De Waele Power Law Research Articles

Computational Analysis of Rheological Secondary Flow in a Pipe-Manifold Containing In-Plane Double Bends

Non-Newtonian fluid flow in pipe bends is inevitable in industrial applications. Previous researchers have extensively explored Newtonian flow through curved ducts. However, the non-Newtonian counterpart gets little attention. We study the turbulent flow of shear-dependent fluids obeying the Power-Law model in a pipe manifold containing an in-plane double bend. Ostwald–de Waele's power law is used to model the fluid's rheology. We utilize computational fluid dynamics (CFD) to solve Reynolds-averaged Navier–Stokes (RANS) equations with the k-ε turbulence model. We validate our numerical results with previous experimental results. The in-plane double bend perturbs the flow in the pipe manifold to develop a Prandtl's secondary flow of the first kind. A fully developed flow at the bend upstream is disturbed due to the bend's curvature and regains its fully developed characteristics upon a certain downstream length after the exit of the bend. We study the rheological characteristics of the secondary flow within the bend and the evolution of fluid flow at the bend downstream. We demonstrate that the centrifugal force-dominated secondary flow increases with a decrease of the non-Newtonian power-law index. We capture the camel's-back-shaped velocity profiles within the bend due to accelerating-decelerating flow. The study reveals that the average flow velocity increases along the bend with a corresponding pressure head loss. We quantify this velocity rise by a newly introduced non-dimensional number, viz. enhancement ratio. The double bend's enhancement ratio decreases with an increase in n.

Capillary imbibition of inelastic non-Newtonian fluids in an asymmetric flow assay

Inelastic non-Newtonian fluids are imbibed in a non-uniform conical-shaped microfluidic channel, discussed quantitatively using a theoretical model. We consider the Ostwald–de Waele power-law model to describe the rheology of the inelastic non-Newtonian fluids. Consistent with the reduced order model, the theoretical framework developed here accounts for the coupled effects of fluid rheology, flow configuration, and surface wettability to describe the transient progression of the filling length in the capillary under varied cases. Considering the simultaneously intervening interactions resulting due to rheological effect, geometrical modulation, and surface wetting condition during the imbibition phenomenon, we demonstrate the temporal advancement of the filling phase fluid length in the non-uniform capillary for a set of involving parameters pertinent to this analysis. Non-uniformity in the capillary cross-section gives rise to an alteration in the viscous force acting at the fluid–fluid–solid interface, which in turn, leads to a control over the temporal progression of fluid length, retaining a balance between the capillary and viscous forces. From the predicted relation between filing length and time of filling, we identify three distinct regimes of filling and establish the befitting scaling laws describing the imbibition phenomenon in the respective regimes.

Asymptotically consistent analytical solutions for the non-Newtonian Sakiadis boundary layer

The Sakiadis boundary layer induced by a moving wall in a semi-infinite fluid domain is a fundamental laminar flow field relevant to high speed coating processes. This work provides an analytical solution to the boundary-layer problem for Ostwald–de Waele power law fluids via a power series expansion and extends the approach taken for Newtonian fluids [Naghshineh et al. “On the use of asymptotically motivated gauge functions to obtain convergent series solutions to nonlinear ODEs,” IMA J. of Appl. Math. 88, 43 (2023)] in which variable substitutions (which naturally determine the gauge function in the power series) are chosen to be consistent with the large distance behavior away from the wall. Contrary to prior literature, the asymptotic behavior dictates that a solution only exists in the range of power law exponents, α, lying in the range of 0.5<α≤1. An analytical solution is obtained in the range of approximately 0.74≤α<1, using a convergent power series with an asymptotically motivated gauge function. For power laws corresponding to 0.5<α<0.74, the gauge function becomes ill-defined over the full domain, and an approximate analytical solution is obtained using the method of asymptotic approximants [Barlow et al. “On the summation of divergent, truncated, and underspecified power series via asymptotic approximants,” Q. J. Mech. Appl. Math. 70, 21–48 (2017)]. The approximant requires knowledge of two physical constants, which we compute a priori using a numerical shooting method on a finite domain. The utility of the power series solution is that it can be solved on the entire semi-infinite domain and—in contrast to a numerical solution—does not require a finite domain length approximation and subsequent domain length refinement.

Computational study of bioconvection rheological nanofluid flow containing gyrotactic microorganisms: a model for bioengineering nanofluid fuel cells

ABSTRACT The recent advancements in using micro-organisms effectively for biofuels are the key reason for conducting this study. Therefore, the current theoretical work addresses the Oswald – DeWaele power-law models of bioconvection nanofluid flow containing both nanoparticles and gyrotactic microorganisms. This study has examined two major elements of gyrotactic microorganisms: swimming behavior and the development of bioconvection structures. In addition, the Lie-group approach of transformations is used to establish similarity representations of the governing equations rather than the existing transformations. To achieve the model’s aims, the Legendre-based collocation method (LBCM) was adopted. The findings are analyzed using graphs to examine the patterns of the relevant parameters in the problem. Compared with existing results, the present analysis is effective, fast-converging, and accurate. The bioconvection parameters have a significant impact on the density of motile microorganisms, nanoparticle volume fraction, flow momentum, and temperature distribution. Furthermore, the current work finds application in nanofluid fuel.

Cooling of a periodic heat-generating solid element in an electronic cabinet using a non-Newtonian pseudoplastic nanofluid and a heat-conducting substrate

PurposeNowadays, the most important challenge in mechanical engineering, power engineering and electronics is a development of effective cooling systems for heat-generating units. Taking into account this challenge, this study aims to deal with computational investigation of thermogravitational energy transport of pseudoplastic nanoliquid in an electronic chamber with a periodic thermally producing unit placed on the bottom heat-conducting wall of finite thickness under an influence of isothermal cooling from vertical side walls.Design/methodology/approachThe control equations formulated using the Boussinesq approach, Ostwald–de Waele power law and single-phase nanofluid model with experimentally based correlations of Guo et al. for nanofluid dynamic viscosity and Jang and Choi for nanofluid thermal conductivity have been worked out by the in-house computational procedure using the finite difference technique. The impact of the Rayleigh number, nanoadditives concentration, frequency of the periodic heat generation from the local element and thickness of the bottom solid substrate on nanoliquid circulation and energy transport has been studied.FindingsIt has been found that a raise of the nanoadditives concentration intensifies the cooling of the heat-generating element, while a growth of the heat-generation frequency allows reducing the amplitude of the heater temperature.Originality/valueMathematical modeling of a pseudoplastic nanomaterial thermogravitational energy transport in an electronic cabinet with a periodic thermally generating unit, a heat-conducting substrate and isothermal cooling vertical surfaces to identify the possibility of intensifying heat removal from a heated surface.

Evaluation of OKRA (Abelmoschus esculentus) Macromolecular Solution for Enhanced Oil Recovery in Kazakhstan Carbonate Reservoir

Natural polymers have been investigated as part of the endeavors of green chemistry practice in the oil field. However, natural polymer studies are still preliminary. The current study examines okra’s (natural polymer) efficiency for polymer flooding, particularly in Kazakhstan. The evaluation targets the heavy oil trapped in carbonate reservoirs. SEM and FTIR were used to characterize morphology and chemical composition. A rheology study was conducted under different shear rates for three plausible concentrations: 1 wt.%, 2 wt.% and 5 wt.%. The core flooding was challenged by the low porosity and permeability of the core. The results showed that okra’s size is between 150–900 μm. The morphology can be described by rod-like structures with pores and staking as sheet structures. The FTIR confirmed that the solution contains a substantial amount of polysaccharides. During the rheology test, okra showed a proportional relationship between the concentration and viscosity increase, and an inversely proportional relationship with the shear rate. At reservoir temperature, the viscosity reduction was insignificant, which indicated good polymer stability. Okra showed shear-thinning behavior. It was fitted to the Ostwald–de Waele power-law model by a (90–99)% regression coefficient. The findings confirm okra’s pseudo-plasticity, and that it is proportional to the solution concentration. The incremental oil recovery was 7%. The flow was found to be restricted due to the mechanical entrapment resulting from the large molecule size and the low porosity–permeability of the system. This study proves that the dominating feature of natural polysaccharide derivatives is their applicability to moderate reservoir conditions. The current study is a positive attempt at natural polymer application in Kazakhstan and similar field conditions.

Characterization of wood‐filled thermoplastic polyurethanes for the injection molding process

AbstractFiber fillings of wood plastic composites (WPC) are almost exclusively limited to standard plastics such as polyethylene and polypropylene. At the Kunststofftechnik Paderborn of the University of Paderborn the wood fiber filling of engineering plastics is being promoted. WPC with different fiber types and fiber contents based on two thermoplastic polyurethanes (TPU) were compounded and subsequently characterized. We found that the physicochemical properties of the materials differ from standard plastic‐based WPC. Wood filling with increasing fiber content did not immediately correlate with an increase in density. A decrease in density and swelling of the compound was detected with reaching a critical fiber content. Our compounds showed an increased water absorption at high‐fiber contents over time, which can be described logarithmically. The observed viscosity curves obey the Ostwald and de Waele power law, but an increased viscosity at increased fiber content was not apparent for both TPU matrices.

Pressure Losses in Power-Law Fluid Flow through a Tube of Variable Cross-Section

Steady laminar flow of an incompressible power-law fluid is a tube with an obstacle of given shape is numerically simulated. The mathematical description of the process is based on the vortex transport equation and the Poisson equation for the stream function, while the rheological properties of the medium are described by the Ostwald—de Waele power law. The steady solution of the problem is obtained using a time-dependent method based on the finite-difference approximation of the governing equations. The pressure distribution is determined by numerically solving the Poisson equation. A parametric investigation of the kinematic and dynamic flow parameters, as functions of the control parameters of the problem, is performed for non-Newtonian media. The effect of the Reynolds number, the nonlinearity exponent, and the obstacle geometry on the coefficient of the local fluid resistance is demonstrated.

Capillary imbibition of non-Newtonian fluids in a microfluidic channel: analysis and experiments

We have presented an experimental analysis on the investigations of capillary filling dynamics of inelastic non-Newtonian fluids in the regime of surface tension dominated flows. We use the Ostwald–de Waele power-law model to describe the rheology of the non-Newtonian fluids. Our analysis primarily focuses on the experimental observations and revisits the theoretical understanding of the capillary dynamics from the perspective of filling kinematics at the interfacial scale. Notably, theoretical predictions of the filling length into the capillary largely endorse our experimental results. We study the effects of the shear-thinning nature of the fluid on the underlying filling phenomenon in the capillary-driven regime through a quantitative analysis. We further show that the dynamics of contact line motion in this regime plays an essential role in advancing the fluid front in the capillary. Our experimental results on the filling in a horizontal capillary re-establish the applicability of the Washburn analysis in predicting the filling characteristics of non-Newtonian fluids in a vertical capillary during early stage of filling (Digilov 2008 Langmuir 24 , 13 663–13 667 ( doi:10.1021/la801807j )). Finally, through a scaling analysis, we suggest that the late stage of filling by the shear-thinning fluids closely follows the variation x ~ t . Such a regime can be called the modified Washburn regime (Washburn 1921 Phys. Rev. 17 , 273–283 ( doi:10.1103/PhysRev.17.273 )).

How dimensional analysis allows to go beyond Metzner–Otto concept for non-Newtonian fluids

Abstract The concept of Metzner and Otto was initially developed for correlating power measurements in stirred vessels for shear-thinning fluids in the laminar regime with regard to those obtained for Newtonian liquids. To get this overlap, Metzner and Otto postulated and determined an “effective shear rate” which was proportional to the rotational speed of the impeller Although it was not based on a strong theoretical background, it was rapidly admitted as a practical engineering approach and was extended for seeking out a “Newtonian correspondence” with non-Newtonian results (i.e. different classes of fluids). This was applied in a variety of tank processes even for predicting heat transfer or mixing time, which stretches far away from the frame initially envisaged by Metzner and Otto themselves. This paper aimed to show how dimensional analysis offers a theoretically founded framework to address this issue without the experimental determination of effective quantities. This work also aimed to enlarge the underlying questions to any process in which a variable material property exists and impacts the process. For that purpose, the pending questions of Metzner and Otto concept were first reminded (i.e. dependence of the Metzner–Otto constant to rheological parameters, physical meaning of the effective shear rate, etc.). Then, the theoretical background underlying the dimensional analysis was described and applied to the case of variable material properties (including non-Newtonian fluids), by introducing in particular the concept of material similarity. Finally, two examples were proposed to demonstrate how the rigorous framework associated with the dimensional analysis is a powerful method to exceed the concept of Metzner and Otto and can be adapted beyond the Ostwald–de Waele power law model to a wide range of non-Newtonian fluids in various processes, without being restricted to batch reactor and laminar regime.

Formulation of a combined low saline water and polymer flooding for enhanced oil recovery

The study aims at formulating a new Enhanced Oil Recovery (EOR) slug comprising a synthetic polymer, (Polyacrylamide (PAM)) and a biopolymer, (Xanthan Gum (XG)). In this case, the EOR will be known as XPS, where low saline water flooding was assisted by PAM and XG. The polymer flooding application in the depleted oil fields of Upper Assam Basin has faced a number of challenges, as it was unable to enhance the viscosity of the displacing fluid and reduce the effective permeability of the displacing fluid, which led to the failure of the macroscopic sweep efficiency (ES ) of the reservoir. The rheological properties were found to be Non-Newtonian mostly shear thinning and followed the Herschel–Bulkley (HB) and Ostwald–de Waele power law (OWP) models. The appropriate slugs were further flooded in a core flooding. Low saline water flooding followed by polymer flooding increases the overall recovery efficiency.

Rheological characteristics of platy kaolin

Platy kaolin can provide significant value in the coating of paper and paperboard. It can be used in multiple applications and can provide benefits such as titanium dioxide (TiO2) extension, smoothness improvement, improved print gloss or ink set rates, calendering intensity reduction, and improved barrier properties. It is not a pigment that can be simply substituted for traditional hydrous kaolin without some adjustment to the coating formulation. These adjustments can be as simple as reducing solids, but may require binder changes as well. The coater setup may need to be adjusted because of the unique rheological behaviors these pigments exhibit. The unique rheological characteristics of platy kaolin are explored here. Measurements of the water retention of platy kaolin containing coatings confirm that water retention is not reduced in comparison to more blocky kaolin pigments, despite the lower coating solids at which they need to be run. This means that the rheological characteristics are the most important in understanding the runnability. An extensive analysis reveals some unique behaviors that need to be understood when utilizing these materials. Viscoelastic measurements indicate that, for this binder system, Tan δ is mainly a function of solids. This may explain how weeping is initiated on a blade coater. The degree of shear thinning behaviors is investigated using the Ostwald de-Waele power law. The immobilization point was determined using the Dougherty-Krieger equation and related to the work of Weeks at the University of Maine on blade coater runnability. An indirect measure of particle shape and size synergy is also demonstrated using the Dougherty- Krieger equation parameters.

A study of the transient shear flow behavior of the incompressible and isothermal thermoplastic polymer materials

In this article, a feasible mathematical method was proposed to describe the instantaneous shear flow behavior of the incompressible and isothermal thermoplastic materials. Based on the molecular kinetics, thermodynamics, and continuum mechanics method, a viscoelastic model was presented about the intricate rheological phenomena of polymer macromolecules, which is simplified as the generalized Newton’s law. The rheological experiments were carried out with the isotactic polypropylene material. The analysis of variance method was performed to determine the fitness of the model. By comparing with the Ostwald–de Waele power law model and Cross model, the theoretical model was studied. The experimental results reveal that the theoretical model is more accurate than the Ostwald–de Waele power law model and shows a roughly equivalent fitness with the Cross model, which indicates that the theoretical model is effective and robust. Moreover, the results of this study may have potential industrial applications and benefit the users of certain plastic components.

Volume scavenging of networked droplets

A system of N spherical-cap fluid droplets protruding from circular openings on a plane is connected through channels. This system is governed by surface tension acting on the droplets and viscous stresses inside the fluid channels. The fluid rheology is given by the Ostwald–de Waele power law, thus permitting shear thinning. The pressure acting on each droplet is caused by capillarity and given in terms of the droplet volume via the Young–Laplace law. Liquid is exchanged along the network of fluid conduits due to an imbalance of the Laplace pressures between the droplets. In this way some droplets gain volume at the expense of others. This mechanism, christened “volume scavenging,” leads to interesting dynamics.Numerical experiments show that an initial droplet configuration is driven to a stable equilibrium exhibiting 1 super-hemispherical droplet and N−1 sub-hemispherical ones when the initial droplet volumes are large. The selection of this “winning” droplet depends not only on the channel network and the fluid volume, but also notably on the fluid rheology. The rheology is also observed to drastically change the transition to equilibrium. For smaller droplet volumes the long-term behavior is seen to be more complicated since the types of equilibria differ from those arising for larger volumes. These observations motivate our analytical study of equilibria and their stability for the corresponding nonlinear dynamical system. The identification of equilibria is accomplished by locating the zeros of a mass polynomial, defined through the constant volume/mass constraint. The key tool in our stability analysis is a pressure–volume work functional, related to the total surface area, which serves as a Lyapunov function for the dynamical system. This functional is useful since equilibria are typically not hyperbolic and linearization techniques not available. Equilibria will be shown to be hierarchically organized in terms of size of the pressure–volume work functional. For larger droplet volumes this ordering exhibits one hierarchy of equilibria. Two hierarchies exist when the volumes are smaller. The minimizing equilibria in either case are asymptotically stable.

Non-isothermal flow of power-law fluid in a pipe with sudden expansion

In this paper, numerical simulation of steady power-law fluid flow through axisymmetric sudden expansion under non-isothermal conditions is implemented. Mathematical model of the flow accounts for viscous dissipation effects and temperature-dependent consistency coefficient in the Ostwald de Waele power law. The problem formulated is solved using the finite-difference method. Two-dimensional flow zones are found in the vicinity of expansion plane including recirculating region in the corner. Variation in the length of these zones is analysed in respect to power-law index and dimensionless criteria of the problem. The effect of viscous dissipation on the flow structure formation and viscosity and temperature distribution is estimated in a wide range of the main parameters.

Electroosmotically induced alterations in peristaltic microflows of power law fluids through physiological vessels

A mathematical model to analyze the effects of electric double layer and applied external electric field on peristaltic transport of non-Newtonian aqueous solution through a microchannel is presented. Ostwald–de Waele power law model is employed to describe the non-Newtonian fluid, in an effort to capture the essential biofluid dynamics. The governing equations of physical problem are simplified using low Reynolds number and long wavelength approximations. Poisson Boltzmann equations are also solved under Debye Huckel linearization. Following non-dimensional transformation of the linearized boundary value problem, closed-form analytical solutions are presented for the velocity components, pressure gradient, average flow rate, and stream function subject to physically appropriate boundary conditions. Validation with existing results is also made. A comparative discussion between shear thinning fluid and shear thickening fluids under the influences of Debye length and Helmholtz–Smoluchowski velocity are presented numerically. Trapping phenomenon for dilatant fluids and pseudoplastic fluids under the electrokinetic phenomenon is also computed. This model can help toward designing artificial biomedical devices based on microfluidic devices which can also be applicable to control the physiological transport.

Influence of viscous dissipation and thermo-diffusion on double diffusive convection over a vertical cone in a non-Darcy porous medium saturated by a non-Newtonian fluid with variable heat and mass fluxes

Abstract This paper emphasizes the thermo-diffusion and viscous dissipation effects on double diffusive natural convection heat and mass transfer characteristics of non-Newtonian power-law fluid over a vertical cone embedded in a non-Darcy porous medium with variable heat and mass flux conditions. The Ostwald–de Waele power-law model is employed to describe the behavior of non-Newtonian fluid. Local non-similarity procedure is applied to transform the set of non-dimensional partial differential equations into set of ordinary differential equations and then the resulting system of equations are solved numerically by Runge-Kutta fourth order method together with a shooting technique. The influence of pertinent parameters on temperature and concentration, heat and mass transfer rates are analyzed in opposing and aiding buoyancy cases through graphical representation and explored in detail.

Pressure driven laminar flow of a power-law fluid in a T-channel

Planar flow of a non-Newtonian fluid in a T-channel is investigated. The viscosity is determined by the Ostwald–de Waele power law. Motion of the fluid is caused by pressure drop given in boundary sections of the T-channel. On the solid walls, the no slip boundary condition is used. The problem is numerically solved with using a finite difference method based on the SIMPLE procedure. As a result of this study, characteristic flow regimes have been found. Influence of main parameters on the flow pattern has been demonstrated. Criteria dependences describing basic characteristics of the flow under conditions of the present work have been shown.

Nonisothermal filling of a planar channel with a power-law fluid

In this paper, the fountain flow of a power-law fluid during the filling of a planar vertical channel is investigated taking into account a dissipative heating. The rheological properties of the medium are defined by Ostwald de Waele power-law with an exponential temperature dependence of effective viscosity. The problem is solved numerically by means of a finite-difference method based on the SIMPLE algorithm and method of invariants for compliance with the natural boundary conditions on a free surface. The parametrical investigations of the shape of free surface as a function of the basic dimensionless criteria and nonlinearity degree of the fluid are implemented. The effect of viscous dissipation on the formation of the kinematic characteristics of the flow is shown.

Rheological Characterisation of the Flow Behaviour of Wood Plastic Composites in Consideration of Different Volume Fractions of Wood

In this study, the rheological properties of wood plastic composites (WPC) with different polymeric matrices (LDPE, low-density polyethylene and PP, polypropylene) and with different types of wood filler (hardwood flour and softwood flour) have been investigated by means of high pressure capillary rheometry. The volume fraction of wood was varied between 0 and 60 %. The shear thinning behaviour of the WPC melts can be well described by the Ostwald - de Waele power law relationship. The flow consistency index K of the power law shows a good correlation with the volume fraction of wood. Interparticular interaction effects of wood particles can be mathematically taken into account by implementation of an interaction exponent (defined as the ratio between flow exponent of WPC and flow exponent of polymeric matrix). The interaction exponent shows a good correlation with the flow consistency index. On the basis of these relationships the concept of shear-stress-equivalent inner shear rate has been modified. Thus, the flow behaviour of the investigated wood filled polymer melts could be well described mathematically by the modified concept of shear-stress-equivalent inner shear rate. On this basis, the shear thinning behaviour of WPC can now be estimated with good accuracy, taking into account the volume fraction of wood.