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The purpose of this paper is to investigate the combined effects of slip and rheological parameters on the flow and heat transfer of the Herschel-Bulkley fluid.

The combinative dimensionless parameter method is introduced into the equations of the slip flow and heat transfer to make the discussion more comprehensive. More specifically, the slip and rheological parameters are transformed into the dimensionless slip number as well as Herschel-Bulkley number. We solve the dimensionless equations and then focus on the effects of these parameters on the slip flow and heat transfer.

The results show that, for a given value of Herschel-Bulkley number, there is a finite critical value of slip number at which the pressure gradient reaches the lowest value and both the dimensionless yield radius and slip velocity become 1. Meanwhile, the Nusselt number tends to be infinite at this critical value of slip number. For the case of slip, the Nusselt number also approaches infinity at a finite critical value of Herschel-Bulkley number. Furthermore, the dimensionless velocity as well as temperature of the yield pseudoplastic fluid with higher slip number is lower within a small radius but becomes higher near the wall. Meanwhile, from the velocity and temperature profiles, the effect of Herschel-Bulkley number on these two parameters of the Bingham fluid at the smaller radius is opposite.

These associated expressions can be generalized to the flow and heat transfer of a Herschel-Bulkley fluid under slip boundary condition. It can provide a reference for the engineering application relating to the heat transfer and flow of a Herschel-Bulkley fluid. Meanwhile, it also suggests some revelations for dealing with this similar problem.

The purpose of this paper is to analyze two-dimensional steady-state Rayleigh–Bénard convection within rectangular enclosures in different aspect ratios filled with yield stress fluids obeying the Herschel–Bulkley model.

In this study, a numerical method based on the finite element has been developed for analyzing two-dimensional natural convection of a Herschel–Bulkley fluid. The effects of Bingham number Bn and power law index n on heat and momentum transport have been investigated for a nominal Rayleigh number range (5 × 10³ < Ra < 10⁵), three different aspect ratios (ratio of enclosure length:height AR = 1, 2, 3) and a single representative value of nominal Prandtl number (Pr = 10).

Results show that the mean Nusselt number Nu¯ increases with increasing Rayleigh number due to strengthening of convective transport. However, with the same nominal value of Ra, the values of Nu¯ for shear thinning fluids n < 1 are greater than shear thickening fluids n > 1. The values of Nu¯ decrease with Bingham number and for large values of Bn, Nu¯ rapidly approaches unity, which indicates that heat transfer takes place principally by thermal conduction. The effects of aspect ratios have also been investigated and results show that Nu¯ increases with increasing AR due to stronger convection effects.

This paper presents a numerical study of Rayleigh–Bérnard flows involving Herschel–Bulkley fluids for a wide range of Rayleigh numbers, Bingham numbers and power law index based on finite element method. The effects of aspect ratio on flow and heat transfer of Herschel–Bulkley fluids are also studied.

This paper aims to investigate the flow of two-layered non-Newtonian fluids with different viscosities in an axisymmetric elastic tube.

A mathematical model was considered for this study to describe the flow characteristics of two-layered non- Newtonian Jeffrey fluids in an elastic tube. Because Jeffrey fluid model is a better model for the description of physiological fluid motion. Further, this model is a significant generalization of Newtonian fluid model. Analytical expressions for flux, stream functions, velocities and interface velocity have been derived in terms of elastic parameters, inlet, outlet and external pressures. The effects of various pertinent parameters on the flow behavior have been studied.

The volumetric flow rate was calculated by different models of Mazumdar (1992) and Rubinow and Keller (1972); from this it was found that the flux of Jeffrey fluid is more in the case of Rubinow and Keller model than Mazumdar. A comparative study is made between single-fluid and two-fluid models of Jeffrey fluid flows and it was observed that more flux and higher velocities were observed in the case of two-fluid model rather than single-fluid model. Furthermore, when both the Jeffrey parameter tends to zero and ratios of viscosities and radii are unity, the results in this study agree with those of Rubinow and Keller (1972).

To describe the fluid flow in an elastic tube with two-layered systems, the models and solutions developed here are very important. These results will be highly suitable in analyzing the rheological characteristics of blood flow in a small blood vessel because of their elastic nature.

The screw extruder is applied in cement-three-dimensional (3D) printing. The cement paste flow in 3D printing is the typical Herschel–Bulkley fluid. To understand the flow in the channel, the improved lattice Boltzmann method (LBM) is proposed.

For Herschel–Bulkley flow, an improved LBM is presented to avoid the poor stability and accuracy. The non-Newtonian effect is regard as a special forcing term. The Poiseuille flow is taken to discuss the detailed process of the method. With the method, the analytical solution and numerical solution are obtained and compared. Then, the effect of the initial yield stress on the numerical solution is both explored by the shear-thickening fluid and the shear-thinning fluid. Moreover, the variations of the relative errors under different lattice nodes and different power-law indexes are analyzed. Finally, the method is applied into the simulation of the flow in the extruder of cement-3D printing.

The results show that the improved method is effective for Herschel–Bulkley fluids, which can simulate the flow in the extruder stably and accurately.

The simulation can contribute to understand the cement paste flow in the screw extruder, which helps to optimize the structure of the extruder in the following periods.

The improve method provide a new way to analyze the flow in the extruder of cement-3D printing. Also, in the past research, LBM for Herschel–Bulkley fluid is ignored, whereas the study can provide the reference for the numerical simulation.

This paper is concerned with the peristaltic transport of an incompressible non-Newtonian fluid in a porous elastic tube. The impacts of slip and heat transfer on the Herschel-Bulkley fluid are considered. The impacts of relevant parameters on flow rate and temperature are examined graphically. The examination incorporates Newtonian, Power-law and Bingham plastic fluids. The paper aims to discuss these issues.

The administering equations are solved utilizing long wavelength and low Reynolds number approximations, and exact solutions are acquired for velocity, temperature, flux and stream functions.

It is seen that the flow rate in a Newtonian fluid is high when contrasted with the Herschel-Bulkley model, and the inlet elastic radius and outlet elastic radius have opposite effects on the flow rate.

The analysis carried out in this paper is about the peristaltic transport of an incompressible non-Newtonian fluid in a porous elastic tube. The impact of slip and heat transfer on a Herschel-Bulkley fluid is taken into account. The impacts of relevant parameters on the flow rate and temperature are examined graphically. The examination incorporates Newtonian, Power-law and Bingham plastic fluids.

This paper sets out to present a fully explicit smoothed particle hydrodynamics (SPH) method to solve non‐Newtonian fluid flow problems.

The governing equations are momentum equations along with the continuity equation which are described in a Lagrangian framework. A new treatment similar to that used in Eulerian formulations is applied to viscous terms, which facilitates the implementation of various inelastic non‐Newtonian models. This approach utilizes the exact forms of the shear strain rate tensor and its second principal invariant to calculate the shear stress tensor. Three constitutive laws including power‐law, Bingham‐plastic and Herschel‐Bulkley models are studied in this work. The imposition of the incompressibility is fulfilled using a penalty‐like formulation which creates a trade‐off between the pressure and density variations. Solid walls are simulated by the boundary particles whose positions are fixed but contribute to the field variables in the same way as the fluid particles in flow field.

The performance of the proposed algorithm is assessed by solving three test cases including a non‐Newtonian dam‐break problem, flow in an annular viscometer using the aforementioned models and a mud fluid flow on a sloping bed under an overlying water. The results obtained by the proposed SPH algorithm are in close agreement with the available experimental and/or numerical data.

In this work, only inelastic non‐Newtonian models are studied. This paper deals with 2D problems, although extension of the proposed scheme to 3D is straightforward.

This study shows that various types of flow problems involving fluid‐solid and fluid‐fluid interfaces can be solved using the proposed SPH method.

Using the proposed numerical treatment of viscous terms, a unified and consistent approach was devised to study various non‐Newtonian flow models.

This paper aims to investigate the flow characteristics of lubricating grease in extremely cold weather in which it is difficult to convey the grease due to a huge pressure drop.

The rheological behavior of grease at various temperatures is studied by a rotary rheometer to determine the constitutive equation of lubricating grease. Based on the Herschel–Bulkley (H–B) model, the flow pattern of grease is then simulated by computational fluid dynamics and compared with the test results.

The yield stress increased dramatically when the shear rate was less than 1s⁻¹ in the rheological experiments of continuous shear mode, and the phenomenon was more significant with the decrease in temperature. The rheological data obtained from the continuous shear mode agrees with the H–B equation after the shear thinned. In extremely cold conditions, there is a large yield stress in the lubricating grease; the numerical results show that the viscosity of lubricating grease increased with an increase in temperature, and the viscosity and velocity of lubricating grease showed uneven distribution leading to difficulty of lubricating grease delivery.

This paper focuses on the flow characteristics of lubricating grease in extremely cold area conditions which is studied rarely. In addition, the continuous shear model and oscillatory model are combined to establish the constitutive equations. Experiment and numerical simulation method are all used by establishing the H–B models.

The purpose of this paper is to analyze the effects of complex boundary conditions on natural convection of a yield stress fluid in a square enclosure heated from below (uniformly and non-uniformly) and symmetrically cooled from the sides.

The governing equations are solved numerically subject to continuous and discontinuous Dirichlet boundary conditions by Galerkin’s weighted residuals scheme of finite element method and using a non-uniform unstructured triangular grid.

Results show that the overall heat transfer from the heated wall decreases in the case of non-uniform heating for both Newtonian and yield stress fluids. It is found that the effect of yield stress on heat transfer is almost similar in both uniform and non-uniform heating cases. The yield stress has a stabilizing effect, reducing the convection intensity in both cases. Above a certain value of yield number Y, heat transfer is only due to conduction. It is found that a transition of different modes of stability may occur as Rayleigh number changes; this fact gives rise to a discontinuity in the variation of critical yield number.

Besides the new numerical method based on the finite element and using a non-uniform unstructured grid for analyzing natural convection of viscoplastic materials with complex boundary conditions, the originality of the present work concerns the treatment of the yield stress fluids under the influence of complex boundary conditions.

This paper aims to use a fractional constitutive model with a nonlocal velocity gradient for replacing the nonlinear constitutive model to characterize its complex rheological behavior, where non-linear characteristics exist, for example, the inherent viscous behavior of the crude oil. The feasibility and flexibility of the fractional model are tested via a case study of non-Newtonian fluid. The finite element method is non-Newtonian used to numerically solve both momentum equation and energy equation to describe the fluid flow and convection heat transfer process.

This paper provides a comprehensive theoretical and numerical study of flow and heat transfer of non-Newtonian fluids in a pipe based on the fractional constitutive model. Contrary to fractional order a, the rheological property of non-Newtonian fluid changes from shear-thinning to shear-thickening with the increase of power-law index n, therefore the flow and heat transfer are hindered to some extent.

This paper discusses two dimensionless parameters on flow regime and thermal patterns, including Reynolds number (Re) and Nusselt number (Nu) in evaluating the flow rate and heat transfer rate. Analysis results show that the viscosity of the non-Newtonian fluid decreases with the rheological index (order α) increasing. While large fractional (order α) corresponds to the enhancement of heat transfer capacity.

First, it is observed that the increase of the Re results in an increase of the local Nusselt number (Nul). It means the heat transfer enhancement ratio increases with Re. Meanwhile, the increasement of the Nul indicating the enhancement in the heat transfer coefficient, produces a higher speed flow of crude oil.

This study presents a new numerical investigation on characteristics of steady-state pipe flow and forced convection heat transfer by using a fractional constitutive model. The influences of various non-dimensional characteristic parameters of fluid on the velocity and temperature fields are analyzed in detail.

The purpose of this paper is to obtain a constitutive model of cement-based material in the rheological stage, which owns the different water-cement ratio (w/c) and temperature and have a significant impact on the workability of concrete materials.

It is introduced a modified Arrhenius equation into the Herschel–Bulkley model, which is widely applied in rheological analysis and constructed an ordinary differential equation (ODE) of w/c from the Navier–Stokes equation. By solving the ODE, an approximate constitutive relation of cement-based materials included w/c and temperature is derived. Compared with the experimental results, the present model is validated.

The shear stress and shear rate curves with different w/c and temperature are simulated by the present method, and the present model can be applied to analyze the changes of apparent viscosity in cement-based material slurry as the w/c and temperature varying.

This work gives a mathematical model, which can effectively approximate the shear stress–shear rate relation with different w/c and temperature in the rheological stage of cement-based material.