effects. Therefore, a novel mathematical model is developed to employ an artificial neural network technique with a different approach that has not been examined before. The governing equations of the problem are simplified using long wavelength and low Reynolds number approximations, and the resulting system is numerically solved using the BVP4c scheme in MATLAB based on the shooting algorithm. Furthermore, a dataset is constructed through the BVP4c technique for the proposed LM-BNN, considering eight scenarios of peristaltic motion of the Reiner–Philippoff fluid model by varying the Bingham number, the Brinkman number, the Grashof number, the R–Ph fluid parameter, and the Hartmann number. The numerical dataset is divided into 15% for testing, 15% for training, and 70% for validation, which are utilized in LM-BNN to analyze the numerical solutions and Levenberg–Marquardt neural networks (LM-NNs) predicted results. The consistency and effectiveness of LM-BNN are validated through regression analysis, stresses at the wall, error histogram, correlation index, heat transfer, and mean squared error based fitness curves, which vary from 10−3→10−11. Variations in several flow parameters affecting temperature and velocity profiles are explained physically through graphs. Additionally, an analysis of heat transfer and stresses at the wall, including absolute errors, is provided through tables. The outcomes reveal that the improving Grashof number and the Hartmann number tend to increase the temperature profile. Tabular results indicate that rates of heat transfer improve when assigning higher values to the Hartmann number, the Brinkman number, and the Grashof number, whereas stresses at the wall decrease for the Reiner–Philippoff fluid parameter and the Bingham number. The error analysis of numerical simulations is a valuable step in determining whether the data obtained are reliable and accurate. In terms of absolute error, the disagreement between numerical solutions and those predicted by LM-NNs is approximately 10−5→10−11. It is clear from this error analysis that LM-NNs predicted results are consistent and reliable.

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.

Effect of the Magnetic Field and Rotation on Peristaltic Flow of A Bingham Fluid in Asymmetric Channel with Porous Medium

In this paper, we investigate the influence of the rotation and magnetic field on the peristaltic flow of the Bingham fluid in an asymmetric channel with a porous medium under the long wavelength and low Reynolds number approximation assumptions. The perturbation method and the Mathematica program for solving nonlinear partial differential equations are used to couple the momentum equations with the rotational and magnetic field equations. The fluid is considered to be subject to a magnetic field and to flow within a porous medium. Graphs are used to display expressions for speed, stress gradient, magnetic subject, current density, rotation impact, and drift function. The findings reveal that the rotation, density, permeability, coupling diversity, and non-dimensional wave amplitude all play significant roles in the phenomenon. The quantities flow has been tested for variant parameters. The impact of the Bingham, Hartman and Darcy numbers are also tested for different values to indicate the effect on the movement of flow fluid. The applications can be seen through many graphics.