Power-law Liquids Research Articles

Non-Newtonian two-phase stratified flow with curved interface through horizontal and inclined pipes

A model which predicts liquid holdup (ɛTPL), pressure gradient and frictional multiplier (ϕL) for non-Newtonian liquid–gas flow with different interfacial shapes is presented. The interfacial shape is calculated by solving the Young–Laplace equation, while the non-Newtonian liquid is a power-law shear-thinning liquid. The effects of interfacial tension and interfacial shape are included in the model. The results show that the interface shape can be assumed as an arc shape when Bond number is lower than 10. Furthermore the model can predict liquid holdup, two-phase pressure gradient and frictional multiplier for uniform stratified flows. In addition, the result indicate that the interfacial effects on the holdup and pressure difference between two phases are significant, especially for small liquid holdups.

SPRAY OF POWER-LAW FLUID FROM A SWIRL INJECTOR WITH NONTANGENTIAL INLET CHANNELS

The spray formation of power-law non-Newtonian liquids from swirl injectors with nontangential inlet channels was described by carrying out experiments in a spray test facility and compared with the spray behavior from common swirl injectors. The most evident difference between the two types of injectors is that the spray cone pulsated for the swirl injectors with nontangential inlets. The pulsation is caused by the compressible air remaining in the swirl chamber, which can reduce the sheet breakup length under high pressure drop. The nontangential entry of liquids into the swirl chamber eventually reduces the fluid swirl radius, and the equivalent geometric characteristics constant decreases. As a result, the spray angle would decrease, and the mass flow rate would increase.

Dynamics of Drainage of Power-Law Liquid into a Deformable Porous Material

In this study we explore the one-dimensional drainage of a power-law fluid into a deformable porous material. Initially, the fluid is imbibed into the dry undeformed material due to capillary suction which in turn deforms the porous material and forms liquid and solid interfaces. Mixture theory is employed to study the movement of the liquid and solid phases. The zero-gravity model contains the similarity solution that is solved numerically. The stress gradient within the deformable porous material is induced from a pressure gradient that produces an evolving solid fraction and hence deformation. In the absence of gravity effects, the deformation of the solid seems in the same direction of imbibition. This is because of attraction of gravity. Note that these liquid and solid dynamics depend on both the power-law indexes n and μ. We performed the experiments to measure the drainage and deformations of deformable porous materials for two samples of silicon oil (polydimethylsiloxane) in a polyurethane foam. Our experiments show that the silicon with high viscosity drains slower than silicon oil with low viscosity. The theoretical and experimental results show the same qualitative trend.

Momentum and Heat Transfer Phenomena of Confined Spheroid Particles in Power-Law Liquids

Effects of wall confinement and the power-law fluid behavior index on momentum and heat transfer phenomena of confined spheroid particles are numerically investigated over wide ranges of conditions such as the Reynolds number, Re, 1–100; Prandtl number, Pr, 1–1000; particle aspect ratio, e, 0.5–2.5; wall confinement factor, λ, 2–5; and power-law fluid behavior index, n, 0.4–1.8. Some of the key findings include that the size of the recirculation wake at the rear end of the particle increases with the increasing particle aspect ratio and/or Reynolds number and/or wall factor. The total drag coefficient (Cd) decreases with the increasing wall factor and/or Reynolds number and/or the decreasing power-law index. Regardless of values of the Reynolds and Prandtl numbers and regardless of the shape of the spheroid particles, the average Nusselt number increases with the decreasing power-law index and/or with the decreasing wall factor.

Wall polymer depletion effects on electrokinetic diffusioosmosis of power-law liquids in cylindrical capillaries

We investigate electrokinetic diffusioosmotic flows of power-law liquids including the effects of a polymer-depleted Newtonian liquid layer near the wall boundaries in circular cylindrical capillaries. Semi-analytical solutions to the flow velocity distribution and volume flow rate are obtained for conditions involving finite double layer effects on the induced electric field of the electrokinetic diffusioosmosis. Results show that the flow behavior and responses of the electrokinetic diffusioosmotic flow depend not only on the wall zeta potential, diffusivity difference parameter, and flow behavior index, but also on the depletion layer thickness to Debye thickness ratio, the ratio of the flow consistency parameter of the power-law liquid core to the viscosity of the Newtonian depletion layer, as well as the exact numeric values of the flow consistency parameter and the Newtonian viscosity. Including the Newtonian depletion layer gives rise to wiggled-shaped zero flow rate border curves on the zeta potential versus diffusivity difference parameter map when the depletion layer thickness to Debye thickness ratio and the ratio of the flow consistency parameter of the power-law liquid core to the viscosity of the Newtonian depletion layer are close to one. These results are not identified in previous Newtonian or non-Newtonian electrokinetic diffusioosmosis literature and may likely open new possibilities and suggest new ideas in the analysis and design of diffusiophoretic separation and diffusioosmotic flow operations.

Hele-Shaw rheometry

In this paper, we describe a novel approach to determine the flow behavior index of a power-law fluid by means of a microfluidic device. The concept of this method is based on a mathematical analysis by Aronsson and Janfalk [Eur. J. Appl. Math. 3, 343–366 (1992)] of Hele-Shaw flow of power-law liquids. We implement this approach by driving a non-Newtonian fluid through a glass microfluidic chip with a 100:1 contraction. The flow in this chip satisfies the Hele-Shaw flow conditions in most of the device. Two conjugate p-Laplace equations describe the pressure and stream function in such flows. These equations depend on the flow behavior index, n. Therefore, by fitting the p-Laplace equation to the velocity field obtained from a micro particle image velocimetry measurement of the flow, the flow behavior index of the fluid in the chip can be determined. Because in practice, fluids rarely show perfectly inelastic power-law behavior, conditions under which the assumption of inelastic flow is valid were derived by analyzing Hele-Shaw flow of an Oldroyd-B fluid. The concept was tested using three different classes of model fluids, a Newtonian fluid, an inelastic power-law fluid, and a Boger fluid. In all three cases, satisfactory results were obtained, with values of n deviating at most 4% from values measured using conventional rheometry. The method presented here is expected to be potentially useful in online quality control in, for example, polymer or food processing.

A simple model for predicting the pressure drop and film thickness of non-Newtonian annular flows in horizontal pipes

A model of two-phase non-Newtonian horizontal annular flows, which predicts film thickness and pressure gradient from flowrates only, is presented. In the model, the gas and non-Newtonian liquid flows are calculated separately based on the independent governing equations. The shear stress balance at the gas–liquid interface is calculated in order to link two phases together. The non-Newtonian fluid is assumed as a power-law shear-thinning liquid. The logarithmic velocity distribution is chosen to calculate the turbulent velocity profile in the gas core. The influences of entrainment and aeration are included in the model. The pressure drop, film thickness, void fraction, the frictional multiplier, and Lockhart–Martinelli parameter are predicted. The analytical model is compared with the published experimental investigations, and the results show that the model can predict the film thickness and pressure gradient simultaneously based on the flowrates of liquid and gas. The frictional multiplier and Lockhart–Martinelli parameter are calculated at the same time, and the predicted values are comparable with the experimental data. The difference between the analytical model and the experiments is lower than 10%.

MHD Flow and Heat Transfer in a Power-Law Liquid Film at a Porous Surface in the Presence of Thermal Radiation

In this paper, the effects of variable thermal conductivity and thermal radiation on the MHD flow and heat transfer of a non-Newtonian power-law liquid film at a horizontal porous sheet in the presence of viscous dissipation is studied. The governing time dependent boundary layer equations are transformed to coupled, non-linear ordinary differential equations with power-law index, unsteady parameter, film thickness, magnetic parameter, injection parameter, variable thermal conductivity parameter, thermal radiation parameter, the Prandtl number and the Eckert number. These coupled non-linear equations are solved numerically by an implicit, finite difference scheme known as the Keller box method. The obtained numerical results for velocity and temperature profiles are presented graphically. Also, the obtained results of our study for some special cases are compared with the previously published results, and the results are found to be in very good agreement. The effects of unsteady parameter on the skin friction, walltemperature gradient and the film thickness are explored for different values of the power-law index and the magnetic parameter. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena.

Natural convection heat transfer from two vertically aligned circular cylinders in power-law fluids

In this work, laminar natural convection in power-law liquids from a pair of vertically aligned heated cylinders has been studied numerically over the following ranges of conditions: Prandtl number (0.72⩽Pr⩽102), Grashof number (10⩽Gr⩽104), power-law index (0.3⩽n⩽1.5) thereby embracing both shear-thinning and shear-thickening fluid behaviour and the center-to-center gap between the two cylinders was varied over the range (2⩽(S/D)⩽20). The heat transfer characteristics of the lower cylinder are little influenced by the presence of the upper heated cylinder in line with the previous studies. On the other hand, the average Nusselt number for the upper cylinder could be higher or lower than that of the lower cylinder depending upon the value of (S/D), Grashof number and power-law index. Overall, all else being equal, shear-thinning behaviour can enhance heat transfer by up to 100% whereas shear-thickening has adverse influence on the rate of heat transfer. The present results on Nusselt number and drag coefficient have been correlated using simple analytical forms which permit the estimation of these parameters for the two cylinders in a new application with acceptable levels of accuracy. In addition to the overall heat transfer characteristics, the detailed structure of the velocity and temperature fields is visualised in terms of the streamline and isotherm contours over the preceding ranges of parameters.

Instability analysis of a power law liquid jet

The temporal instability of a power law liquid jet injected into a static inviscid gas medium is investigated theoretically for axisymmetric disturbances. The corresponding dispersion relation between the growth rate and the wavenumber of disturbed waves is derived after using a linear approximation. It is available for both shear-thinning and shear-thickening liquids. The effects of several dimensionless parameters including the generalized Reynolds number, the Weber number, the density ratio of gas to liquid and the power law exponent, on the instability of the jet are studied. The results reveal that the jet breakup can be classified into Rayleigh Mode and Taylor Mode. And the instability characteristics are different for different modes and power law exponents. For Rayleigh Mode, surface tension promotes the breakup of the jet and the liquid viscosity prevents the jet from breaking up; for Taylor Mode, both surface tension and the viscosity prevent the jets from breaking up, while the interaction between the gas and the liquid significantly promotes the breakup process. Moreover a liquid jet with a smaller power law exponent is easier to disintegrate.

Soret Effect on the Natural Convection From a Vertical Plate in a Thermally Stratified Porous Medium Saturated With Non-Newtonian Liquid

The paper highlights the application of a recent seminumerical successive linearization method (SLM) in solving highly coupled, nonlinear boundary value problem. The method is presented in detail by solving the problem of free convection flow due to a vertical plate embedded in a non-Darcy thermally stratified porous medium saturated with a non-Newtonian power-law liquid. Thermal-diffusion (Soret) and variable viscosity effects are taken into consideration. The Ostwald–de Waele power-law model is used to characterize the non-Newtonian behavior of the fluid. The governing partial differential equations are transformed into a system of ordinary differential equations and solved by SLM. The accuracy of the SLM has been tested by comparing the results with those obtained using the shooting technique. The effect of various physical parameters such as power-law index, Soret number, variable viscosity parameter, and thermal stratification parameter on the dynamics of the fluid is analyzed through computed results. Heat and mass transfer coefficients are also shown graphically for different values of the parameters.

Effects of variable fluid properties on the thin film flow of Ostwald-de Waele fluid over a stretching surface

We investigate, in this paper, the effects of thermo-physical properties on the flow and heat transfer in a thin film of a power-law liquid over a horizontal stretching surface in the presence of a viscous dissipation. The fluid properties, namely the fluid viscosity and the fluid thermal conductivity, are assumed to vary with temperature. Using a similarity transformation, the governing partial differential equations with a time dependent boundary are converted into coupled non-linear Ordinary Differential Equations (ODEs) with variable coefficients. Numerical solutions of the coupled ODEs are obtained by a finite difference scheme known as the Keller-box method. Results for the velocity and temperature distributions are presented graphically for different values of the pertinent parameters. The effects of unsteady parameter on the skin friction, the wall temperature gradient and the film thickness are presented and analyzed for zero and non-zero values of the temperature-dependent thermo-physical properties. The results obtained reveal many interesting features that warrant further study on the non-Newtonian thin film fluid flow phenomena, especially the shear-thinning phenomena.

Laminar natural convection in power-law liquids from a heated semi-circular cylinder with its flat side oriented downward

In this work, laminar free convection heat transfer from a heated semi-circular cylinder, with its flat base facing downward, submerged in quiescent power-law fluids has been studied. The coupled momentum and energy equations have been solved numerically in the following ranges of dimensionless parameters: Grashof number (10⩽Gr⩽105), Prandtl number (0.72⩽Pr⩽100) and power-law index (0.2⩽n⩽1.8). The detailed flow and temperature fields in the vicinity of the heated object are visualized in terms of the streamline and isotherm contours respectively. The rate of heat transfer is described in terms of the local Nusselt number variation along the surface of the cylinder together with its average value over the above-mentioned ranges of parameters. As expected, the value of the local Nusselt number increases from the front stagnation point up to the sharp corner and then decreases all the way up to the rear stagnation point. Broadly, over the range of conditions spanned here, the flow remains attached to the surface of the cylinder. The average Nusselt number increases with both the Grashof and Prandtl numbers and it decreases with the increasing power-law index. Furthermore, shear-thinning behavior (n<1) enhances heat transfer whereas shear-thickening behavior (n>1) impedes it with reference to that in Newtonian fluids otherwise under identical conditions. The numerical results obtained herein are correlated using a composite parameter consistent with the boundary layer analysis and these are contrasted with the results for the other two-dimensional shapes thereby suggesting the general usefulness of the composite parameter.

Finite double layer and non-Newtonian power-law effects on electrokinetic diffusioosmotic flows in parallel plate microchannels

Diffusioosmotic flows of electrolytic non-Newtonian power-law liquids in parallel plate microchannels are theoretically investigated by considering the finite thickness of the interfacial electrical double layers. Semi-analytical or numerical solutions to the shear stress, flow velocity, effective viscosity, and volume flow rate are obtained as functions of the spatial coordinate, half channel width to Debye length ratio, interfacial zeta potential, diffusivity difference parameter, and the power-law flow behavior index. The border curves of zero flow rate obtained herein delineating the parametric regimes in which the non-Newtonian diffusioosmotic flow rate is directed towards downstream or upstream are distributed differently on the zeta potential versus diffusivity difference parameter map as compared to those obtained using thin double layer theory. As compared to Newtonian liquids, the rheology of dilatant and pseudoplastic liquids respectively enlarges and reduces the parametric regimes in which flow rate towards downstream occurs, which is a result exactly opposite to that obtained based on the thin double layer theory reported in recent literature.

First-harmonic balance for fast evaluation of power-law fluid flow enhancement under periodic pressure gradient

This work considers rectilinear laminar flow of power-law liquids under a periodic pressure gradient. In contrast to Newtonian flow, previous studies have shown that the interaction between nonlinearities by non-Newtonian rheology and exogenous periodic forcing can yield important changes in the flow patterns, such as mean flowrate enhancement for shear-thinning fluid whereas flowrate reduction for shear-thickening fluid relative to non-forcing conditions. The aim of this work was to evaluate the ability of first-harmonic approximation for estimating mean flowrate enhancement/reductions under periodic pressure gradient oscillations. The first-harmonic balance involves the projection of nonlinearities on a first-harmonic Fourier basis, which reduces the governing partial differential equation into three ordinary boundary-value problems that should be solved numerically. The results showed that the predictions from the first-harmonic balance were in good agreement, both qualitative and quantitative, with direct numerical simulations of the governing partial differential equation. The advantage of the former over the latter is the computational burden since numerical computations for the first-harmonic equations does not depend of the forcing frequency.

Hydrodynamic stability of a layer of viscous gravity-driven power-law fluid flow under rotating centrifugal force

This article investigates the stability of a thin power-law liquid film flowing down along the outside surface of a rotating vertical cylinder. The result is improved through the introduction of the flow index and the rotating centrifugal force into the governing equation. In contrast to most previous studies presented in the literature, the solution scheme employed in this study is based on a power series approximation approach rather than an analytical method. The growth rate of the amplitudes, the wave speeds, and the threshold conditions are obtained subsequently as the by-products of stability solutions. When the fluid film flows down the outside surface of a stationary vertical cylinder, the degree of instability in the film flow is further intensified by the lateral curvature of the cylinder. As the cylinder starts to rotate, the flow becomes relatively more unstable as the cylinder radius is increased at larger values of the Rossby and Reynolds numbers. Notably, this finding is the opposite of that observed for film flows along a stationary vertical cylinder. Furthermore, the size of the supercritical region increases significantly as the cylinder rotates. The analysis results also indicate that by increasing the flow index, the film flow can become relatively less unstable while traveling down along the rotating vertical cylinder.

Effect of aspect ratio on natural convection in power-law liquids from a heated horizontal elliptic cylinder

The influence of aspect ratio and shear-dependent viscosity on free convection heat transfer from a horizontal heated elliptic cylinder in power-law fluids has been investigated. In particular, the coupled momentum and energy equations have been solved numerically over the following ranges of conditions: Grashof number, 10⩽Gr⩽105; Prandtl number, 0.72⩽Pr⩽100; power-law index, 0.3⩽n⩽1.5 and aspect ratio, 0.2⩽E⩽5. The new extensive results demonstrate the influence of the Grashof number (Gr), Prandtl number (Pr), power-law index (n) and aspect ratio (E) on the macroscopic heat and momentum transfer characteristics like local and average values of Nusselt number (Nu) and drag coefficients (CD). Further insights are developed by examining the structure of the flow and temperature fields adjacent to the cylinder. Broadly speaking, all else being equal, shear- thinning fluid behaviour promotes heat transfer whereas shear-thickening viscosity has a deleterious effect on it with reference to that in Newtonian fluids. Also, the rate of heat transfer gradually increases as the cylinder shape passes from blunt to slender with respect to the direction of gravity. Finally, the present numerical values of the Nusselt number are correlated using a simple analytical form which facilitates interpolation of the present results for the intermediate values of the governing parameters. The paper is concluded by presenting detailed comparisons with the previous numerical and experimental results available in the literature, especially in Newtonian fluids.

Modeling of aerated stirred tanks with shear-thinning power law liquids

Detailed simulations of aerated stirred tanks with shear-thinning power law liquids are presented. The lattice-Boltzmann scheme was used to discretize the filtered conservation equations of the liquid phase. The motion of bubbles was tracked based on the Euler–Lagrange approach with a bubble cluster concept. The collision, breakup and coalescence of bubbles were modeled as stochastic events. The predicted flow field of a single-phase stirred tank with shear-thinning power law liquid shows reasonable agreement with experimental data. For aerated systems, qualitatively similar gas holdup distribution was achieved when comparing the predicted result with experiments. Using the proposed modeling approach, it was found that a change in rheology alters the number mean diameter, Sauter diameter and the shape of bubble size distribution.

Heat transfer and entropy generation analyses associated with mixed electrokinetically induced and pressure-driven power-law microflows

In the present study, the first and second law analyses associated with combined pressure-driven and electroosmotic flow of power-law liquids through a uniform microannulus are performed numerically. In the first law analysis, the Poisson–Boltzmann and incompressible Navier–Stokes–Fourier (NSF) equations in the cylindrical polar coordinate reference frame as the governing equations are numerically solved using a finite difference method. The obtained velocity and temperature distributions are then employed to evaluate the associated Nusselt number, while simultaneous effects of viscous dissipation and Joule heating are taken into consideration. In the second law analysis, the relevant general expression for the entropy generation process is derived and the contributions of thermal diffusion, fluid friction and Joule heating irreversibility to the total entropy generation in the micro domain are examined. Finally, the variations of the aforementioned major variables with influential parameters such as the aspect ratio of the annuli, flow behavior index, dimensionless Debye–Hückel parameter, forcing ratio, dimensionless viscous heating parameter, and dimensionless Joule heating parameter are investigated.

Energy-based model for capillary spreading of power-law liquids on a horizontal plane

The capillary controlled spreading of a spherical-cap droplet of power-law fluid is theoretically investigated using a spreading kinetics model based on the classical energy-based approach. The model gives analytical predictions of the dynamic contact angles and spreading radius for shear-thinning, shear-thickening, and Newtonian fluids in complete or partial wetting systems. The model also shows that the non-Newtonian effects in the power-law liquids bring on an explicit effect of droplet volume on the dynamic contact angle. A modified capillary number, Can,V, which includes the volume, liquid properties and spreading speed, is proposed to characterize the spreading process. Experimental results in literature validate the model up to a modified capillary number of 0.1.