Power-law Fluid Model Research Articles

Design of a capillary viscometer with numerical and computational methods

A high temperature and shear rate capillary viscometer has been designed, constructed and recently commissioned. This device will be used to measure the viscosity of semi-solid metals under the high temperature and shear rate conditions, similar to those found in industry. Design criteria for the device included a requirement for a highly controllable temperature (±1°C) up to 650°C, capability for injection shear rates above 10,000s−1 and controllable injection profiles. The design of this viscometer was aided with the use of numerical modelling methods based on a power law thixotropic fluid flow relation. This analysis allowed calculation of required injection speeds and expected system forces. Computational modelling work, based on current power law fluid models, was also performed in order to investigate how the viscosity would be expected to fluctuate with shear rate and fraction solid. This data could then be used to compare with experimental work. The computational model was a 2D two-phase theoretical unsteady state model. This was used to evaluate the viscosity of semi-solid metals passing through the designed capillary viscometer at injection speeds of 0.075, 0.5 and 1 m/sec. The effects of fractions solid (fs) of the metal from 0.25, 0.3, 0.33 and 0.50 were also investigated. Strong correlations between these parameters and the resulting viscosity were noted.

Theoretical Analysis of Turbulent Flow of Power-Law Fluids in Coiled Tubing

Summary A comprehensive theoretical analysis of turbulent flow of a power-law fluid in coiled tubing is conducted with the approach of boundary layer approximation. Equations of momentum integrals for the boundary layer flow are derived and solved numerically. Based on the results of the numerical analysis, a new friction-factor correlation is developed which is applicable to a wide range of flow behavior index of power-law fluid model. The new correlation is verified by comparing it with the published Ito correlation for the special case of Newtonian fluid. For non-Newtonian fluids, there is also a close agreement between the new correlation and the experimental data from recent full-scale coiled tubing flow experiments.

Attenuation of seismic waves and the universal rheological model of the Earth’s mantle

Analysis of results of laboratory studies on creep of mantle rocks, data on seismic wave attenuation in the mantle, and rheological micromechanisms shows that the universal, i.e., relevant to all time scales, rheological model of the mantle can be represented as four rheological elements connected in series. These elements account for elasticity, diffusion rheology, high temperature dislocation rheology, and low temperature dislocation rheology. The diffusion rheology element is described in terms of a Newtonian viscous fluid. The high temperature dislocation rheology element is described by the rheological model previously proposed by the author. This model is a combination of a power-law non-Newtonian fluid model for stationary flows and the linear hereditary Andrade model for flows associated with small strains. The low temperature dislocation rheology element is described by the linear hereditary Lomnitz model.

Theoretical analysis of laminar flow of power‐law fluids in coiled tubing

AbstractA comprehensive theoretical study of the flow of power‐law fluid in coiled tubing by using the boundary layer approximation method is presented. First, a boundary layer approximation analysis was applied to the governing equations of continuity and motion of a power law fluid in the curved tubing flow geometry. Momentum integral equations for the boundary layer flow were then derived and solved numerically. The resulting solutions of the velocity field were used to develop a new friction factor correlation in terms of generalized Dean number, coiled tubing curvature ratio, and flow behavior index of the power law fluid model. The new correlation of this study and a previous correlation by Mashelkar and Devarajan were evaluated using experimental data obtained from full‐scale coiled tubing flow experiments. An excellent agreement was found between the new correlation and the experimental results. The Mashelkar and Devarajan correlation did not result in any acceptable agreement with the experimental data, nor did it match the Ito correlation for the limiting case of Newtonian fluids (n = 1). This work extends the range of applicability of the new correlation to fluids with flow behavior indices as low as 0.25, which would cover most fluids used for coiled tubing operations in the oil and gas industry as well as fluids used in other industries. © 2007 American Institute of Chemical Engineers AIChE J, 2007

An ALE based iterative CBS algorithm for non-isothermal non-Newtonian flow with adaptive coupled finite element and meshfree method

This paper presents a generalized scheme of the fractional step algorithm using Characteristic Based Split (CBS) in ALE framework for the numerical simulation of incompressible non-isothermal non-Newtonian fluid flows and applies it, combined with the adaptive coupled finite element and meshfree method for spatial discretizations, to the polymer injection molding process. Based on the idea to discretize the material time derivative instead of the spatial time derivative, the characteristic Galerkin method is derived in a more general way and is used to discretize the non-Newtonian momentum and the energy conservation equations. An iterative procedure which makes the conservation equations satisfied in an implicit sense is introduced into the scheme to enhance its stability in the time domain. Numerical experiments with the power-law fluid model demonstrate and validate the performance and the capability of the proposed scheme.

Effects of non-Newtonian rheology on the film-height history between non-parallel sliding-squeezing surfaces

The effects of flow rheology of non-Newtonian fluids on the time-dependent film-thickness history between non-parallel sliding-squeezing surfaces are studied. On the basis of the power-law fluid model, the hydrodynamic film pressure is obtained from the non-Newtonian dynamic Reynolds-type equation by considering the transient squeezing-action effect. The hydrodynamic film force is then applied to derive the non-linear motion equation of the squeezing part. Using the non-linear transient method, the film-height trajectory of the squeezing part is presented. The results of the present study provide a reference for engineer applications in machine tools, the matched gears, the rolling elements, and the piston-cylinder systems when viscosity-pressure dependence, viscosity-temperature variation, and elastic deformations of the contacting surfaces are neglected.

Pressure-driven laminar flow of a non-Newtonian fluid in a slit with wall suction or injection

The one-dimensional approximate equation in the rectangular Cartesian coordinates governing pressure-driven flow of a non-Newtonian fluid in a thin slit whose walls are porous, with the lower plate is stationary while the upper plate is moving with the constant velocity in the x-direction, is derived by accounting of the order of magnitudes of terms as well as the accompanying approximations to the full-blown three-dimensional equations by using scaling arguments, asymptotic techniques and assuming the cross-flow velocity is much less than the axial velocity. The non-Newtonian fluid behavior is described by power-law fluid model and the one-dimensional governing equation for a power-law fluid flow under the influence of pressure gradient in the narrow gap between two closely spaced parallel porous plates, with the lower plate is stationary while the top plate is exposed to sudden acceleration with a constant velocity in the x-direction, is solved analytically for a Newtonian fluid case ( n = 1) and numerically for various values of power-law index to determine the transient and thus the overall transient velocity distributions. The effects of the transverse mass suction/injection rates at the bottom porous plate on the pressure-driven flows of non-Newtonian fluids are examined for various values of time, dynamic pressure drops in the axial direction and power-law indexes. The results obtained from the present analysis are compared with the data available in the literature.

An iterative stabilized CNBS–CG scheme for incompressible non-isothermal non-Newtonian fluid flow

An iterative stabilized fractional step scheme abbreviated as I-CNBS–CG is developed, in which the Crank–Nicolson method based split (CNBS) scheme and the characteristic-Galerkin (CG) method are, respectively, used to discretize and solve the non-Newtonian momentum–mass conservation equations and the energy conservation equation in consideration of their convective character. Owing to introduction of an iterative procedure into the scheme the stability of the proposed scheme in time domain is greatly enhanced and much larger time step sizes are allowed to be used than those limited in existing explicit and semi-implicit ones. The proposed I-CNBS–CG scheme particularly suits to numerically model the non-isothermal non-Newtonian fluid flows with moderate or high viscosity and low thermal conductivity, such as molten polymer flow process in a mould cavity. Numerical experiments with the power-law fluid model demonstrate the improved performances of the proposed scheme.

Non-Darcy natural convection of a non-Newtonian fluid in a porous cavity

A numerical study of non-Darcy natural convection in a porous enclosure saturated with a power-law fluid is presented. Hydrodynamic and heat transfer results are reported for the configuration in which the enclosure is heated from a side-wall while the horizontal walls are insulated. The flow in the porous medium is modeled using the modified Brinkman–Forchheimer-extended Darcy model for power-law fluids, which accounts for both inertia and boundary effects. The results indicate that when the power law index is decreased, the circulation within the enclosure increases leading to a higher Nusselt number and these effects are enhanced as the Darcy number is increased. Consequently as the power law index decreases, the onset of the transitions from Darcy regime to Darcy–Forchheimer–Brinkman regime to asymptotic convection (boundary layer) regime shift to higher corresponding values of the Darcy number. An increase in Rayleigh number produces similar effects as a decrease in power law index.

A FRACTAL ANALYSIS OF PERMEABILITY FOR POWER-LAW FLUIDS IN POROUS MEDIA

A fractal analysis of permeability for power-law fluids in porous media is presented based on the fractal characters of pore size distributions and tortuous flow paths/streamlines in the media. The proposed permeability model for power-law fluids in porous media is expressed as a function of the fractal dimensions of pore size distributions and tortuous flow paths/streamlines, porosity and microstructural parameters, as well as power exponent, and there is no empirical constant in the proposed model and every parameter in the model has clear physical meaning. The results predicted by the present fractal permeability model show that the model predictions (as the power exponent is 1) are in agreement with the available experimental data, and the predicted permeabilities (as the power exponent is not equal to 1) increase with the power exponent, which is also consistent with the physical situation.

Numerical simulation of natural convection of latent heat phase-change-material microcapsulate slurry packed in a horizontal rectangular enclosure heated from below and cooled from above

A two-dimensional numerical simulation of natural convection in a rectangular enclosure heated from below and cooled from above has been conducted with non-Newtonian phase-change-material (PCM) microcapsulate slurry with latent heat capacities. The formulation of the mathematical model in dimensionless co-ordinates and discretization of the governing equations have been done using the finite volume method. Both natural convection and heat transfer characteristics are discussed about natural convection with PCM microcapsulate slurry, which exhibits the pseudoplastic non-Newtonian fluid behavior and a peak value in the specific heat capacity with latent heat. The viscosity of the present PCM microcapsulate slurry is assumed to follow the Ostwald-de Waele power law fluid model with the power-law index n and the consistency coefficient K. The effects of phase-change material, the mass concentration, and the aspect ratio Ar on the natural convection heat transfer are described, respectively. By comparing with the results of microcapsule slurry without phase change, the enhancement in heat transfer is found in microcapsule slurry with phase change during the phase change temperature range. Numerical simulations are performed in the following parametric ranges: the width–height aspect ratio of the enclosure Ar from 2 to 20, the mass concentrations Cm of the slurry from 10 to 40%, power law index n of the slurry from 0.89 to 1.0 and Rayleigh numbers Ra ranges from 103 to 107.

Flow Behavior of Herschel-Bulkley Fluid in a Slot Die

To achieve uniform thickness of coated films using non-Newtonian fluids, we are often required well-designed geometries of flow channel in a slot die. This is the case especially for fluids with the properties of both shear thinning and yield stress which is non-negligible in a range of low shear rate such as in an intermittent die coating. In the present work, flow in the slot die has been studied using Herschel-Bulkley fluid expressing both shear thinning and yield stress. The Herschel-Bulkley fluid includes power-law and Bingham fluid models comprehensively. This fluid model is applicable to many real fluids. We derive simplified equations for the outflow distribution as functions of power-law index, yield stress and the geometry parameters of the die. Furthermore, we propose a useful designing method for predicting the optimum geometry so as to guarantee uniformity of outflow from the die slot, based on the Herschel-Bulkley fluid. The value of this method is confirmed by comparing it with experimental results and with previous theoretical results for the power-law and Bingham fluid models.

Melt Spinning Instability and Transient Disturbance Response for Power Law Fluid Model

Melt Spinning Instability and Transient Disturbance Response for Power Law Fluid Model

Viscoelastic Constitutive Model of Unvulcanized Rubber

Based on rheological test results, a new viscoelastic constitutive equation for unvulcanized rubber has been set up, with mathematical justification to describe its mechanical properties in relation to the yield stress and shear-thinning effect. In this model, every term or coefficient has an explicit physical meaning. The proposed model indicates that the yield stress is one of the main causes for the shear-thinning effect and reveals why some materials possess double-Newtonian regions with the shear viscosity in the first region higher than that in the second region. The yield stress makes the flow index of the power law fluid model vary widely, so that it needs to be eliminated from the power law fluid model. The model can also distinguish the true shear viscosity from the apparent shear viscosity effectively. The parameters of the equation are determined by the step fitting method, which is the precondition for quantitative analysis. However, the equation is a one-dimensional model, so further research is needed.

Roller extrusion of biscuit doughs

Biscuit doughs are dense solid–liquid pastes which exhibit complex rheological behaviour. The rolling behaviour of sheets of commercial short and hard biscuit doughs was investigated using an instrumented counter-rotating roll mill, which allowed the roll torque, separating force and surface pressure to be monitored. The rheological characteristics of the two doughs were quantified by analysing data from ram extrusion experiments in terms of a quasi-plastic model (following the Benbow–Bridgwater approach) and a power law fluid model. The results indicated that the doughs were not ideally suited to the quasi-plastic analysis. The power law parameters varied noticeably between the doughs but both were strongly shear-thinning (power law shear indices of 0.25 and 0.5), with large extensional viscosities. These rheological model parameters were subsequently used to generate predictions of dough behaviour during rolling. A standard sheet metal plasticity model was modified to include strain rate dependency but neither dough’s behaviour could be adequately described by this model. The power law fluid approach based on the lubrication assumption reported by Levine tended to underpredict the work requirement owing to the significant contribution from extensional deformation. Interestingly, the system could be modelled in terms of an apparent power law rheology by fitting data to Levine’s model. Better agreement was obtained for some parameters but not consistently so, indicating that the power law approach is not adequate for these soft-solid materials.

Optimization-based design of polymer sheeting dies using generalized Newtonian fluid models

A polymer sheeting die design methodology is presented, which integrates finite element flow simulations, numerical optimization, and design sensitivity analyses to compute die cavity geometries capable of giving a near-uniform exit velocity. This work extends earlier die design methods to include generalized Newtonian fluid (GNF) models that represent the shear-thinning behavior of polymer melt. Melt flow computations and design sensitivity analyses are provided using the generalized Hele-Shaw flow approximation with isothermal power-law, Carreau-Yasuda, Cross, Ellis, and Bingham fluid models. The nonlinear equations for die cavity pressure are solved using the Newton-Raphson iteration method and design sensitivities are derived with the adjoint variable method. The die design method is applied to an industrial coat hanger die, in which a design parameterization is defined that allows for an arbitrary gap height distribution in the manifold of the die. In addition, die performance is assessed and compared for power-law and Carreau-Yasuda fluid flow over a range of die operating conditions. Pareto optimal die designs are also considered in this study. POLYM. ENG. SCI., 45:953–965, 2005. © 2005 Society of Plastics Engineers

Flows of inelastic non-Newtonian fluids through arrays of aligned cylinders. Part 2. Inertial effects for square arrays

Numerical simulations are presented for flows of inelastic non-Newtonian fluids through periodic arrays of aligned cylinders. The truncated power-law fluid model is used for the relationship between the viscous stress and the rate-of-strain tensor. Results for the drag coefficient for creeping flows of such fluids have been presented in a companion paper [1]. In this second part the effects of finite fluid inertia are investigated for flows through square arrays. It is shown that the Reynolds-number dependence of the drag coefficient of a cylinder in the array is of the form C d≡ F/(ηU) = k 0 + k 2 Re2+ .. for small values of the Reynolds number Re ≡ ρaU/η, where F is the drag force, U is the averaged velocity in the array, η = K (U/a)n-1 is a viscosity scale with K and n the power-law coefficient and index and a the cylinder radius, and k 0 is the drag coefficient for creeping flows. The proportionality constant k 2 depends on the way the drag coefficient and the Reynolds number are defined. It is shown that the observed strong dependence of k 2 on n can almost be eliminated by using length scales different from a in the viscosity scales η used in the definition of Re and in the definition of the drag coefficient. Numerical simulation results are also presented for the velocity variance components. Results for flows at moderate Reynolds number, of order 100, are also presented; these are qualitatively similar to those for Newtonian fluids. The value of the Reynolds number beyond which the flow becomes unsteady was related to the Newtonian fluid case by rescaling. These results for moderate-Reynolds-number flow are compared against previously published experimental data.

Flows of inelastic non-Newtonian fluids through arrays of aligned cylinders. Part 1. Creeping flows

Numerical simulations are presented for flows of inelastic non-Newtonian fluids through periodic arrays of aligned cylinders, for cases in which fluid inertia can be neglected. The truncated power-law fluid model is used to define the relationship between the viscous stress and the rate-of-strain tensor. The flow is shown to be dominated by shear effects, not extension. Numerical simulation results are presented for the drag coefficient of the cylinders and the velocity variance components, and are compared against asymptotically valid analytical results. Square and hexagonal arrays are considered, both for crossflow in the plane perpendicular to the alignment vector of the cylinders (flows along the axes of the array as well as off-axis flows), and for flow along the cylinders. It is shown that the observed strong dependence of the drag coefficient on the power-law index (through which the stress tensor is related to the rate-of-strain tensor) can be described at all solid area fractions by scaling the drag on a cylinder with appropriate velocity and length scales. The velocity variance components show only a weak dependence on the power-law index. The results for steady-state drag and velocity variances are used in an approximate theory for flows accelerated from rest. The numerical simulation data for unsteady flows agree very well with the approximate theory.

Transient buoyant convection of a power-law non-Newtonian fluid in an enclosure

Transient buoyant convection in a square enclosure of a non-Newtonian fluid is studied. A simple power-law fluid model, with the power-law index n and the consistency coefficient K, is adopted. Flow is initiated from the motionless isothermal initial state by abruptly raising the temperature at one vertical sidewall and lowering the temperature at the opposite vertical sidewall. The appropriately defined overall Rayleigh number Ra is large to render a boundary layer-type flow. A scale analysis is performed to gain a rudimentary understanding of the evolution process. Principal force balances are considered in each of the transient stages. Comprehensive numerical solutions are acquired to the governing equations. The transient flow and thermal characteristics, both in the boundary layers and in the interior, are portrayed. The effects of Ra and of n on the behavior of the Nusselt number are delineated. The system-wide heat transport at the steady state is calculated. Based on the numerical results, the Nusselt number correlations are proposed. The numerical solutions are in broad qualitative agreement with the descriptions obtainable from the scale analysis.

TEHD Behavior of Non-Newtonian Dynamically Loaded Journal Bearings in Mixed Lubrication for Direct Problem

A transient non-Newtonian TEHD analysis for dynamically loaded journal bearings in mixed lubrication is developed for direct problem. A mass conserving cavitation is included and viscoelasticity subsuming shear thinning is characterized by the upper converted Maxwell fluid and the power law fluid models. The Reynolds equation, the transient two-dimensional energy equation in the film, the quasi-steady-state two-dimensional heat conduction equation in the bushing and heat balance in the journal are solved using finite difference formulations. The thermal and elastic deformation equations are solved using the finite element method. The relation between the average contact pressure and the average gap is numerically determined in micro scale. Effects of roughness texture, degree of asperity contact, shear index, relaxation time, thermal and elastic deformations on bearing behavior are investigated.