Generalized Power Law Model Research Articles

GENERAL SOLUTION OF TWO-DIMENSIONAL PROJECTILE MOTION WITH AIR RESISTANCE

In this study, two-dimensional projectile motion is considered under the effect of a general power law model of air resistance. Classically, a projectile is treated as a point mass with mass m moving in a uniform gravitational field. The projectile is launched from the ground with an angle α to horizon. the drag force is assumed to be proportional to the speed raised to the power n. The analysis of the problem is performed using Cartesian coordinates. A general exact parametrical solution (with respect to the angle of motion) is derived for any power n, following simple steps: 1) find the speed in the direction of the axis x (horizontal – no gravity) 2) find the vertical component of the speed 3) find the time 4) find the horizontal position of the projectile and finally,5) find the vertical position of the projectile. Steps 1) and 2) give explicit closed form equations and the rest are given by exact integrals which can be solved numerically. In this study spreadsheet calculation are performed using trapezoidal rule of integration. The cases of motion in avacuum and linear drag law are used to check the accuracy of the numerical calculations.The importance of the proposed study is three-fold: a) The method of the derived solution is new, and couldnt be found elsewhere b) The derived equations make it possible to use spreadsheets for presenting the subject (no programming is required), and thus, serve as a tool to enhance teaching c) The derived equations are general for any power n thus, the same procedure could be used to find the position of the projectile at any time.

Generalized power law model of blood flow in a stenosed bifurcated artery

Stenosis is a blockage caused by atherosclerosis literally lead to serious circulatory complaints, by narrowing the vessel wall and causing an alternation in the flow structure which consequently reduced the fluid flow passing to the other organs and tissues. This study, the geometry of the bifurcated is considered with presence of stenosis at mother and daughter artery. The blood vessel is modelled as a two-dimensional (2D) rigid wall since the wall of a disease artery is reported to be less compliant. The blood will be assumed as incompressible, laminar, steady and characterized as the generalized power-law model. Simulation result is obtained by using COMSOL Multiphysics 5.2, a software based on the FEM to solve the problem. Results concerning the effect of different blood rheology and severity of stenosis on the streamlines pattern are discussed.

Numerical investigation of capture efficiency of carrier particles in a Y-shaped vessel considering particle-particle interaction and Non-Newtonian behavior

Chemotherapy kills the cancer cells by disrupting cell division. Although chemotherapy is an efficient treatment of cancer, it has many side effects since drugs circulate throughout the body. A promising approach to reduce these side effects is targeted drug delivery. The present study investigates the capture efficiency of carrier particles in a Y-shaped vessel under the effect of an external magnetic field produced by a current-carrying wire. Different parameters including magnetic, drag, buoyancy, and Lorentz forces have been considered. In addition, the effect of particle-particle interaction and non-Newtonian behavior of blood have been included. The generalized power-law (GPL) and modified Casson models are used for modeling non-Newtonian behavior. Results show that capture efficiency (CE) is increased as the blood velocity and the particle diameter are reduced and increased, respectively. Besides, it was found that considering particle-particle interaction has led to a reduction in CE in all cases. For example, the CE of nanoparticles for the case without interaction force is 51.66%, while for the case with interaction force is 43.33%, showing a nearly 16% reduction. Besides, it was observed that modified Casson provides a more accurate prediction for blood behavior than the GPL model.

Effect of Different Types of Stenosis on Generalized Power Law Model of Blood Flow in a Bifurcated Artery

This study is focus on generalized power law model of blood flow in a stenosed bifurcated artery under the effect of different types of stenosis. Stenosis can cause the narrowing of the artery that may reduce the flow of blood supply to the heart, and this may lead to the heart attacks. The geometry of the bifurcated artery with different classification of stenosis locations is considered in order to shows four possible morphologies formation of plaque from healthy artery to disease artery. The bifurcated artery is modelled as a two-dimensional rigid wall since the wall of a disease artery is reported to be less flexibility. Few assumptions are considered such as blood are incompressible, laminar, steady and characterized as the generalized power-law model. Simulation results are obtained using COMSOL Multiphysics 5.2, which is a software that based on the finite element method to solve this problem. Results concerning the effect of different locations of stenosis on generalized power law model of the blood flow characteristic such as streamlines pattern are discussed.

Dynamic strain aging of an AISI 1025 steel coil and its relationship with macroscopic responses during the upsetting process

In this paper, dynamic strain aging (DSA) of an AISI 1025 steel coil and related strain hardening behavior are described, including their macroscopic effects. A cylinder compression test was conducted for various sample temperatures and strain rates ranging from 20°C to 500°C and 1/s to 20/s, respectively. The experimental flow stress data were indicative of typical DSA, which caused serrated flow; this was verified by the strain hardening rate obtained from the Kocks-Mecking hardening model. The experimental data were mathematically fitted by a simple but general power law model, with piecewise bi-linear functions of sample temperature and strain for both the strength constant and strain rate sensitivity exponent. The flow stresses extracted from JMatPro were used as the theoretical flow stresses for comparison with fitted flow stresses associated with DSA, to characterize the effects. The fitted flow stresses were validated experimentally by compression tests conducted at various sample temperatures and strain rates; the varying conditions had a negligible effect despite the complexities in the flow stress due to DSA. An instability index was proposed to evaluate the macroscopic effect of DSA on upsetting processes in non-isothermal cylinder compression tests. This index was used to quantify the macroscopic plastic deformation behavior of the material with DSA.

Hemodynamic analysis for stenosis microfluidic model of thrombosis with refined computational fluid dynamics simulation

Disturbed blood flow has been increasingly recognized for its critical role in platelet aggregation and thrombosis. Microfluidics with hump shaped contractions have been developed to mimic microvascular stenosis and recapitulate the prothrombotic effect of flow disturbance. However the physical determinants of microfluidic hemodynamics are not completely defined. Here, we report a refined computational fluid dynamics (CFD) simulation approach to map the shear rate (γ) and wall shear stress (τ) distribution in the stenotic region at high accuracy. Using ultra-fine meshing with sensitivity verification, our CFD results show that the stenosis level (S) is dominant over the bulk shear rate (γ0) and contraction angle (α) in determining γ and τ distribution at stenosis. In contrast, α plays a significant role in governing the shear rate gradient (γ′) distribution while it exhibits subtle effects on the peak γ. To investigate the viscosity effect, we employ a Generalized Power-Law model to simulate blood flow as a non-Newtonian fluid, showing negligible difference in the γ distribution when compared with Newtonian simulation with water medium. Together, our refined CFD method represents a comprehensive approach to examine microfluidic hemodynamics in three dimensions and guide microfabrication designs. Combining this with hematological experiments promises to advance understandings of the rheological effect in thrombosis and platelet mechanobiology.

Develop Boltzmann equation to simulate non‐Newtonian magneto‐hydrodynamic nanofluid flow using power law magnetic Reynolds number

The single relaxation D2Q9 lattice Boltzmann method (LBM) is run in the current research beside the generalized power law model for simulation of non‐Newtonian magneto‐hydrodynamics (MHD) laminar flow field inside a channel with local symmetric constriction. Analytical results of non‐Newtonian fluid flow in a channel without magnetic field, as well as Newtonian fluid flow at various Hartmann No., are used to validate the numerical model. Then, fluid flow simulation is performed for non‐Newtonian fluid with different power law index at various Hartmann No. (Ha) whereas Reynolds No. are set to be constant in all cases. The present non‐Newtonian fluid can be achieved by adding various nanoparticles such as MWCNT to the base fluid. To explore the effect of magnetic Reynolds No. (Rem), the fluid flows with different magnetic resistivity are also simulated. Results show that the separation can be controlled by a magnetic field with the penalty of larger friction coefficient and pressure loss along the channel length. In fact, for a specified Rem, the higher the Ha, the larger the pressure loss. It is also observed that the pressure loss is larger for fluids flow with higher power law index and lower Rem.

Considerations for readdressing theoretical descriptions of particle-reinforced composite food gels.

The aim of this work was to address the ability of established theoretical models to describe the small deformation mechanical properties of particle-filled food protein gels. To this end, the effect of incorporating glass microspheres on the elastic modulus of heat-set whey protein isolate/xanthan gum gels is reported. Filler size and polydispersity strongly influenced the observed reinforcement with increasing filler content; however, these effects were also strongly correlated to the ionic strength of the gelator phase (0-200 mM NaCl). Fillers with greater polydispersity provided less reinforcement at high filler content, which was associated with improved packing efficiency. Increasing ionic strength reduced the extent of filler/matrix binding, drastically reducing the impact of the smaller glass microspheres (4 μm, 7-10 μm). Larger particles increased the elastic modulus at high salt content due to interfacial stress concentration and particle-particle contacts. Theoretical fits could not satisfactorily describe the general trend in reinforcement observed with increasing filler content, despite employing various methods to account for the effects of filler self-crowding. Using an empirical approach, we propose an alternative functional form which provides improved fits over the entire range of filler content investigated. This general power law (GPL) model provided physically reasonable values for the maximum packing fraction through an empirically-derived expression for the scaling exponent. A weighted average approach was also proposed to incorporate effects of imperfect filler/matrix adhesion. This method incorporates contributions of both bound and unbound fillers, providing a means to model the effect of increasing ionic strength.

The Discrepancy between Einstein Mass and Dynamical Mass for SIS and Power-law Mass Models

Abstract We investigate the discrepancy between the two-dimensional projected lensing mass and the dynamical mass for an ensemble of 97 strong gravitational lensing systems discovered by the Sloan Lens ACS Survey, the BOSS Emission-Line Lens Survey (BELLS), and the BELLS for GALaxy-Lyα EmitteR sYstems Survey. We fit the lensing data to obtain the Einstein mass and use the velocity dispersion of the lensing galaxies provided by the Sloan Digital Sky Survey to get the projected dynamical mass within the Einstein radius by assuming the power-law mass approximation. The discrepancy is found to be obvious and quantified by Bayesian analysis. For the singular isothermal sphere mass model, we obtain that the Einstein mass is 20.7% more than the dynamical mass, and the discrepancy increases with the redshift of the lensing galaxies. For the more general power-law mass model, the discrepancy still exists within a 1σ credible region. We suspect the main reason for this discrepancy is mass contamination, including all invisible masses along the line of sight. In addition, the measurement errors and the approximation of the mass models could also contribute to the discrepancy.

Generalized Power-Law Model of Magnetohydrodynamic Blood Flow with Heat Transfer

Unsteady non-Newtonian blood fow characterized by the generalized power-law model subjected to an external magnetic feld together with heat transfer through stenosed artery has been developed. The arterial segment is assumed to be a cylindrical tube and the arterial wall is considered to be fexible having cosine shaped stenoses a condition due to the abnormal narrowing of a blood vessel. The full equations comprising the governing equations of motion, heat equation, the initial and the boundary conditions are solved using numerical procedure involves the discretization of the equations using the Marker and Cell (MAC) method, where pressure along the artery is calculated iteratively using the Successive-Over-Relaxation (SOR) technique. The results demonstrated through the simulations that under the infuence of magnetic feld, the blood temperature distribution over the entire arterial segment increases. The present results also predict the effects of the generalized power-law index and the Prandtl number on the distribution of blood temperature.

Blood flow through an inclined stenosed artery

The present work is to investigate the effect of the inclination angle on blood flow through the flexible stenosed artery. The mathematical model of blood considered laminar, incompressible, unsteady and fully developed, is represented by the generalized power law model. The numerical Marker and Cell method (MAC) has been utilized to solve the continuity and momentum equations in cylindrical coordinate. The Poisson equation for the pressure has been solved using Successive-Over-Relaxation method (SOR) and the pressure–velocity correction formula has been derived. The effects of inclination angle ( ) and power law index (n) on the flow characteristics of blood such as velocity, pressure drop, and wall shear stress are presented by their representation graphs.

Relaxation rates in the Maxwellian collision model and its variable hard sphere surrogate

The variable hard sphere and related models have proven to be accurate and computationally convenient replacements for the inverse power law model of classical kinetic theory in direct simulation Monte Carlo calculations. We attempt to provide theoretical support for this remarkable success by comparing the relaxation rates in the linearized Boltzmann equation for the Maxwellian collision model with those of its variable hard sphere surrogate. The comparison demonstrates that the linearized collision operator with variable hard sphere interactions can accurately approximate the linearized collision operator with Maxwellian inverse power law interactions under well-defined and broadly applicable conditions. Extensions of the analysis to the general inverse power law model and to more realistic intermolecular potentials are briefly discussed.

Physical Properties of Aqueous Solutions of Pectin Containing Sunflower Wax

AbstractThe aim of this study is to investigate the physical properties of aqueous solutions of pectin (PA) containing sunflower wax (SFW), which are used as a basis for producing edible films. The stability and the rheological and microstructural characteristics of SFW/PA mixtures were evaluated. SFW/PA mixtures formed oil‐in‐water emulsions that were milky and opaque in appearance and were stable towards phase separation. Polarized micrographs revealed the presence of wax crystals, whose size decreased as pectin concentration increased. The rheological behavior of the aqueous solutions of pectin containing different amounts of SFW were best described by the generalized power law model of Herschel–Bulkley (H–B), which gave the best fit in all the range of shear rate values. Apparent viscosities and yield stress were determined using this model, and both properties increased with increasing pectin content. The apparent viscosity values were between 0.0095 and 0.1031 Pa s. SFW addition resulted in a small decrease in viscosity for emulsions formulated with 1 and 2 % PA, but the opposite effect was observed for emulsions formulated with 3 % PA. In addition, shear stress values were higher for emulsions with higher PA content, but were not affected by SFW addition.

A MODEL FOR DIELECTRIC CHARACTERIZATION OF NANOCOMPOSITE POLYMERIC INDUSTRIAL MATERIALS

Interphase Power Law IPL model is based on a simple extension of a general power law model in which composite system containin g filler interphase and matrix regions may be treated as a unique three-com ponent composite system comprising two primary components (matrix and fille r) and an interphase region that is inextricably dependent upon the char acteristics of the filler and matrix components. In this paper, it can be investi gated that all possible causes and effects of the interphase region on a variety o f complex composite systems and so, the effects of the composite filler types and filler surfaces areas as well as the dielectric characteristics of the interphase region have been investigated. Effects of the composite filler types and filler surface areas as well as the dielectric characteristics of the interphase region have been explored on new industrial materials. Finally, this research explains the dielectric characteristics of new nanocomposite pol ymeric materials by IPL.

High-temperature and high-concentration SCR of NO with NH3: Application in a CCS process for removal of carbon dioxide

This study investigates several commercial selective catalytic reduction (SCR) catalysts (A–E) for application in a high-temperature (approximately 525°C) and high-concentration (5000ppm NO) system in combination with CO2 capture. The suggested process for removing high concentrations of NOx seems plausible and autothermal operation is possible for very high NO concentrations. A key property of the catalyst in this system is its thermal stability. This was tested and modelled with the general power law model using second-order decay of the BET surface area with time. Most of the materials did not have very high thermal stability. The zeolite-based materials could likely be used, but they too need improved stability. The SCR activity and the possible formation of the by-product N2O were determined by measurement in a fixed-bed reactor at 300–525°C. All materials displayed sufficiently high activity for a designed 96% conversion in the twin-bed SCR reactor system proposed. The amount of catalyst needed varied considerably and was much higher for the zeolithic materials. The formation of N2O increased with temperature for almost all materials except the zeolithic ones. The selectivity to N2 production at 525°C was 98.6% for the best material and 95.7% for the worst with 1000ppm NOx in the inlet; at 5000ppm NOx, the values were much better, i.e., 98.3 and 99.9%, respectively.

Effect of Non-Newtonian Pulsatile Blood Flow on Temperature Distribution in Atherosclerotic Coronary Artery

Activated inflammatory cells placed in atherosclerotic plaques release heat while the plaque is cooled by blood flow. Temperature heterogeneity in plaque can cause thermal stress, and speeds up plaque growth or rupture process. In the present work, arterial wall temperature distribution of atherosclerotic Right Coronary with nonNewtonian blood flow is investigated by numerical methods. The rheology of the flowing blood is modeled by a generalized Power law model. An advanced algorithm with coupled FEM-FVM is used to determine temperature distribution inside the artery. Transient Navier-Stokes and energy equations in 2D idealized arterial model of a bending artery are discretized using the Finite-Volume Method and solved by SIMPLE algorithm in curvilinear coordinate to analyze pulsatile blood flow, whereas the transient heat conduction equation in the plaque is solved simultaneously with these equations using Finite-Element Method. The plaque temperature, Nusselt Number and heat flux at the plaque/lumen interface is obtained for various moments of cardiac cycle and different Power law indices (n) to investigate influence of blood viscosity on cooling effect of blood. It is observed that how blood dilution modifies the temperature heterogeneity of plaque and decreases probability of rupture of Atherosclerotic plaque.

Wave propagation in an inhomogeneous transversely isotropic material obeying the generalized power law model

The analytical solutions for body-wave velocity in a continuously inhomogeneous transversely isotropic material, in which Young’s moduli (E, E′), shear modulus (G′), and material density (ρ) change according to the generalized power law model, (a+b z) c , are set down. The remaining elastic constants of transversely isotropic media, ν, and ν′ are assumed to be constants throughout the depth. The planes of transversely isotropy are selected to be parallel to the horizontal surface. The generalized Hooke’s law, strain-displacement relationships, and equilibrium equations are integrated to constitute the governing equations. In these equations, utilizing the displacement components as fundamental variables, the solutions of three quasi-wave velocities (V SV , V P , V SH ) are generated for the present inhomogeneous transversely isotropic materials. The proposed solutions are compared with those of Daley and Hron (Bull Seismol Soc Am 67:661–675, (1977)), and Levin (Geophysics 44:918–936, (1979)) when the inhomogeneity parameter c = 0. The agreement between the present results and previously published ones is excellent. In addition, the parametric study results reveal that the magnitudes of wave velocity are remarkably affected by (1) the inhomogeneity parameters (a, b, c); (2) the type and degree of material anisotropy (E/E′, ν/ν′, G/G′); (3) the phase angle (θ); and (4) the depth of the medium (z). Consequently, it is imperative to consider the effects of inhomogeneity when investigating wave propagation in transversely isotropic media.

NUMERICAL MODELING OF PULSATILE FLOW OF BLOOD THROUGH A STENOSED TAPERED ARTERY UNDER PERIODIC BODY ACCELERATION

A mathematical model is developed for the pulsatile flow of blood through a stenosed tapered artery under the influence of externally imposed body acceleration. The artery is assumed as a cylindrical tube with time-dependent radius having mild stenosis and the non-Newtonian behavior of blood is characterized by generalized Power-law model. The governing equations are transformed by using a radial transformation and solved numerically by a suitable finite difference scheme in order to obtain the velocity, fluid acceleration, wall shear stress, and flow rate. The effect of stenosis severity, tapering, and externally imposed body acceleration on the blood flow in artery is discussed with the help of graph. It is found that all flow characteristics are affected by the stenosis severity, tapering, and periodic acceleration applied on the body.

EFFECT OF ASYMMETRY AND ROUGHNESS OF STENOSIS ON NON-NEWTONIAN FLOW PAST AN ARTERIAL SEGMENT

Numerical investigations of non-Newtonian blood flow are carried out through an asymmetric arterial constriction (stenosis) obtained from casting of mildly stenosed artery [Back et al. [1984] Effect of mild atherosclerosis on flow resistance in a coronary artery casting by man, J. Biomech. Eng., Trans. ASME106, 48]. The Marker and Cell method, for governing equations of motion for the flow in primitive variables formulations is developed in a staggered grid to discretize the momentum equations representing the non-Newtonian viscous incompressible flow characterized by the generalized Power-law model in cylindrical coordinates system under axial symmetric conditions so that the problem effectively becomes two-dimensional. The modified pressure equation has been solved by Successive-Over-Relaxation method and the pressure–velocity correction formulae have been derived. Satisfactory level of convergence namely, the mass conservation of the order of 0.5 × 10-12 and consequently the steady-state criteria have been achieved. The separation points, reattachment points, pressure drop, and the wall shear stress distribution resulting from the present simulation agree well with the available numerical and experimental results. Secondary separation has also been predicted at higher Reynolds numbers. Further, in-depth study of the flow patterns reveals that shear-thickening model of generalized Power-law fluid experiences excess pressure drop more than that of shear-thinning model as in the case of flow past through cosine and smooth-shaped constrictions than irregular ones. The efficiency of the numerical code is illustrated by applying it to a test problem in order to validate the applicability of the technique as well as the simulation under consideration.

The Unsteady Power Law Blood Flow Through a Multi-irregular Stenosed Artery

A non-Newtonian pulsatile model of blood flow through multiple stenoses with irregular surfaces is considered. The model chosen is the generalized power law model of blood viscocity where the flow is assumed to be unsteady, laminar, two- dimensional and axisymmetric. The governing equations of motion in terms of the viscous shear stress and the boundary conditions in the cylindrical coordinate system are first transformed using a radial coordinate transformation before they are dis- cretized using a finite difference scheme based on central difference approximations on non-uniform grids. The numerical results obtained in terms of blood flow character- ictics show that the values of the axial velocity and flow rate in the power-law model are lower while the resistance to flow and the wall shear stress are higher compared to the Newtonian model. These features concur with the general observations of blood flowing through small stenosed arteries.