Flow visualization in rheometry

Electroosmotic flow of Power-law fluids over a surface with arbitrary zeta potentials is analyzed. The governing equations including the nonlinear Poisson-Boltzmann equation, the Cauchy momentum equation and the continuity equation are solved to seek exact solutions for the electroosmotic velocity, shear stress, and dynamic viscosity distributions inside the electric double layer. Specifically, an expression for the general Smoluchowski velocity is obtained for electroosmosis of Power-law fluids in a fashion similar to the classic Smoluchowski velocity for Newtonian fluids. The existing Smoluchowski slip velocities under two special cases, (i) for Newtonian fluids with arbitrary zeta potentials and (ii) for Power-law fluids with small zeta potentials, can be recovered from our derived formula. It is interesting to note that the general Smoluchowski velocity for non-Newtonian Power-law fluids is a nonlinear function of the electric field strength and surface zeta potentials; this is due to the coupling electrostatics and non-Newtonian fluid behavior, which is different from its counterpart for Newtonian fluids. This general Smoluchowski velocity is of practical significance in determining the flow rates in microfluidic devices involving non-Newtonian Power-law fluids.

Foam drilling has grown tremendously in the last few years. Foam is often used as a circulating fluid in underbalanced drilling operations (i.e. to keep the pressure of the drilling fluid less than the pressure of the formation fluid). Foam drilling has been shown to provide significant advantages, such as higher penetration rate, effective cuttings transport, less formation damage, etc. However, although, a few studies have been conducted on vertical wellbores, there is little detailed published information related to its use in horizontal wellbores. In this study, the cuttings behavior in horizontal foam drilling was numerically investigated. First of all, the experimental results of foam rheology and particle settling test were shown. Experimental results showed that the foam behaves like power law fluid and foam volumetric quality has a major impact on the settling phenomena. The next, a three layer model was developed to predict and interpret the cutting transport behavior in horizontal foam drilling. This model consists of a three-layer configuration, i.e. stationary bed of drilled cuttings at the bottom; a moving bed above it; and a heterogeneous suspension layer at the top. This model is based on conservation laws of mass and linear momentum as well as on constitutive relations that describe the interactions between each phase and the walls. The model was solved numerically to predict the cuttings bed height, pressure drop at different foam flow rates under several conditions. Furthermore, numerical simulations were carried out in order to investigate some factors affecting the cutting transport in horizontal foam drilling. The results of numerical simulation are presented.

Introduction Generalized Transport Phenomena Approach to Problem Analysis General Content Conservation Laws and Field Equations Concentrations, Velocities, and Fluxes Thermodynamics of Purely Viscous Fluid Mixtures Conservation of Mass for a One-Component System Conservation of Mass for a Mixture Modification of Field Equations for Mass Transfer Conservation of Linear Momentum for One-Component Systems Conservation of Linear Momentum for a Mixture Conservation of Moment of Momentum for One-Component Systems Conservation of Moment of Momentum for a Mixture Strategies for the Solution of Mass Transfer Problems Boundary Conditions Definitions Jump Balances for Mass Conservation Jump Balances for Linear Momentum Conservation Postulated Boundary Conditions at Phase Interfaces Boundary Conditions in the Absence of Mass Transfer Utilization of Jump Balances Additional Comments on Boundary Conditions Boundary Conditions and Uniqueness of Solutions Constitutive Equations Constitutive Principles First-Order Theory for Binary Systems Combined Field and Constitutive Equations for First-Order Binary Theory First-Order Theory for Ternary Systems Special Second-Order Theory for Binary Systems Viscoelastic Effects in Flow and Diffusion Validity of Constitutive Equations Parameters in Constitutive Equations General Approach in Parameter Determination Diffusion in Polymer-Solvent Mixtures Diffusion in Infinitely Dilute Polymer Solutions Diffusion in Dilute Polymer Solutions Diffusion in Concentrated Polymer Solutions - Free-Volume Theory for Self-Diffusion Diffusion in Concentrated Polymer Solutions - Mutual Diffusion Process Diffusion in Crosslinked Polymers Additional Properties of Diffusion Coefficients Special Behaviors of Polymer-Penetrant Systems Volumetric Behavior of Polymer-Penetrant Systems Sorption Behavior of Polymer-Penetrant Systems Antiplasticization Nonequilibrium at Polymer-Penetrant Interfaces Mathematical Apparatus Basic Definitions Classification of Second-Order Partial Differential Equations Specification of Boundary Conditions Sturm-Liouville Theory Series and Integral Representations of Functions Solution Methods for Partial Differential Equations Separation of Variables Method Separation of Variables Solutions Integral Transforms Similarity Transformations Green's Functions for Ordinary Differential Equations Green's Functions for Elliptic Equations Green's Functions for Parabolic Equations Perturbation Solutions Weighted Residual Method Solution Strategy for Mass Transfer Problems Proposed Solution Methods Induced Convection Solutions of a General Set of Mass Transfer Problems Mixing of Two Ideal Gases Steady Evaporation of a Liquid in a Tube Unsteady-State Evaporation Analysis of Free Diffusion Experiments Dissolution of a Rubbery Polymer Bubble Growth from Zero Initial Size Stability Behavior and Negative Concentrations in Ternary Systems Analysis of Impurity Migration in Plastic Containers Efficiency of Green's Function Solution Method Mass Transfer in Tube Flow Time-Dependent Interfacial Resistance Laminar Liquid Jet Diffusion Analysis Analysis of the Diaphragm Cell Dissolved Organic Carbon Removal from Marine Aquariums Unsteady Diffusion in a Block Copolymer Drying of Solvent-Coated Polymer Films Flow and Diffusion Past a Flat Plate with Solid Dissolution Gas Absorption in Vertical Laminar Liquid Jet Utilization of Polymers in Drug Delivery Gas Absorption and Diffusion into a Falling Liquid Film Perturbation Solutions of Mass Transfer Moving Boundary Problems Dissolution of a Plane Surface of a Pure Gas Phase Bubble Dissolution Singular Perturbations in Moving Boundary Problems Dropping Mercury Electrode Sorption in Thin Films Numerical Analysis of Mass Transfer Moving Boundary Problems Diffusion and Reaction Design of a Tubular Polymerization Reactor Transport Effects in Low-Pressure CVD Reactors Solution of Reaction Problems with First-Order Reactions Plug Flow Reactors with Variable Mass Density Bubble Dissolution and Chemical Reaction Danckwerts Boundary Conditions for Chemical Reactors Transport in Nonporous Membranes Assumptions Used in the Theory for Membrane Transport Steady Mass Transport in Binary Membranes Steady Mass Transport in Ternary Membranes Unsteady Mass Transport in Binary Membranes Phase Inversion Process for Forming Asymmetric Membranes Pressure Effects in Membranes Analysis of Sorption and Desorption Derivation of a Short-Time Solution Form for Sorption in Thin Films Sorption to a Film from a Pure Fluid of Finite Volume A General Analysis of Sorption in Thin Films Analysis of Step-Change Sorption Experiments Integral Sorption in Glassy Polymers Integral Sorption in Rubbery Polymers Oscillatory Diffusion and Diffusion Waves Dispersion and Chromatography Formulation of Taylor Dispersion Problem Dispersion in Laminar Tube Flow for Low Peclet Numbers Dispersion in Laminar Tube Flow for Long Times Dispersion in Laminar Tube Flow for Short Times Analysis of an Inverse Gas Chromatography Experiment Effects of Pressure Gradients on Diffusion: Wave Behavior and Sedimentation Wave Propagation in Binary Fluid Mixtures Hyperbolic Waves Dispersive Waves Time Effects for Parabolic and Hyperbolic Equations Sedimentation Equilibrium Viscoelastic Diffusion Experimental Results for Sorption Experiments Viscoelastic Effects in Step-Change Sorption Experiments Slow Bubble Dissolution in a Viscoelastic Fluid Transport with Moving Reference Frames Relationships Between Fixed and Moving Reference Frames Field Equations in Moving Reference Frames Steady Diffusion in an Ultracentrifuge Material Time Derivative Operators Frame Indifference of Material Time Derivatives Frame Indifference of Velocity Gradient Tensor Rheological Implications Appendix: Vector and Tensor Notation General Notation Conventions Vectors Tensors Results for Curvilinear Coordinates Material and Spatial Representations Reynolds' Transport Theorem

The theoretical analysis for the mass transfer process of an oscillatory electroosmotic flow (EOF) in the fractional Jeffrey fluid model is studied through a polyelectrolyte layer (PEL) coated cylindrical annulus with reversible and irreversible wall reactions. The ion partitioning effect is observed due to the difference in permittivity of the PEL and the electrolyte solution, which is accounted for by the Born energy. Considering ion partitioning effects, analytical solutions for induced potential and axial velocity are presented, respectively in both the PEL and electrolyte region from the modified Poisson-Boltzmann equation and the Cauchy momentum equation with a proper constitutive equation, respectively. The Maxwell fluid and classical viscous Newtonian fluid models can be achieved separately by adjusting the relaxation and retardation time in the constitutive equation of this model. The analytical solution of the convection-diffusion equation for solute transport is established in the full domain. The separation of species is found to be dependent mainly on the Damköhler number, absorption parameter, phase partitioning coefficient, etc. It is observed that the osmotic pressure increases with the thickness and fixed charge density of the PEL. The velocity decreases with an increase in the permittivity difference of these layers. Our results suggest that the separation may be achieved through a difference in absorption kinetics.

Electroosmotic flow (EOF) is one of the most important techniques in a microfluidic system. Many microfluidic devices are made from a combination of different materials, and thus asymmetric electrochemical boundary conditions should be applied for the reasonable analysis of the EOF. In this study, the EOF of power-law fluids in a slit microchannel with different zeta potentials at the top and bottom walls are studied analytically. The flow is assumed to be steady, fully developed, and unidirectional with no applied pressure. The continuity equation, the Cauchy momentum equation, and the linearized Poisson-Boltzmann equation are solved for the velocity field. The exact solutions of the velocity distribution are obtained in terms of the Appell’s first hypergeometric functions. The velocity distributions are investigated and discussed as a function of the fluid behavior index, Debye length, and the difference in the zeta potential between the top and bottom.

A comprehensive theoretical study has been carried out on the flow behavior of both single and multiple phase non-Newtonian fluids in porous media. This work is divided into three parts: (1) development of numerical and analytical solutions; (2) theoretical studies of transient flow of non-Newtonian fluids in porous media; and (3) applications of well test analysis and displacement efficiency evaluation to field problems. A fully implicit, integral finite difference model has been developed for simulation of non-Newtonian and Newtonian fluid flow through porous media. Several commonly-used rheological models of power-law and Bingham plastic non-Newtonian fluids have been incorporated in the simulator. A Buckley-Leverett type analytical solution for one-dimensional, immiscible displacement involving non-Newtonian fluids in porous media has been developed. Based on this solution, a graphic approach for evaluating non-Newtonian displacement efficiency has been developed. The Buckley-Leverett-Welge theory is extended to flow problems with non-Newtonian fluids. An integral method is also presented for the study of transient flow of Bingham fluids in porous media. In addition, two well test analysis methods have been developed for analyzing pressure transient tests of power-law and Bingham fluids, respectively. Applications are included to demonstrate this new technology. The physical mechanisms involved in immiscible displacement with non-Newtonian fluids in porous media have been studied using the Buckley-Leverett type analytical solution. The results show that this kind of displacement is a complicated process and is determined by the rheological properties of the non-Newtonian fluids and the flow conditions, in addition to relative permeability data. In another study, an idealized fracture model has been used to obtain some insights into the flow of a power-law fluid in a double-porosity medium. For flow at a constant rate, non-Newtonian flow behavior in a fractured medium is characterized by two-parallel straight lines on a log-log plot of injection pressure versus time. Transient flow of a general pseudoplastic fluid has been studied numerically and it has been found that the long time pressure responses tend to be equivalent to that of a Newtonian system.

Chapter 5 - Constitutive Modeling in Mechanics and Thermomechanics

Summary An adaptive three-dimensional mesh refinement method based on the law of mass conservation has been introduced and tested using some analytical velocity fields as accurate in identifying singular point, asymptotic plane and drawing closed streamlines. This paper further investigates the adaptive mesh refinement method using a velocity field that is due to a uniform strain and a point vortex. Similar results have been obtained.

1R1. Tensor Analysis and Continuum Mechanics. - YR Talpaert (Dept of Sci and Eng, Algiers Univ, Algeria). Kluwer Acad Publ, Dordrecht, Netherlands. 2002. 591 pp. ISBN 1-4020-1055-9. $161.00.Reviewed by JG Simmonds (Dept of Civil Eng, Univ of Virginia, Thornton Hall, Charlottesville VA 22903).This is a carefully written treatise that begins by (over?) developing the tensor machinery necessary to state and manipulate various global and local forms of the general equations of continuum mechanics (conservation of mass, translational and rotational momentum, first and second laws of thermodynamics, excluding chemical and electromagnetic effects). Each of the book’s six chapters ends with a good number of worked-out exercises. Although considerable attention is given to the kinematics and kinetics of deforming continua, the important topic jumps and shocks are not treated and the only specific constitutive relations considered are those of linear elasticity, mostly for isotropic bodies. However, no matter how well done, one must ask in such a well-plowed field as continuum mechanics if a new book supercedes in any way—novelty of presentation, new insights into well-established principles and equations, range of phenomena, or variety of constitutive relations—some of the older, standard treatments, such as those of Truesdell and Toupin (1960), Truesdell and Noll (1965), Chadwick (1976), Spenser (1980), Gurtin (1981), or Truesdell (1984, 1991). Unfortunately, in my opinion, the answer is “No.” (Incidentally, none of the books just cited appears in the author’s list of references.) By no means am I implying that continuum mechanics is a closed field. But it seems to me that today’s focus is on the establishment, analysis, and implications of ever more comprehensive constitutive relations, an area that the author of the book under review barely touches. In summary, Tensor Analysis and Continuum Mechanics is a sound, well-written book, suitable for a first-year graduate course, but researchers in the various branches of continuum mechanics will find little that is new.

High-resolution laser Doppler velocimetry and flow-induced birefringence measurements were used to determine velocity and stress fields and to calculate a transient planar elongation viscosity profile for viscoelastic flow on the shear-free centerplane of a planar contraction geometry. A novel, two-component Boger fluid was developed that enabled Weissenberg numbers, We, as high as 2.9 to be attained and allowed the stress-optical coefficients of the fluid components to be measured. The experimental apparatus was designed to subject the fluid traveling on the centerplane to Hencky strains, ε, as great as 3.5. Despite the high We and ε, the flow induced only a weakly nonlinear viscoelastic stress response in the fluid. A spectral decomposition of predictions of linear and nonlinear constitutive equations was used to demonstrate that the weakly nonlinear response was a result of the nonhomogeneous form of the centerplane strain-rate profile in conjunction with the finite response time of the individual relaxation modes. The degree of nonlinear stress response in this experiment cannot be increased by increasing the contraction ratio to increase the total Hencky strain experienced by a fluid element traveling along the centerplane, increasing the flow rate to increase the strain rate at a given point along the centerplane, or introducing a taper to the contraction geometry to change the strain rate profile. Thus, the experimental configuration of flow through a contraction appears to be of limited value in evaluating the accuracy of nonlinear constitutive models in predicting the stress response of a viscoelastic polymer solution to a generalized elongational flow.

Electrorheological fluids have numerous potential applications in vibration dampers, brakes, valves, clutches, exercise equipment, etc. The flows in such applications are complex three-dimensional flows. Most models that have been developed to describe the flows of electrorheological fluids are one-dimensional models. Here, we discuss the behavior of two fully three-dimensional models for electrorheological fluids. The models are such that they reduce, in the case of simple shear flows with the intensity of the electric field perpendicular to the streamlines, to the same constitutive relation, but they would not be identical in more complicated three-dimensional settings. In order to show the difference between the two models, we study the flow of these fluids between eccentrically placed rotating cylinders kept at different potentials, in the setting that corresponds to technologically relevant problem of flow of electrorheological fluid in journal bearing. Even though the two models have quite a different constitutive structure, due to the assumed forms for the velocity and pressure fields, the models lead to the same velocity field but to different pressure fields. This finding illustrates the need for considering the flows of fluids described by three-dimensional constitutive models in complex geometries, and not restricting ourselves to flows of fluids described by one-dimensional models or simple shear flows of fluids characterized by three-dimensional models.

Turbulent jet flows require significant computational resources at high Reynolds numbers when simulated using large-eddy simulation (LES). To address this challenge, a convolutional neural network (CNN) was used to reproduce small-scale fluctuations from large-scale velocity fields in turbulent jet flows. High-resolution LES data were used as ground truth, and the fast Fourier transform was applied to separate large-scale and subgrid-scale (SGS) components. The CNN was trained to predict SGS velocity fields using the large-scale components extracted from coarse-grid LES data. The performance of the model was evaluated under a 20% and 40% decrease in data resolution, and with a 50% increase in Reynolds number. The CNN-generated velocity fields reproduced energy spectra and turbulent kinetic energy, which closely agreed with those of the high-resolution LES. Although turbulence production and dissipation were less accurately predicted, due to complex energy transfer mechanisms, the CNN outputs demonstrated physical consistency in terms of mass and momentum conservation.

A harmonic scalar field has a Laplacian (i.e., both source-free and curl-free) gradient vector field and vice versa. Despite the good performance of spherical harmonic series on modeling the gravitational field generated by spheroidal bodies (e.g., the Earth), the series may diverge inside the Brillouin sphere enclosing all field-generating mass. Divergence may realistically occur when determining the gravitational fields of asteroids or comets that have complex shapes, which is known as the complex-boundary value problem (CBVP). To overcome this weakness, we propose a new spatial-domain numerical method based on the equivalence transformation which is well known in the fluid dynamics community: a potential-flow velocity field and a gravitational force vector field are equivalent in a mathematical sense, both referring to a Laplacian vector field. The new method abandons the perturbation theory based on the Laplace equation, and, instead, derives the governing equation and the boundary condition of the potential flow from the conservation laws of mass, momentum and energy. Correspondingly, computational fluid dynamics (CFD) techniques are introduced as a numerical solving scheme. We apply this novel approach to the gravitational field of the comet 67P/Churyumov–Gerasimenko which has an irregular shape. The method is validated in a closed-loop simulation by comparing the result with a direct integration of Newton’s formula. Both methods are consistent with a relative magnitude discrepancy at the percentage level and with a small directional difference root-mean-square value of 0.78^{circ }. Moreover, the Laplacian property of the potential flow’s velocity field is proved mathematically. From both theoretical and practical points of view, the new numerical method is able to overcome the divergence problem and, hence, has a good potential for solving CBVPs.

Finite element methdos for calculation of steady, viscoelastic flow using constitutive equations with a Newtonian viscosity

One-dimensional reductions and functional separable solutions to unsteady plane and axisymmetric boundary-layer equations for non-Newtonian fluids