Decrease In Power-law Index Research Articles

Propagation prediction of asymmetrically originated fractures by use of displacement discontinuity method

In existing fracture propagation simulation studies, the fracture propagation simulation is based on the assumption that the fracture originates symmetrically with respect to the wellbore. However, the drilling treatments can induce stress reorientation near the wellbore, resulting in asymmetrically originated fractures, which is different from the assumption of existing studies. In such cases, it can be difficult to predict the fracture geometries. In this work, the authors propose a fracture propagation model to simulate the propagation of asymmetrically originated fractures with the displacement discontinuity method. With the aid of the proposed model, the authors investigate the impact of the redistribution of stress field on the propagation of asymmetrically originated fractures and conduct a comprehensive study to investigate the effects of various geological and fluid parameters on the propagation of asymmetrically originated fractures. The results show that the redistribution of the stress field renders the newly generated fracture segments to deviate from the initial originations so that although the initial fracture origination is asymmetric, the asymmetrically originated fractures rapidly propagate along an approximately symmetrical path and propagate parallel to the direction of maximum principal stress as it goes away from the wellbore. The closer the two wings of the fracture are to the symmetric origin, the faster it is for the change of propagation direction to the direction of the maximum principal stress. A large principal stress difference tends to cause asymmetrically originated fractures to quickly deviate from the initial originations and propagate parallel to the direction of maximum principal stress. As the fluid power law index decreases, the fluid viscosity is significantly reduced due to the effect of shear thinning, causing the asymmetrically originated fractures to propagate parallel to the direction of maximum principal stress. The fluid leak-off coefficient only influences the fracture length and width, whereas its influence on the propagation trajectory can be slight.

Shape stability of a microbubble in a power–law liquid

The onset of non-spherical oscillations of a microbubble in an unbounded power–law liquid, important for biomedical ultrasound applications, is studied. Two sets of evolution equations are obtained from the equation of motion: a Rayleigh Plesset-type equation for the spherical oscillations and an equation for the non-spherical oscillations. The non-spherical oscillations are modeled using the perturbation method via the Legendre polynomials. Two kinds of instabilities, namely parametric and Rayleigh-Taylor instabilities, are investigated. A higher power–law index causes the damping of the oscillations for both spherical and non-spherical oscillations. The power–law index damping effect depends on the ultrasonic drive frequency. At natural frequency, the amplitude of the perturbations is high compared to the non-resonant cases. At a low consistency index, the damping effect of the power–law index decreases. Unlike Newtonian liquids, the viscosity of power–law liquids is affected by the frequency of the acoustic field, thereby affecting Rayleigh-Taylor instability.

Molecular dynamics simulation of the flow mechanism of shear-thinning fluids in a microchannel

Shear-thinning fluids have been widely used in microfluidic systems, but their internal flow mechanism is still unclear. Therefore, in this paper, molecular dynamics simulations are used to study the laminar flow of shear-thinning fluid in a microchannel. We validated the feasibility of our simulation method by evaluating the mean square displacement and Reynolds number of the solution layers. The results show that the change rule of the fluid system’s velocity profile and interaction energy can reflect the shear-thinning characteristics of the fluids. The velocity profile resembles a top-hat shape, intensifying as the fluid’s power law index decreases. The interaction energy between the wall and the fluid decreases gradually with increasing velocity, and a high concentration of non-Newtonian fluid reaches a plateau sooner. Moreover, the velocity profile of the fluid is related to the molecule number density distribution and their values are inversely proportional. By analyzing the radial distribution function, we found that the hydrogen bonds between solute and water molecules weaken with the increase in velocity. This observation offers an explanation for the shear-thinning phenomenon of the non-Newtonian flow from a micro perspective.

Numerical simulation of non-Newtonian Carreau fluid in a lid-driven cavity

In this work, the 2D lid-driven cavity flow of non-Newtonian Carreau fluids has been studied by finite difference method on a staggered grid. A finite-difference algorithm on staggered grid based on projection method is adopted to solve the lid-driven cavity flow, which includes a second-order central difference scheme for the non-Newtonian viscous stress term. This study has been conducted for the certain pertinent parameters of Reynolds number (Re=100-1000), power-law index (n=0.6-1.4). The results show that as the Reynolds number increases, the influence of the power-law index on the flow increases. As the power-law index decreases, the flow field becomes more complicated.

Effects of a centered conducting body on natural convection of a non-Newtonian fluid in a square enclosure with time-periodic temperature distribution

In this paper, a modified lattice Boltzmann model that incorporates the effect of heat capacity is adopted to study the effects of a centered conducting body on the natural convection of a non-Newtonian fluid in a square cavity with time-periodic temperature distribution. The effects of power-law index, Rayleigh number, heat capacity ratio, thermal conductivity ratio, body size, temperature pulsating period, and temperature pulsating amplitude on fluid flow and heat transfer are analyzed in detail. The results showed that the increase of Rayleigh number and thermal conductivity ratio as well as the decrease of power-law index can strengthen both transient and global heat transfer, while the increase of heat capacitance of the fluid to the solid wall can enhance the transient heat transfer and has little effect on the overall heat transfer. Furthermore, the increase of body size will reduce both the transient heat transfer ratio and the overall heat transfer ratio. In addition, the decrease of temperature pulsating period can enhance the transient heat transfer, but will slightly weaken the overall heat transfer. Finally, the results show that both the transient and overall heat transfer ratios increase with the increase of temperature pulsating amplitude.

Numerical simulation of shear-thinning droplet impact on surfaces with different wettability

Controlling impact dynamics of droplets on solid surfaces is a significant problem in a variety of applications, such as inkjet printing, spray cooling and coating and so on. Most of fluids used in industries always contain various kinds of additives such as surfactants, polymers and particles. Therefore, these fluids exhibit non-Newtonian behaviors, for instance, yield-stress, viscoelastic, shear-thickening and shear-thinning. The impact dynamics of Newtonian droplets on solid surfaces has been extensively investigated. However, the number of researches about fluids with non-Newtonian properties is comparatively very small. In this work, we employ the finite element scheme coupled with level set method to simulate the impact process of droplets on solid surfaces. The numerical simulation models the presence of shear-thinning viscosity by using the truncated power-law rheological model. We first conduct a mesh convergence study and verify the numerical model. The simulation results are found to be in good agreement with experimental data in the literature. By performing extensive numerical simulations and varying the rheological parameters and surface wettabilities, the influences of these parameters on the impact dynamics are evaluated, and the dominant effects that govern the spreading and receding process are determined. The simulation results show that for the case of droplet impacting on surface with contact angle <i>θ</i> = 55°, the spreading is stronger with power-law index decreasing as evidenced by larger shape deformation and faster interface moving speed. As power-law index decreases, we expect the maximum dimensionless diameter to increase and the minimum dimensionless height to decrease during inertial spreading. For the case of droplet with lower power-law index (<i>m</i> = 0.85 and 0.80), which indicates lower viscous dissipation during impact, the dimensionless parameters have significant differences. After first receding, the impacting droplet is not balanced any more and it starts to spread again until its kinetic energy is completely damped by fluid viscous dissipation. For the case of droplet (<i>m</i> = 0.80) impacting on surface, the center breakage can be observed during droplet spreading, which results from the effect of strong shear-thinning property. When a shear-thinning droplet impacts on a surface with contact angle <i>θ</i> = 100°, the oscillation behavior can be observed and the oscillation amplitude increases as power law index decreases. Bouncing phenomenon can be observed when a droplet impacts on surface with contact angle <i>θ</i> = 160°, regardless of rheological property. Finally, we propose an empirical model to predict the maximum dimensionless diameter of shear-thinning droplet impacting on the surface with contact angle <i>θ</i> = 55° as a function of non-Newtonian Reynolds number <i>Re</i><sub>n</sub>.

Numerical study on combined buoyancy–Marangoni convection heat and mass transfer of power-law nanofluids in a cubic cavity filled with a heterogeneous porous medium

In this paper, the combined buoyancy–Marangoni convection of non-Newtonian power-law nanofluids in a 3D heterogeneous porous cubic cavity is investigated in detail with the compact high order finite volume method. Special attentions are given to detect the effects of heterogeneity level, Marangoni number, thermal Rayleigh number, buoyancy ratio and nanoparticle volume fraction on the fluid flow as well as on rates of heat and mass transfer. It is observed that as a result of the exponential distribution of the permeability, the heat and mass transfer rates reduce as the level of heterogeneity enhances. The effect of the surface tension on the heat and mass transfer intensity becomes insignificant when the buoyancy force is strengthened. Further, the average Nusselt and Sherwood numbers increase as the Marangoni number and thermal Rayleigh number increase due to the combined effects of buoyancy and surface tension. The augmentation of the buoyancy ratio causes convection heat and mass transfer to increase for the thermal dominated flow while decrease in buoyancy ratio also augments it for the solutal dominated flow. The heat transfer (mass transfer) rate is found to increase (decrease) with increasing the nanoparticle volume fraction. Moreover, for all above studied parameters, an intensification of the flow and an increase in average Nusselt and Sherwood numbers occur with the decrease in power-law index.

THEORETICAL STUDY ON STRESS SENSITIVITY OF FRACTAL POROUS MEDIA WITH IRREDUCIBLE WATER

The couple flow deformation behavior in porous media has drawn tremendous attention in various scientific and engineering fields. However, though the coupled flow deformation mechanism has been intensively investigated in the last decades, the essential controls on stress sensitivity are not determined. It is of practical significance to use analytic methods to study stress sensitivity of porous media. Unfortunately, because of the disordered and extremely complicated microstructures of porous media, the theoretical model for stress sensitivity is scarce. The goal of this work is to establish a novel and reasonable quantitative model to determine the essential controls on stress sensitivity. The predictions of the theoretical model, derived from the Hertzian contact theory and fractal geometry, agree well with the available experimental data. Compared with the previous models, our model takes into account more factors, including the influence of the water saturation and the microstructural parameters of the pore space. The proposed models can reveal more mechanisms that affect the coupled flow deformation behavior in fractal porous media. The results show that the irreducible water saturation increases with the increase of effective stress, and decreases with the increased rock elastic modulus (or increased power law index) at a given effective stress. The effect of stress variation on porosity is smaller than that on permeability. Under a given effective stress, the normalized permeability (or the normalized porosity) becomes smaller with the decrease of rock elastic modulus (or the decrease of power law index). And a lower capillary pressure will correspond to an increased rock elastic modulus (or an increased power law index) under a given water saturation.

A Computational Study about the Effect of Turbines Pitched Blade Attack Angle on the Power Consumption of a Stirred Tank

In this study, the stirring mechanism of shear-thinning fluids benefiting from four blades in turbulent flow is considered. The fluid is studied inside a stirred cylindrical tank with a flat bottom. The height of fluid is equal to the cylinder’s diameter and the impeller is positioned centrally. A CFD simulation has been carried out and three-dimensional turbulent flow is numerically analyzed using the Shear Stress Transport k-ω (k- ω SST) model. The parameters related to power consumption including attack angle and flow index were studied. The power consumed during the mixing of the shear thinning liquids under a specific Reynolds number and attack angle is less than that consumed when the fluid used is water, which is a Newtonian fluid. As the power law index decreases, the corresponding power consumption also declines. At a certain attack angle and power law index, an increase in the Reynolds number first significantly decreases power consumption; beyond a given range, the consumption plateaus. To validate the numerical simulation results, the findings derived on the basis of the power number used in this work were compared with the test results of other studies, and good agreement was observed.

The effect of conductive baffles on natural convection in a power-law fluid-filled square cavity

This research involves the numerical study of the conjugate natural convection of a power-law fluid in a two-dimensional square cavity with a pair of conducting baffles. The lower wall of the cavity contains a heat source with constant heat flux, while a pair of conductive baffles is embedded on its upper wall. The left and right side walls and some parts of the lower wall of the cavity are thermally insulated and its upper wall is kept at a low temperature. The governing equations along with the corresponding boundary conditions are solved using the numerical finite difference method based on the control volume formulation and SIMPLE algorithm. The effects of pertinent parameters on flow and temperature fields and heat transfer rate are investigated. From the results of the numerical solution, increasing the Rayleigh number and baffles thermal conductivity enhance the thermal performance of the cavity. So that for $$ Ra = 10^{6} , $$ with an increase in the thermal conductivity ratio from $$ 1 $$ to $$ 100, $$ the average Nusselt number increases by about 14% for both cases $$ n = 0.6 $$ and $$ n = 1.2 $$ . As well as a decrease in power-law index, except in low Rayleigh numbers, increases the heat transfer rate. Based on the results, the increase of the average Nusselt number is about 283% for a power-law index reduction in the range of $$ 0.6 \le n \le 1.2 $$ at $$ Ra = 10^{6} $$ . Also, an increase of the length of the baffles deteriorates the thermal performance of the cavity at high Rayleigh numbers, while enhances it at low Rayleigh numbers.

Torque rheological properties of polypropylene/cocoa pod husk composites

The torque rheological properties of plastic wood composites are important to practical processing, but the research in this field is rare. In present, a Brabender plastrograph torque rheometer was used to analyse the rheological behavior of polypropylene (PP)/cocoa pod husk (CPH) composites. The effect of processing parameter, filler content, and addition of maleated polypropylene (MAPP) on torque rheological properties was investigated. The torque rheological data found that the processing torque increased with the increases of rotor speed, filler content, and addition of MAPP. The PP/CPH composites melt behavior as pseudoplastics and shear thinning occurred at higher shear rate. The decrease of power law index ( n) evidenced the pseudoplasticity of PP/CPH composites increased at higher filler content and presence of MAPP. The increase of viscosity on PP/CPH was due to filler–filler interaction at higher filler content and strong filler–matrix adhesion after addition of MAPP. The activation energy of PP/CPH composites also increased with higher amount of CPH and addition of MAPP.

Unsteady Convective Heat Transfer of Power-Law Fluid with Variable Fluid Properties in a Concentric Annulus Originating from a Polymer Flooding Process

We study the unsteady convective heat transfer of power-law fluid with variable fluid properties in a concentric annulus with isothermal surface. The problem is originated from the polymer flooding process between a sucker rod and oil well. A new power-law rheological model is proposed, which takes the effects of temperature on fluid viscosity and thermal conductivity into account. Numerical solutions are presented for velocity and temperature fields using the Chebyshev spectral method coupled with the strong stability-preserving Runge–Kutta time discretization. The exponential convergence is verified by accuracy testing between a smooth exact solution of the Partial Differential Equations (PDEs) with source terms and the numerical approximation of manufactured solutions. It is found that heat transfer is enhanced in the variable power-law index model, and a decrease in power-law index of pseudoplastic fluids promotes heat transfer due to the increased Nusselt number. Moreover, the influences of other parameters on convective heat transfer behaviors are discussed in detail.

Slip-Flow and Heat Transfer in a Porous Microchannel Saturated with Power-Law Fluid

This study aims to numerically examine the fluid flow and heat transfer in a porous microchannel saturated with power-law fluid. The governing momentum and energy equations are solved by using the finite difference technique. The present study focuses on the slip flow regime, and the flow in porous media is modeled using the modified Darcy-Brinkman-Forchheimer model for power-law fluids. Parametric studies are conducted to examine the effects of Knudsen number, Darcy number, power law index, and inertia parameter. Results are given in terms of skin friction and Nusselt number. It is found that when the Knudsen number and the power law index decrease, the skin friction on the walls decreases. This effect is reduced slowly while the Darcy number decreases until it reaches the Darcy regime. Consequently, with a very low permeability the effect of power law index vanishes. The numerical results indicated also that when the power law index decreases the fully-developed Nusselt number increases considerably especially, in the limit of high permeability, that is, nonDarcy regime. As far as Darcy regime is concerned the effects of the Knudsen number and the power law index of the fully-developed Nusselt number is very little.

말단캡핑제를 이용한 폴리카보네이트의 분자량 특성 조절

말단캡핑제(end capping agent) 6종류에 대해 폴리카보네이트를 용융중합법으로 제조하였다. 합성된 폴리카보네이트의 화학구조는 FT-IR 스펙트럼을 이용하여 확인하였다. 평균분자량 및 분자량분포도, 유리전이 온도 및 열분해 온도는 GPC, DSC와 TGA를 이용하여 측정하였다. 말단캡핑제의 화학구조에 따라 평균분자량이 증가 또는 감소를 나타내었으며, 말단캡핑제로 4-tert-butylphenol (TBP)이 사용되었을 때 최적의 분자량 조절 결과를 나타내었다. 말단캡핑제의 함량이 증가할수록 평균분자량은 감소하였고, 4-tert-butylphenol이 0.05 - 0.1 mol%가 첨가될 때 폴리카보네이트의 대형 사출물 가공에 적합한 20,000 - 30,000 정도의 수평균분자량을 나타내었다. 폴리카보네이트의 용융점도와 유리전이온도는 분자량이 감소할수록 감소하였다. 말단캡핑제의 투입방법 변경은 PC의 분자량분포도에 영향을 주었고, power law index의 감소를 보여주었다. Polycarbonates (PCs) with six different end capping agents were synthesized from melt polymerization. Chemical structure of the synthesized PC was determined by FT-IR spectroscopy. The average molecular weight and distribution, glass transition and thermal degradation temperatures were determined by GPC, DSC and TGA. Average molecular weight changed with the chemical structure of end capping agent, and 4-tert-butylphenol was estimated as the optimum end capping agent. The average molecular weights of PCs decreased with the concentration of the agent, the number average molecular weight was observed as 20,000 - 30,000 when 0.05-0.15 mol% of 4-tert-butylphenol added in PCs. The melt viscosities and glass transition temperature of the PCs decreased with molecular weight. The change for adding method of the agent affected on both the molecular weight distribution and decrease in power law index.

Non-Newtonian power-law flow across a confined triangular bluff body in a channel

Wall effects on the flow of incompressible non-Newtonian power-law fluids across an equilateral triangular cylinder confined in a horizontal plane channel have been investigated for the range of conditions: Reynolds number, Re=1–40, power-law index, n=0.4–1.8 (covering shear-thinning, Newtonian and shear-thickening behaviors) and blockage ratio=0.125–0.5. Extensive numerical results on flow pattern, wake/recirculation length, individual and overall drag coefficients, variation of pressure coefficient on the surface of the triangular cylinder and so forth are reported to elucidate the combined effect of power-law index, blockage ratio and Reynolds number. The size of vortices decreases with an increase in the value of the blockage ratio and/or power-law index. For a fixed value of the Reynolds number, individual and overall drags decrease with decrease in power-law index and/or blockage ratio in steady confined flow regime. Simple correlations of wake length and drag are also obtained for the range of settings considered.

FEM analysis for different configurations of non‐recessed hole‐entry hybrid journal bearings for non‐Newtonian lubricants

PurposeThe present work aims to predict accurately the bearing design data for non‐recessed hybrid journal bearings, considering the effect of non‐Newtonian behavior of lubricant for different symmetric and non‐symmetric bearing geometric configurations.Design/methodology/approachThe simultaneous solution of generalized Reynold's equation governing the laminar flow of incompressible lubricant and the equation of flow of lubricant through the capillary restrictor, considering variable viscosity of lubricant following the “Power law”, has been carried out using FEM. For a given set of bearing geometric, operating parameters and for given external vertical load, the values of various performance characteristics have been obtained for a range of values of power law index, after establishing the journal center equilibrium position, the analysis for which has been elaborately explained.FindingsThe results obtained have been presented graphically for various bearing performance characteristics. It has been observed that with decrease in power law index “n”(0<n≤1), the value of h¯min and load carrying capacity decreases, while bearing flow rate increases for all configuration. The load‐carrying capacity of asymmetric configurations is better and stable over entire range of restrictor design parameter. Bearing configuration with land width ratio = 0.25 and aspect ratio = 1.0, having two rows of holes and six holes in each row, will be better suited for high‐load support, as it has maximum value of minimum fluid film thickness, moderate value of bearing flow and value of attitude angle is almost constant.Originality/valueThe performance characteristics of journal bearing have been presented for a wide range of values of power law index and for different values of restrictor design parameter for capillary restrictor, after establishing the journal center equilibrium position. The comparison of the different symmetric and non‐symmetric journal‐bearing configurations to find the best geometric configuration at different operating conditions, considering the effect of non‐Newtonian behavior of lubricant, represents the originality of the work.

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.

Fluid Flow and Heat Transfer in Power-Law Fluids Across Circular Cylinders: Analytical Study

An integral approach of the boundary layer analysis is employed for the modeling of fluid flow around and heat transfer from infinite circular cylinders in power-law fluids. The Von Karman-Pohlhausen method is used to solve the momentum integral equation whereas the energy integral equation is solved for both isothermal and isoflux boundary conditions. A fourth-order velocity profile in the hydrodynamic boundary layer and a third-order temperature profile in the thermal boundary layer are used to solve both integral equations. Closed form expressions are obtained for the drag and heat transfer coefficients that can be used for a wide range of the power-law index, and generalized Reynolds and Prandtl numbers. It is found that pseudoplastic fluids offer less skin friction and higher heat transfer coefficients than dilatant fluids. As a result, the drag coefficients decrease and the heat transfer increases with the decrease in power-law index. Comparison of the analytical models with available experimental/numerical data proves the applicability of the integral approach for power-law fluids.

Roll-over-web Coating of Pseudoplastic and Viscoplastic Sheets Using the Lubrication Approximation

The Lubrication Approximation Theory (LAT) is used to provide numerical results in roll coating over a moving flat web. The Herschel–Bulkley model of viscoplasticity is used, which reduces with appropriate modifications to the Bingham, power-law, and Newtonian models. Results are obtained for such quantities as coating thickness, separation point, and the volumetric flow rate required for various values of the power-law index (in the case of pseudoplasticity) and of the Bingham number (in the case of viscoplasticity). All these values increase substantially with the increasing non-Newtonian character of the fluid. Yielded and unyielded areas are quantitatively shown for several cases of viscoplasticity. Pressure gradient and pressure distributions are given for all cases. Integrated quantities of engineering interest are also calculated. These include the maximum pressure, the roll/sheet separating force, and the power input to the roll. These quantities increase substantially and monotonically in a dimensionless form, as the power-law index decreases or the Bingham number increases.

Forced convection of a power-law fluid in a porous channel - numerical solutions

A detailed numerical study of laminar forced convection in a porous channel which contains a fibrous medium saturated with a power-law fluid was performed. Hydrodynamic and heat transfer results are presented for a configuration that has uniform heat flux or uniform temperature heating at the walls. The flow in the porous medium was modeled using the modified Brinkman-Forchheimer-extended Darcy model for power law fluids in which the non-Darcy effects of inertia and boundary were considered. Parametric studies were conducted to examine the effects of Darcy number, power law index, inertia parameter and Prandtl number. The results indicate that when the power law index is decreased, the velocity gradient near the walls increases but these effects are reduced gradually as the Darcy number decreases until the Darcy regime (Da≤10−6) is reached in which case the effects of power law index become negligible. As the power law index is decreased, the thermal boundary layer thickness decreases significantly only in the non-Darcy regime. Consequently, as the power law index decreases, the fully developed Nusselt number increases considerably in the non-Darcy regime whereas in the Darcy regime the change in Nusselt number is very small. As the Prandtl number increases, the local Nusselt number increases and this effect is more significant for shear thinning fluids (n<1.0).