Values Of Power-law Index Research Articles

Laminar forced convection in power-law and Bingham plastic fluids in ducts of semi-circular and other cross-sections

In this work, the effects of the power-law and Bingham plastic viscosity on the flow and heat transfer characteristics of laminar forced convection through non-circular ducts of a range of cross-sections have been investigated numerically over the wide ranges of Peclet number, 10⩽(Pe=Re·Pr)⩽104, power-law index, 0.2⩽n⩽2 and Bingham number, 0⩽Bn⩽15. Two different thermal boundary conditions (constant wall temperature and constant wall heat flux) on the duct wall have been implemented. Extensive results for the flow and temperature fields in terms of the dimensionless velocity and temperature profiles, pressure drop and local and average Nusselt number are presented and discussed both in the developing and fully developed regions. For the sake of conciseness, the results for three duct shapes, namely, semi-circular, four cusped and rectangular, are presented and discussed in detail whereas that for the other shapes are summarised in a tabular form. The dependence of the Poiseuille number (f·Re) and asymptotic Nusselt number (Nu) on the power-law index (n) and Bingham number (Bn) is studied under fully developed (hydrodynamically and thermally) conditions. It is shown that the fanning friction factor has a positive dependence on the increasing values of power-law index while an opposite dependence is observed for the Nusselt number. Finally, simple predictive correlations for hydrodynamic entrance length and Nusselt number are developed as functions of the generalised Reynolds number (Reg), Graetz number (Gz), power-law index or the Bingham number. The paper is concluded by comparing the predictions of the power-law model with that of the extended modified power-law (EMPL) model for a semi-circular duct in order to ascertain the influence of the deficiencies of the power-law equation in capturing the zero- and infinite-shear viscosities on the present results.

Free Convection in Confined Power-Law Fluids From Two Side-by-Side Cylinders in a Square Enclosure

Laminar free convection heat transfer in power-law fluids from two side-by-side cylinders (one hot and one cold) confined in a square duct has been studied numerically in the two-dimensional flow regime. For a fixed value of the ratio of cylinder radius to size of the enclosure, the effect of geometrical placement of the cylinders is studied on the resulting velocity and temperature fields in the laminar free convection regime by considering six asymmetric locations of the two cylinders. In particular, extensive results reported herein span the range of conditions of Grashof number, 10 to 105; Prandtl number, 0.7 to 100, thereby yielding the range of the Rayleigh number as 7 to 107; power-law index, 0.3 to 1.8; and the relative positions (dimensionless) of the cylinders with respect to the centerline, –0.25 to 0.25. The heat transfer characteristics are analyzed in terms of the local Nusselt number along the surfaces of the two cylinders and the enclosure walls. Overall, the average Nusselt number shows a positive dependence on both the Grashof number and the Prandtl number irrespective of the values of power-law index and relative positioning of the cylinders. Also, all else being equal, shear-thinning fluid behavior promotes heat transfer with reference to that in Newtonian fluids. When the two cylinders are situated close to the bottom wall, the rate of heat transfer is augmented with reference to that for the symmetric positioning of the cylinders along the horizontal mid-plane of the enclosure. Conversely, heat transfer deteriorates as the cylinders are located above the centerline of the enclosure. The present numerical results have been consolidated via the use of a modified Rayleigh number, thereby enabling the estimation of the average Nusselt number in a new application.

Effect of buoyancy-assisted flow on convection from an isothermal spheroid in power-law fluids

In this work, the coupled momentum and energy equations have been solved to elucidate the effect of aiding-buoyancy on the laminar mixed-convection from a spheroidal particle in power-law media over wide ranges of the pertinent parameters: Richardson number, 0≤Ri≤5; Reynolds number, 1≤Re≤100; Prandtl number, 1≤Pr≤100; power-law index, 0.3≤n≤1.8, and aspect ratio, 0.2≤e≤5 for the case of constant thermo-physical properties. New results for the velocity and temperature fields are discussed in terms of the streamline and isotherm contours, surface pressure and vorticity contours, drag coefficient, local and surface averaged Nusselt number. The effect of particle shape on the flow is seen to be more pronounced in the case of oblates (e 1). The propensity for wake formation reduces with the rising values of power-law index, Richardson number and slenderness of the body shape (e > 1). Also, the drag coefficient is seen to increase with the Richardson number and power-law index. All else being equal, the Nusselt number shows a positive dependence on the Richardson number and Reynolds number and an inverse dependence on the power-law index and aspect ratio of the spheroid. Limited results were also obtained by considering the exponential temperature dependence of the power-law consistency index. This factor can increase the values of the average Nusselt number by up to ~10-12% with reference to the corresponding values for the case of the constant thermo-physical properties under otherwise identical conditions. Finally, the present values of the Nusselt number have been consolidated in the form of Colburn j-factor as a function of the modified Reynolds and Prandtl numbers for each value of the aspect ratio (e). The effect of the temperature dependent viscosity is included in this correlation in terms of a multiplication factor.

Dual solutions of slip flow past a nonlinearly shrinking permeable sheet

The aim of the paper is to investigate the flow of an incompressible viscous fluid past a nonlinearly shrinking permeable sheet. Partial slip condition is considered instead of no slip condition at the boundary. The self similar equations are obtained and then solved numerically by a shooting technique. Dual solutions are obtained for the flow past a nonlinearly shrinking sheet with slip condition in the presence of suction. It is found that for the first solution the momentum boundary layer thickness decreases with slip and suction parameters; but it increases with the power-law index of the shrinking velocity. The dual solutions for the velocity field are obtained for the positive values of power-law index n and for certain values of the other parameters in the study. Velocity slip controls the boundary layer separation. However, the power-law index acts to accelerate the boundary layer separation.

A statistical study of magnetospheric ion composition along the geomagnetic field using the Cluster spacecraft for L values between 5.9 and 9.5

AbstractUsing ion density data obtained by the CODIF (ion Composition and Distribution Function analyser) instrument on board the Cluster spacecraft, for the interval spanning 2001–2005, an empirical model describing the average ion mass distribution along closed geomagnetic field lines is determined. The empirical model describes the region spanning 5.9≤L < 9.5, with dependences on L shell and magnetic local time included, and represents ions in the energy range of 0.025 to 40 keV/charge. The data reduction process involves the identification and rejection of CODIF data contaminated by penetrating energetic radiation belt particles, found to frequently occur for L < 5.9. Furthermore, a comparison of data with observations of the cold plasma population in the region provides evidence that the CODIF data set is representative of the full plasma population. The variations in average ion mass along the field lines were modeled using a power law form, which maximizes toward the magnetic equatorial plane, with observed power law index values ranging between approximately −2.0 and 0.0. The resulting model illustrates some key features of the average ion mass spatial distribution, such as an average ion mass enhancement at low L in the evening sector, indicating the transport of high‐latitude heavy ion outflows to the closed inner magnetosphere.

Deformation of a Capsule in a Power-Law Shear Flow.

An immersed boundary-lattice Boltzmann method is developed for fluid-structure interactions involving non-Newtonian fluids (e.g., power-law fluid). In this method, the flexible structure (e.g., capsule) dynamics and the fluid dynamics are coupled by using the immersed boundary method. The incompressible viscous power-law fluid motion is obtained by solving the lattice Boltzmann equation. The non-Newtonian rheology is achieved by using a shear rate-dependant relaxation time in the lattice Boltzmann method. The non-Newtonian flow solver is then validated by considering a power-law flow in a straight channel which is one of the benchmark problems to validate an in-house solver. The numerical results present a good agreement with the analytical solutions for various values of power-law index. Finally, we apply this method to study the deformation of a capsule in a power-law shear flow by varying the Reynolds number from 0.025 to 0.1, dimensionless shear rate from 0.004 to 0.1, and power-law index from 0.2 to 1.8. It is found that the deformation of the capsule increases with the power-law index for different Reynolds numbers and nondimensional shear rates. In addition, the Reynolds number does not have significant effect on the capsule deformation in the flow regime considered. Moreover, the power-law index effect is stronger for larger dimensionless shear rate compared to smaller values.

New concept about existence of Hartmann boundary layer in peristalsis through curved channel-asymptotic solution

The main objective of this paper is to investigate boundary layer character of the velocity in peristaltic flow of a Sisko fluid in a curved channel under the influence of strong imposed radial magnetic field. The Sisko fluid model falls in the category of generalized Newtonian fluid models. The constitutive equation of Sisko model is described in terms of three material constants namely; power-law index (n), infinite shear rate viscosity (a) and consistency index (b). This model is capable of predicting shear-thinning and shear-thickening effects for n 1, respectively. The equation governing the flow is first derived under the assumptions of long wavelength and low Reynolds number, and then made dimensionless by defining appropriate parameters. In dimensionless form it contains three dimensionless parameters namely; generalized ratio of infinite-shear rate viscosity to consistency index, power-law index and Hartmann number characterizing strength of the imposed magnetic field. It is found that the governing equation of flow becomes singular for large values of Hartmann number. Asymptotic solutions representing flow velocity at large values of Hartmann number are reported for two specific values of power-law index (namely n = 1 and n = 1/2) using singular perturbation technique. The flow velocity in either case exhibits qualitatively similar behavior. In fact, it exhibits boundary layer character i.e., it varies sharply in thin layer near the walls and varies linearly over rest of the cross-sections. This is contrary to what that is observed for flow velocity in straight channel (where except in thin layer near the channel walls the velocity over rest of the cross-section is uniform). The estimates of boundary layer thickness at upper and lower walls in either case are different. Moreover, the boundary layer thickness in either case is found to be inversely proportional to the Hartmann number.

Size-frequency distribution of boulders ≥7 m on comet 67P/Churyumov-Gerasimenko

We derive for the first time the size-frequency distribution of boulders on a comet, 67P/Churyumov-Gerasimenko (67P), computed from the images taken by the Rosetta/OSIRIS imaging system. We highlight the possible physical processes that lead to these boulder size distributions. We used images acquired by the OSIRIS Narrow Angle Camera, NAC, on 5 and 6 August 2014. The scale of these images (2.44−2.03 m/px) is such that boulders ≥7 m can be identified and manually extracted from the datasets with the software ArcGIS. We derived both global and localized size-frequency distributions. The three-pixel sampling detection, coupled with the favorable shadowing of the surface (observation phase angle ranging from 48◦ to 53◦ ), enables unequivocally detecting boulders scattered all over the illuminated side of 67P. We identify 3546 boulders larger than 7 m on the imaged surface (36.4 km 2 ), with a global number density of nearly 100/km 2 and a cumulative size-frequency distribution represented by a power-law with index of −3.6 +0.2/−0.3. The two lobes of 67P appear to have slightly different distributions, with an index of −3.5 +0.2/−0.3 for the main lobe (body) and −4.0 +0.3/−0.2 for the small lobe (head). The steeper distribution of the small lobe might be due to a more pervasive fracturing. The difference of the distribution for the connecting region (neck) is much more significant, with an index value of −2.2 +0.2/−0.2. We propose that the boulder field located in the neck area is the result of blocks falling from the contiguous Hathor cliff. The lower slope of the size-frequency distribution we see today in the neck area might be due to the concurrent processes acting on the smallest boulders, such as i) disintegration or fragmentation and vanishing through sublimation; ii) uplifting by gas drag and consequent redistribution; and iii) burial beneath a debris blanket. We also derived the cumulative size-frequency distribution per km 2 of localized areas on 67P. By comparing the cumulative size-frequency distributions of similar geomorphological settings, we derived similar power-law index values. This suggests that despite the selected locations on different and often opposite sides of the comet, similar sublimation or activity processes, pit formation or collapses, as well as thermal stresses or fracturing events occurred on multiple areas of the comet, shaping its surface into the appearance we see today.

Entropy analysis on MHD pseudo-plastic nanofluid flow through a vertical porous channel with convective heating

This paper is concerned with the entropy generation in a magnetohydrodynamic (MHD) pseudo-plastic nanofluid flow through a porous channel with convective heating. Three different types of nanoparticles, namely copper, aluminum oxide and titanium dioxide are considered with pseudo-plastic carboxymethyl cellulose (CMC)–water used as base fluids. The governing equations are solved numerically by shooting technique coupled with Runge–Kutta scheme. The effects of the pertinent parameters on the fluid velocity, temperature, entropy generation, Bejan number as well as the shear stresses at the channel walls are presented graphically and analyzed in detail. It is possible to determine optimum values of magnetic parameter, power-law index, Eckert number and Boit number which lead to a minimum entropy generation rate.

Free Vibration and Dynamic Stability of Functionally Graded Material Plates on Elastic Foundation

The study of parametric resonance characteristics of functionally-graded material (FGM) plates on elastic foundation is proposed under biaxial in plane periodic load. Finite element method in conjunction with Hamilton’s principle is utilised to establish the governing equations in a discrete form, Floquet’s theory was applied to determine the instability regions of FGM plate resting on elastic foundation. The effects of power law index, temperature rise, and foundation coefficients on the natural frequencies and dynamic stability of the plate have been examined in detail through parametric studies. The first two natural frequencies decrease with increase in temperature and power law index values, on the contrary, these two frequencies increase with increase in the foundation constants. Increase in power law index enhances the instability of the FGM plate. Increased foundation stiffness enhances the stability of the plate. Influence of shear layer constant is more dominant compared to the Winkler foundation constant. Defence Science Journal, Vol. 65, No. 3, May 2015, pp.245-251, DOI: http://dx.doi.org/10.14429/dsj.65.8621

Asymptotic Power-Law Index in Aggregation and Weighted-Chipping Processes

The asymptotic power-law behavior of the distribution function P(X) for X clusters is analyzed for aggregation and weighted-chipping processes. Here, chipping off of a cluster of size n takes place with the probability function ∼1/nσ. An exact value of a noninteger power-law index is obtained, that is, P(X) ∼ 1/Xα with α = σ in the case of 1 < σ ≤ 2. This gives thresholds of σ = 2 for the emergence of a condensed cluster and σ = 1 for the appearance of a power-law distribution. Numerical analysis supports well this result.

Viscosity and viscoelasticity measurements of low density polyethylene/poly(lactic acid) blends

The rheological properties and the viscoelastic behaviour of blends of polyethylene with different percentages of poly(lactic acid), ranging from 0 to 100 wt%, were studied. In a first part, all blends were examined under steady conditions using a capillary rheometer (at 180, 190 and 200 °C) and dynamic conditions using a parallel plate rheometer. The results showed that all blends behaved like pseudoplastic fluids, with the power–law index values varying between those of polyethylene and polylactide (0.45–0.75 at 180 °C, 0.49–0.77 at 190 °C and 0.54–0.81 at 200 °C). It was also observed that at low shear rate, pure poly(lactic acid) and polyethylene possessed, respectively, the highest and the lowest flow activation energy (66.9 and 48.3 kJ/mol); however, at high shear rate, the greater the content of poly(lactic acid), the lower the activation energy. In addition, poly(lactic acid) exhibited lower viscosity and lower melt elasticity than either polyethylene or the blends. The dynamic rheological study demonstrated that all formulations displayed shear thinning behaviour and only virgin poly(lactic acid) exhibited a clear Newtonian plateau. Also, mainly at low frequencies, polyethylene had the higher values of storage modulus (325 Pa), loss modulus (937 Pa) and complex viscosity (9,740 Pa.s). However, blends had values lying between those of the two homopolymers without any improvement in the storage modulus, loss modulus or complex viscosity. In a second part, the viscoelastic characteristics were investigated using dynamic mechanical thermal analysis (DMTA). DMTA spectra showed an increase in the storage modulus with the increase of poly(lactic acid) content but the opposite was observed for the loss modulus. A cold crystallization of poly(lactic acid) is observed around 87–100 °C and the temperature of glass transition of poly(lactic acid) did not depend on the composition of the blend. These results indicate that LDPE and PLA are immiscible in all proportions either in the melt state or in the solid state.

KKL correlation for simulation of nanofluid flow and heat transfer in a permeable channel

Hydrothermal behavior of nanofluid fluid between two parallel plates is studied. One of the plates is externally heated, and the other plate, through which coolant fluid is injected, expands or contracts with time. The effective thermal conductivity and viscosity of nanofluid are calculated by KKL correlation. The effects of the nanoparticle volume fraction, Reynolds number, Expansion ratio and power law index on Hydrothermal behavior are investigated. Results show that heat transfer enhancement has direct relationship with Reynolds number when power law index is equals to zero but opposite trend is observed for other values of power law index.

Extension of the dielectric breakdown model for simulation of viscous fingering at finite viscosity ratios

Immiscible displacement of viscous oil by water in a petroleum reservoir is often hydrodynamically unstable. Due to similarities between the physics of dielectric breakdown and immiscible flow in porous media, we extend the existing dielectric breakdown model to simulate viscous fingering patterns for a wide range of viscosity ratios (μ(r)). At low values of power-law index η, the system behaves like a stable Eden growth model and as the value of η is increased to unity, diffusion limited aggregation-like fractals appear. This model is compared with our two-dimensional (2D) experiments to develop a correlation between the viscosity ratio and the power index, i.e., η = 10(-5)μ(r)(0.8775). The 2D and three-dimensional (3D) simulation data appear scalable. The fingering pattern in 3D simulations at finite viscosity ratios appear qualitatively similar to the few experimental results published in the literature.

Surface Instabilities on a Thin Power-Law Fluid During Spin Coating

AbstractThe phenomena of surface instabilities in a thin power-law fluid during spin coating are investigated. The set of Navier-Stokes equations with non-Newtonian behavior in the region of low Reynolds number serves as a mathematical description of the physical systems. Long-wave perturbation analysis is proposed to derive an evolution equation of the Ostwald de-Waele type fluid to govern the propagation of surface waves. Weakly nonlinear dynamics of film flow is studied by the multiple scales method. The amplitude of instability is determined by a Ginzburg-Landau equation. The study reveals that the degree of power-law index plays a vital role in stabilizing the film flow. The shear-thinning fluid is more unstable than the shear-thickening fluid in the stability analysis. Further, the nonlinear wave speed in the supercritical stability region decreases with increasing values of power-law index.

Magneto-thermo-elastic analysis of a functionally graded conical shell

In this paper, magneto-thermo-elastic problem of a thick truncated conical shell immersed in a uniform magnetic field and subjected to internal pressure is investigated. Material properties of the shell including the elastic modulus, magnetic permeability, coefficients of thermal expansion and conduction are assumed to be isotropic and graded through the thickness obeying the simple power law distribution, while the poison's ratio is assumed to be constant. The temperature distribution is assumed to be a function of the thickness direction. Governing equations of the truncated conical shell are derived in terms of components of displacement and thermal fields and discretised with the help of differential quadrature (DQ) method. Results are obtained for different values of power law index of material properties and effects of thermal load on displacement, stress, temperature and magnetic fields are studied. Results of the present method are compared with those of the finite element method.

Magneto-thermo-elastic analysis of a functionally graded conical shell

In this paper, magneto-thermo-elastic problem of a thick truncated conical shell immersed in a uniform magnetic field and subjected to internal pressure is investigated. Material properties of the shell including the elastic modulus, magnetic permeability, coefficients of thermal expansion and conduction are assumed to be isotropic and graded through the thickness obeying the simple power law distribution, while the poison\'s ratio is assumed to be constant. The temperature distribution is assumed to be a function of the thickness direction. Governing equations of the truncated conical shell are derived in terms of components of displacement and thermal fields and discretised with the help of differential quadrature (DQ) method. Results are obtained for different values of power law index of material properties and effects of thermal load on displacement, stress, temperature and magnetic fields are studied. Results of the present method are compared with those of the finite element method.

Rapid heating induced vibration of circular cylindrical shells with magnetostrictive functionally graded material

The vibration and transient response of rapid heating on inner surface of the functionally graded material (FGM) circular cylindrical shells with outer magnetostrictive layer is investigated and computed by using the generalized differential quadrature (GDQ) method. The effects of heat flux value, power law index value, environmental temperature value and control gain value on Terfenol-D FGM circular cylindrical shell subjected to two edges clamped condition due to the not very high temperature fluid rapidly flow into the circular cylindrical shells from one side to the end of axial length direction are analyzed. The higher amplitudes of displacement and thermal stress can be obtained under the higher rapid heat flux value. With suitable product of coil constant and control gain value can reduce the amplitudes of displacement and thermal stress into a smaller value. The displacement of Terfenol-D FGM circular cylindrical shell versus the Terfenol-D thickness is stable for all power law index values. The Terfenol-D FGM circular cylindrical shell can stand against the higher temperature of environment with some values of power law index under rapid heating.

Drag of contaminated bubbles in power-law fluids

Flow and drag phenomena of spherical bubbles in contaminated power-law fluids are numerically investigated using the spherical stagnant cap model over the range of conditions as the Reynolds number (Re: 0.1–200), the power-law index (n: 0.2–1.6) and the degree of contamination (α: 0–180°). The conservation equations of mass and momentum are solved using the semi-implicit method for pressure-linked equations (SIMPLE) algorithm along with the quadratic upstream interpolation for convective kinematics (QUICK) scheme for momentum terms. The solver has been thoroughly validated with the existing literature results over wide range of pertinent conditions. The new results show that for Re>20 and α≥60°, a recirculation wake behind the contaminated bubble is observed for all values of power-law index; and the size of the recirculation wake is found to decrease with the increasing power-law index. Furthermore, a crossover Reynolds number (at Re≈5) exists for drag coefficient vs. Reynolds number behavior with respect to the power-law index; and it is found to be independent of the degree of contamination. Below this crossover Reynolds number, compared to Newtonian fluid results, the normalized drag coefficients are large for shear-thinning fluids (n<1) and are smaller for shear-thickening fluids; and opposite trends are observed beyond the crossover Reynolds number.

Efficient Hybrid Finite Element Method for Flutter Prediction of Functionally Graded Cylindrical Shells

In this work, a hybrid finite element formulation is presented to predict the flutter boundaries of circular cylindrical shells made of functionally graded (FG) materials. The development is based on a combination of linear Sanders thin shell theory and the classic finite element method. Material properties are temperature dependent and graded in the shell thickness direction according to a simple power law distribution in terms of volume fractions of constituents. The temperature field is assumed to be uniform over the shell surface and along the shell thickness. First-order piston theory is applied to account for supersonic aerodynamic pressure. The effects of temperature rise and shell internal pressure on the flutter boundaries of a FG circular cylindrical shell for different values of power law index are investigated. The present study shows efficient and reliable results that can be applied to aeroelastic design and analysis of shells of revolution in aerospace vehicles.