Dimensionless Radius Research Articles

Effects of slip and rheological parameters on the flow and heat transfer of a Herschel-Bulkley fluid

PurposeThe purpose of this paper is to investigate the combined effects of slip and rheological parameters on the flow and heat transfer of the Herschel-Bulkley fluid.Design/methodology/approachThe combinative dimensionless parameter method is introduced into the equations of the slip flow and heat transfer to make the discussion more comprehensive. More specifically, the slip and rheological parameters are transformed into the dimensionless slip number as well as Herschel-Bulkley number. We solve the dimensionless equations and then focus on the effects of these parameters on the slip flow and heat transfer.FindingsThe results show that, for a given value of Herschel-Bulkley number, there is a finite critical value of slip number at which the pressure gradient reaches the lowest value and both the dimensionless yield radius and slip velocity become 1. Meanwhile, the Nusselt number tends to be infinite at this critical value of slip number. For the case of slip, the Nusselt number also approaches infinity at a finite critical value of Herschel-Bulkley number. Furthermore, the dimensionless velocity as well as temperature of the yield pseudoplastic fluid with higher slip number is lower within a small radius but becomes higher near the wall. Meanwhile, from the velocity and temperature profiles, the effect of Herschel-Bulkley number on these two parameters of the Bingham fluid at the smaller radius is opposite.Originality/valueThese associated expressions can be generalized to the flow and heat transfer of a Herschel-Bulkley fluid under slip boundary condition. It can provide a reference for the engineering application relating to the heat transfer and flow of a Herschel-Bulkley fluid. Meanwhile, it also suggests some revelations for dealing with this similar problem.

A simple approach to the elasto-plastic coupling analyses of circular tunnels in confining pressure-dependent strain-softening rock masses

The conventional strain-softening behaviors which assume the strength parameters as the functions of the plastic strain have been studied using various methods. However, rock-like materials have also been found to manifest pressure-dependent behavior for both elastic and plastic rock masses. The realization of the radial stress was found to be continuous, and gradually increased from supporting stress to in situ stress for the circular openings under hydrostatic pressure. The radial stress was spaced as numbers of sections, and each annulus was considered as an ideal plastic rock mass. The close-formed analytical solutions of each annulus can be easily obtained. Therefore, using the displacement continuum and stress boundary conditions, the dimensionless radius corresponding to the spaced radius could be derived. The radial stress and plastic shear strain at the inner radius of the outer adjacent annulus were employed to describe the material property evolution law. Then, the radii could be recursively obtained. Finally, for the conventional strain-softening rock masses, the pressure-dependent elastic rock and elasto-plastic coupling strain-softening rock masses were employed to validate the proposed approach. In this study, having completed the above steps, it was concluded that the results were in accordance with the numerical methods.

Determination of the entropy radial minimum and the various transition velocities in an air-water bubble column

The bubble column hydrodynamics are complex and the macroscopic flow structure is different in the column core and annulus due to both the bubble coalescence and breakup phenomena as well as the gross liquid circulation. In this work, for the first time the local Kolmogorov entropy (KE) minima at different superficial gas velocities Ug were identified. It was found that there is an agreement between the local KE minima (occurring around r/R=0.63) and the inversion point (dimensionless radius=0.7) for the axial liquid velocity reported by both Chen et al. (1994) and Wu and Al-Dahhan (2001).The KE radial profiles were also used to prove the existence of chaotic similarities between the flow patterns in the bubble bed at three different Ug values (0.089, 0.134 and 0.146m/s) belonging to the heterogeneous regime. This finding implies that the flow patterns and the degrees of turbulence at these Ug values are also identical. The same similarity was also found between the KE profiles at Ug=0.056 and 0.067m/s, which belong to the transition flow regime. These results imply that the flow patterns in the bubble bed repeat (especially at different Ug values falling into the same flow regime).Based on the KE profiles as a function of Ug, the effect of the dimensionless radial position on the various transition velocities was studied. It was found that the end of the gas maldistribution regime is shifted to slightly higher Ug value in column core and annulus. In most of the cases, the onset of the churn-turbulent regime occurs at Ug=0.101m/s.

Forming mechanism and radius prediction of hydraulically generated slots based on a two-walled jet

Due to its distinct capability to improve the efficiency of gas drainage, hydraulic slotting has attracted increased attention in recent years. Current attempts to predict the slotting radius are mostly based on the fluid-structure interactions of normal free jets, which is inconsistent with the practical slotting condition. To address this problem, a two-walled jet model with a rotating nozzle was proposed for simulation, focusing on the flow-field characteristics. The hysteresis effect and zone of negative pressure, which account for the forming mechanism of slots, were demonstrated based on the simulation results. Moreover, the decay equations of the centerline velocity were obtained based on the analysis of the jet developments, the effects of the confining height, and the rotating number. As a result, theoretical prediction modeling of the slotting radius was achieved. In addition, experiments using materials similar to the coalbed were also performed to validate the accuracy of the prediction model. The linear relation between the dimensionless radius and the 1/4th power law of the inlet pressure could be obtained in both the theoretical model and the experimental results. The experimental results were in good agreement with the numerical ones, except for the low-pressure condition. Finally, it can be claimed that the theoretical model is, to some extent, suitable to guide practical applications.

An experimental investigation of convective heat transfer enhancement in electronic module using curved deflector

This work investigated experimentally the heat transfer and pressure drop in electronic module using a curved deflector to direct the flow towards the recirculation zone enclosed between the two heat sources. The experiments were carried out to investigate the effect of deflector dimensionless radius (R r ) and both horizontal and vertical distances (R x , R y ) within a range of Reynolds number from 5223 to 11,380. The results show that larger deflector at small vertical distance enhances the heat transfer for upstream and downstream heat sources while the horizontal distance has a contrast effect. Correlations are obtained for the average Nusselt number of both upstream and downstream heat sources utilizing the present measurements within 5223 ≤ Re L ≤ 11,380, 0.02 ≤ R x ≤ 0.4, 0.3 ≤ R y ≤ 0.5 and 0.15 ≤ R r ≤ 0.35.

Existence of Universal Phase Diagrams for Describing General Pore Shape Resulting From an Entrapped Bubble During Solidification

This study shows that there exist a pair of universal and unique phase diagrams to describe general development of the pore shape in solid, resulting from a bubble captured by a solidification front. Like thermodynamics, phase diagrams have advantages to generally identify the states and design processes of a system. Pore formation and its shape in solids influence not only microstructure of materials but also contemporary issues of biology, engineering, foods, geophysics, and climate change, etc. In this study, a pair of phase diagrams is thus found to be under dimensionless coordinate systems of dimensionless apex radius, contact angle, and base radius of the bubble cap, as well as solidification rate, contact angle, and growth rate of base radius. The computed results of the development of the pore shape agree with experimental data. The pore shape in solid thus can be optimistically predicted and controlled by choosing a desired path on phase diagrams.

Modeling of expandable polystyrene expansion

ABSTRACTA fundamental understanding of the expansion kinetics of expandable polystyrene (EPS) is crucial for the design and optimization of processes for EPS‐filled syntactic foams. In this study, a general formulation was developed to model EPS expansion. A semi‐analytical solution was obtained on the basis of the case of a single bubble expansion in an infinite matrix. The dimensionless bubble radius and pressure were defined and found to be exponential functions of the dimensionless expansion time. The characteristic bubble expansion time was able to characterize the timescale of the expansion process. The semi‐analytical solution could qualitatively predict the radial expansion of the EPS microsphere observed in a real‐time experiment. To obtain an accurate prediction, a numerical solution was obtained to the model that coupled the nucleation and expansion of multiple bubbles in a finite matrix at various temperatures. The results show that the numerical solution was able to quantitatively predict the radial expansion of EPS. A parameter sensitivity study was performed to examine the effect of each parameter over the expansion process. © 2016 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2016, 133, 43886.

An experimental investigation of the explosion characteristics of dimethyl ether-air mixtures

In this work, experiments are performed to study the explosion characteristics of DME (dimethyl ether) -air mixtures using a standard 20-L spherical explosion test apparatus. The experimental data reported in this paper includes: the maximum explosion pressure (pmax), flammability limits, maximum rate of pressure rise (dp/dt)max, and combustion properties (i.e., laminar burning velocity, flame radius) of DME-air mixtures at different initial conditions. The experimental results indicate that the variation between pmax and DME concentration (CDME) exhibits a typical inverse “U” shaped behavior, with the peak pmax at slightly larger than the stoichiometric concentration. pmax is also found to decrease as the initial pressure goes down. As the initial pressure decreases from 100 kPa to 40 kPa, the LFL (lower flammability limit) is observed to vary slightly, while the UFL (upper flammability limit) is found to have a more significant drop. The relation between (dp/dt)max and CDME behaves similarly as that of pmax as a function of CDME, and the explosion pressure rises more abruptly at higher initial pressure. A satisfactory agreement is also found between the laminar burning velocity determined experimentally from the pressure measurement and that computed by PREMIX simulations. The present experimental results also show that the increase of the dimensionless radius of the flame is slower at higher initial pressure.

Hawking radiation and the Stefan–Boltzmann law: The effective radius of the black-hole quantum atmosphere

It has recently been suggested [S. B. Giddings, Phys. Lett. B {\bf 754}, 39 (2016)] that the Hawking black-hole radiation spectrum originates from an effective quantum "atmosphere" which extends well outside the black-hole horizon. In particular, comparing the Hawking radiation power of a $(3+1)$-dimensional Schwarzschild black hole of horizon radius $r_{\text{H}}$ with the familiar Stefan-Boltzmann radiation power of a $(3+1)$-dimensional flat space perfect blackbody emitter, Giddings concluded that the source of the Hawking semi-classical black-hole radiation is a quantum region outside the Schwarzschild black-hole horizon whose effective radius $r_{\text{A}}$ is characterized by the relation $\Delta r\equiv r_{\text{A}}-r_{\text{H}}\sim r_{\text{H}}$. It is of considerable physical interest to test the general validity of Giddings's intriguing conclusion. To this end, we study the Hawking radiation of $(D+1)$-dimensional Schwarzschild black holes. We find that the dimensionless radii $r_{\text{A}}/r_{\text{H}}$ which characterize the black-hole quantum atmospheres, as determined from the Hawking black-hole radiation power and the $(D+1)$-dimensional Stefan-Boltzmann radiation law, are a decreasing function of the number $D+1$ of spacetime dimensions. In particular, it is shown that radiating $(D+1)$-dimensional Schwarzschild black holes are characterized by the relation $(r_{\text{A}}-r_{\text{H}})/r_{\text{H}}\ll1$ in the large $D\gg1$ regime. Our results therefore suggest that, at least in some physical cases, the Hawking emission spectrum originates from quantum excitations very near the black-hole horizon.

Three‐dimensional flow structure and patterns of bed shear stress in an evolving compound meander bend

AbstractCompound meander bends with multiple lobes of maximum curvature are common in actively evolving lowland rivers. Interaction among spatial patterns of mean flow, turbulence, bed morphology, bank failures and channel migration in compound bends is poorly understood. In this paper, acoustic Doppler current profiler (ADCP) measurements of the three‐dimensional (3D) flow velocities in a compound bend are examined to evaluate the influence of channel curvature and hydrologic variability on the structure of flow within the bend. Flow structure at various flow stages is related to changes in bed morphology over the study timeframe. Increases in local curvature within the upstream lobe of the bend reduce outer bank velocities at morphologically significant flows, creating a region that protects the bank from high momentum flow and high bed shear stresses. The dimensionless radius of curvature in the upstream lobe is one‐third less than that of the downstream lobe, with average bank erosion rates less than half of the erosion rates for the downstream lobe. Higher bank erosion rates within the downstream lobe correspond to the shift in a core of high velocity and bed shear stresses toward the outer bank as flow moves through the two lobes. These erosion patterns provide a mechanism for continued migration of the downstream lobe in the near future. Bed material size distributions within the bend correspond to spatial patterns of bed shear stress magnitudes, indicating that bed material sorting within the bend is governed by bed shear stress. Results suggest that patterns of flow, sediment entrainment, and planform evolution in compound meander bends are more complex than in simple meander bends. Moreover, interactions among local influences on the flow, such as woody debris, local topographic steering, and locally high curvature, tend to cause compound bends to evolve toward increasing planform complexity over time rather than stable configurations. Copyright © 2016 John Wiley & Sons, Ltd.

Use of Computer Experiments to Study the Current Collected by Cylindrical Langmuir Probes

A particle-in-cell simulation has been developed to study the behaviour of ions in the surroundings of a negatively biased cylindrical Langmuir probe. Here, we report our findings on the transition between radial and orbital behaviour observed by means of the aforementioned code. The influence of the ion to electron temperature ratio on the transition for different dimensionless probe radius is discussed. Two different behaviours have been found for small and large probe radii.

Engine hot spots: Modes of auto-ignition and reaction propagation

Many direct numerical simulations of spherical hot spot auto-ignitions, with different fuels, have identified different auto-ignitive regimes. These range from benign auto-ignition, with pressure waves of small amplitude, to super-knock with the generation of damaging over-pressures. Results of such simulations are generalised diagrammatically, by plotting boundary values of ξ, the ratio of acoustic to auto-ignition velocity, against ɛ. This latter parameter is the residence time of the developing acoustic wave in the hot spot of radius ro, namely ro/a, normalised by the excitation time for the chemical heat release, τe. This ratio controls the energy transfer into the developing acoustic front. A third relevant parameter involves the product of the activation temperature, E/R, for the auto-ignition delay time, τi, normalised by the mixture temperature. T, the ratio, τi/τe, and the dimensionless hot spot temperature gradient, (∂lnT/∂r¯), where r¯ is a dimensionless radius. These parameters define the boundaries of regimes of thermal explosion, subsonic auto-ignition, developing detonations, and non-auto-ignitive deflagrations, in plots of ξ against ɛ.The regime of developing detonation forms a peninsula and contours, throughout the field. The product parameter (E/RT)(τi/τe)/∂lnT/∂r¯ expresses the influences of hot spot temperature gradient and fuel characteristics, and a unique value of it might serve as a boundary between auto-ignitive and deflagrative regimes. Other combustion regimes can also be identified, including a mixed regime of both auto-ignitive and “normal” deflagrative burning. The paper explores the performances of a number of different engines in the regimes of controlled auto-ignition, normal combustion, combustion with mild knock and, ultimately, super-knock. The possible origins of hot spots are discussed and it is shown that the dissipation of turbulent energy alone is unlikely to lead directly to sufficiently energetic hot pots. The knocking characterisation of fuels also is discussed.

Influence of chamber pressure on heat transfer characteristics of a closed loop R134-a spray cooling

A closed loop R134-a spray cooling system was designed using a commercial full-cone nozzle to generate the spray for the cooling of an upward plain copper surface. Given the flow rate, the spray cooling characteristics such as heat flux, critical heat flux (CHF) and spray efficiency were obtained under different spray chamber pressures. The results show the CHF and spray efficiency were enhanced with the increase of the chamber pressure. Due to the dependence of the saturation temperature on the vapor pressure, the cooling temperature of the target surface was directly proportional to the chamber pressure. Moreover, an improved empirical correlation was proposed for the non-dimensional heat flux as a function of the Jacob number, the Weber number, the dimensionless mass flow rate, the dimensionless pressure and the dimensionless bubble departure radius.

Note: A top-view optical approach for observing the coalescence of liquid drops.

We developed a new device that is capable of top-view optical examination of the coalescence of liquid drops. The device exhibits great potential for visualization, particularly for the early stage of liquid bridge expansion, owing to the use of a high-speed shadowgraph technique. The fluid densities of the two approaching drops and that of the ambient fluid are carefully selected to be negligibly different, which allows the size of the generated drops to be unlimitedly large in principle. The unique system design allows the point of coalescence between two drops to serve as an undisturbed optical pathway through which to image the coalescence process. The proposed technique extended the dimensionless initial finite radius of the liquid bridge to 0.001, in contrast to 0.01 obtained for conventional optical measurements. An examination of the growth of the bridge radius for a water and oil-tetrachloroethylene system provided results similar to Paulsen's power laws of the inertially limited viscous and inertial regimes. Furthermore, a miniscule shift in the center of the liquid bridge was detected at the point of crossover between the two regimes, which can be scarcely distinguished with conventional side-view techniques.

Study on the Mechanism of Rock Stress Sensitivity Using a Random Pore Network Simulation

Studies of rock stress sensitivity are mainly focused on experimental and data processing methods, and the mechanism cannot be adequately explained using specific pore shape models. This study, based on a random pore network simulation, explains the rock stress sensitivity mechanism for the first time. Based on the network model theory, the hydraulic conductivity equation, the dimensionless radius equation, and the effective stress equation for partially saturated rock are used to generate a three-dimensional random pore network model based on the QT platform. The simulation results show that the influence of the effective stress on the dimensionless radius becomes more significant as the aspect ratio decreases, and the relationship between dimensionless radius and effective stress can be effectively interpreted through different combinations of pore shapes. Moreover, the mechanism behind permeability stress sensitivity can be explained by establishing the relationship between permeability and effective stress.

Comparative study on the spatial evolution of liquid jet under linear and nonlinear stability theories

In the injecting process of liquid jet, the disturbance wave on jet interface will grow continually, leading to the spatial development and atomization of liquid jet. Studying the spatial evolution of liquid jet will help to deepen the understanding of the mechanism of jet breakup and atomization. In this paper, based on the linear and nonlinear stability theories, the first-order and second-order dispersion equations describing the stability of liquid jet with cavitation bubbles in a coaxial swirling compressible airstream are built, respectively, and the dispersion equation and its solving method are verified by the data in the literature. On this basis, the developments of first-order and second-order disturbance are analyzed, and the spatial evolutions of liquid jet are compared under linear and nonlinear stability theories. The results show that the wavelength and amplitude of the second-order disturbance are much smaller than those of the first-order disturbance. The disturbance development on jet surface is mainly dominated by the development of the first-order disturbance along the axial direction. With the increasing of axial distance, the second-order disturbance gradually begins to play a role in the developing of disturbance. The role of second-order disturbance is mainly reflected in three aspects, i. e., obviously increasing the disturbance amplitude at wave crest, reducing the disturbance amplitude at wave trough (sometimes ups and downs occur), and changing the waveform to a certain degree. The dominant disturbance mode on jet surface will not change under two kinds of theories. By using the nonlinear stability theory, satellite droplets which are found on jet surface in experiments can be reflected, and the shape of main droplet changes obviously from the ellipsoid to sphere. Also, the change of dimensionless radius of liquid jet is greater by nonlinear stability theory than by linear stability theory, which indicates that the oscillation extent of jet surface increases due to considering the second-order disturbance. Therefore, compared with the linear stability theory, the nonlinear stability theory has the advantage that it considers the effects of high-order disturbance on the spatial evolution of liquid jet in addition to the first-order disturbance on jet surface. The nonlinear stability theory can predict the spatial development of liquid jet in more detail than the linear stability theory.

Radial-to-orbital motion transition in cylindrical Langmuir probes studied with particle-in-cell simulations

A particle-in-cell (PIC) simulation of the plasma sheath around a cylindrical Langmuir probe has been developed to evaluate the ion current collected by the probe. The simulation includes the positive ion thermal motion and has been optimized by solely describing the positive ion motion. A transition from the prediction of the radial model to the orbital-motion-limited model is observed. The transition is explained as an effect of the positive ion thermal motion and the radial model is recovered when the positive ion to electron temperature ratio is decreased. The behaviour of this transition strongly depends on the dimensionless probe radius.

A parameter for a family of steady vortex ring cores

Streamlined orthogonal coordinates are derived for the core of Hill’s spherical vortex. Each core boundary of Norbury’s family of vortex rings is a scaled streamline from Hill’s spherical vortex. A formula is given that relates the dimensionless mean core radius of each member of this family to its boundary. A new dimensionless mean core radius for vortex rings is presented that varies between 0 and 1 rather than 0 and the square root of 2, and requires that fewer geometrical parameters be specified in order to uniquely define the core area for each member of the family of vortex rings.

Polarization of Radiation and Basic Parameters of the Circumnuclear Region of Active Galactic Nuclei

The degree of polarization of the radiation in broad emission lines can be used for a unique determination of the dimensionless radius of the broad line emission region, i.e., the ratio of this radius to the gravitational radius. A comparison of the results of this method with others for determining the radius of broad line emission regions yields strong limits on the physical mechanism for the formation of emission line regions in the immediate neighborhood of supermassive black holes. The method developed here offers the prospect of obtaining strong limits on the mass of supermassive black holes in active galactic nuclei.

2D axisymmetric transient inverse heat conduction analysis of air jet impinging on stainless steel plate with finite thickness

A 2D axisymmetric inverse heat-conduction model based on the Levenber–Marquardt method was built. The equivalent Nusselt number Nuequ of air jet impinging on a stainless steel plate with finite thickness were obtained. Radiant heat transfer is significant when the temperature of the target plate is high, which induces the nonlinear feature of the inverse heat-conduction problem. The results of the experiments and the inverse analysis provide the following findings. First, the inverse heat-conduction model is accurate. Second, Nuequ maintains a relatively small value before the onset of the air jet. After the air jet starts, however, the values of Nuequ within the stagnation zone (dimensionless radius R/Dn ≤ 1.0) increase dramatically, reach the peak points in the next 20 s, and then remain nearly constant until the end of the experiments. Third, the radial distribution of Nuequ shows that it decreases rapidly within the region of R/Dn ≤ 8.0. When R/Dn exceeds 10, Nuequ appears unaffected by the Reynolds number and R/Dn. Lastly, Nuequ decreases slowly with the increase in time, which indicates that radiant heat-transfer flux decreases with target plate temperature.