Formula For Velocity Research Articles

Fragmentation behavior and velocity formula for secondary fragments from RC slabs during contact explosions

The failure process of reinforced concrete (RC) slabs under contact explosion is dominated by localized damages, accompanied by numerous flying fragments. These concrete fragments pose a significant threat to personnel and equipment, as well as causing dynamic deformation and failure of the retrofitting materials used to restrain the fragments. The behavior of concrete fragmentation during explosions has been seldom studied experimentally, and few fragment velocity formulas exist that incorporate multiple factors. In this paper, the fragmentation process is recorded using two high-speed cameras, revealing that concrete fragmentation during explosions exhibits notable zoning characteristics, with the maximum velocity observed at the center of the fragment cloud’s front. To predict this maximum velocity, a model is derived from dimensional analysis. Utilizing this model, velocity prediction formulas are proposed and validated by 29 contact explosion tests. The formulas consider the explosive-rebar position, slab thickness, explosive mass and explosive shape, and are further improved by considering the concrete cover. The results indicate that, when the slab is not breached, the deceleration of rebar on fragments is weak. However, once the slab is breached, this deceleration becomes increasingly significant as the defined scaled thickness (TS) decreases. In the limiting case, the fragment velocity when the charge is located above the rebar is 0.636 times that of when the charge avoids the rebar. Fragments when the charge avoids the rebar, should be regarded as the most unfavorable case in secondary fragment hazards assessment and structural design.

On the Expanding Universe – Accurately Calculate the Hubble Constant by Lunar Laser Ranging

Hubble's Law is an empirical formula. We use the product of the speed of light and time instead of distance for differentiation, and obtain the theoretical differential equation dV=Wdt. Then, the expansion velocity, radius, total mass, singularity radius, and redshift formula of the universe were calculated, and the gravitational constant was derived and proved to be proportional to the fourth power of cosmic time. The mathematical relationship between lunar laser ranging data and Hubble constant was obtained. This is a newly discovered precise method for measuring the Hubble constant

Multi-Static Radar System Deployment Within a Non-Connected Region Utilising Particle Swarm Optimization

This paper is mainly devoted to studying the deployment problem of a multi-static radar system (MSRS) within a non-connected deployment region using multi-objective particle swarm optimization (MOPSO). By modeling and reformulating the problem, it can be represented as a multi-objective mixed integer programming (MOMIP), which eliminates the need for additional constraints. To enhance the algorithm performance, integer variables and continuous ones are treated separately employing multiple velocity formulas. The velocity formulas for integer variables are modified using the sigmoid function and genetic operation, leading to the proposal of two MSRS deployment algorithms, namely MOPSO-Sigmoid and MOPSO-Gene. To evaluate the performance of the proposed algorithms, they are compared with two existing MOPSO-based algorithms. The first algorithm is the MSRS deployment algorithm for the non-connected deployment region that addresses the additional constraint problem model. The second algorithm is based on an existing conventional MOPSO algorithm and addresses the equivalent MOMIP problem model. A numerical study demonstrates that MOPSO-Sigmoid and MOPSO-Gene exhibit promising efficiency and effectiveness.

The formation and post-formation processes of vortex rings discharged from laminar starting jets

Motivated by biological systems, vortex rings can enhance the propulsive efficiency in industrial systems. To study the vortex properties during the formation and post-formation stages, impulsively starting jets are investigated by simulations. The effects of the stroke ratio and nozzle geometry are studied at a fixed jet Reynolds number of 2500. The stroke ratio at the formation number is found to be not enough to produce a vortex ring with maximum circulation. The stroke ratio is suggested to be about twice as large as the formation number. An alternative criterion based on the circulation ratio is proposed to describe the onset of pinch-off. This criterion states that the pinch-off would start when the vortex ring attains about 80% of the total jet circulation. During the formation stage, the scaling laws for vortex trajectories and circulation are proposed for continuous formation. By combining the suggested scaling relations and Saffman's velocity formula, the evolution of non-dimensional energy can be predicted. During the post-formation stage, the scaling laws for vortex properties (e.g., the vortex ring diameter, translational velocity, and circulation) are found to be independent of both the nozzle configuration and the vortex Reynolds number. On the grounds of the invariance of impulse in vortex decay, the scaling laws of vortex motion are derived for the non-dimensional energy, circulation, and diffusivity scale of the vortex core. In consequence, the normalized energy and circulation found in experiments can be successfully derived from the similarity model for both nozzle configurations.

Semi-Empirical Model Based on the Influence of Turbulence Intensity on the Wake of Vertical Axis Turbines

In the context of energy shortages and the development of new energy sources, tidal current energy has emerged as a promising alternative. It is typically harnessed by deploying arrays of multiple water turbines offshore. Vertical axis water turbines (VAWTs), as key units in these arrays, have wake effects that influence array spacing and energy efficiency. However, existing studies on wake velocity distribution models for VAWTs are limited in number, accuracy, and consideration of influencing factors. A precise theoretical model (Lam’s formula) for wake lateral velocity can better predict wake decay, aiding in the optimization of tidal current energy array designs. Turbulence in the ocean, serving as a medium for energy exchange between high-energy and low-energy water flows, significantly impacts the wake recovery of water turbines. To simplify the problem, this study uses software ANSYS Fluent 2020 R2 for two-dimensional simulations of VAWT wake decay under different turbulence intensities, confirming the critical role of turbulence intensity in wake velocity decay. Based on the obtained data, a new mathematical approach was employed to incorporate turbulence intensity into Lam’s wake formula for VAWTs, improving its predictive accuracy with a minimum error of 1%, and refining some parameter calculations. The results show that this model effectively reflects the impact of turbulence on VAWT wake recovery and can be used to predict wake decay under various turbulence conditions, providing a theoretical basis for VAWT design, optimization, and array layout.

Three-Dimensional Lorentz-Invariant Velocities

Lorentz invariance underlies special relativity, and the energy formula and relative velocity formula are well known to be invariant under a Lorentz transformation. Here, we determine the functional forms in terms of four arbitrary functions for those three dimensional velocity fields that are automatically invariant under the most general fully three-dimensional Lorentz transformation. For general three-dimensional motion, using rectangular Cartesian coordinates (x,y,z), we determine the first-order partial differential equations for the three velocity components u(x,y,z,t), v(x,y,z,t) and w(x,y,z,t) in the x−, y− and z−directions respectively. These partial differential equations and the associated partial differential relations connecting energy and momentum are fully compatible with the Lorentz-invariant energy–momentum relations and appear not to have been given previously in the literature. We determine the spatial and temporal dependence of the functional forms for those three-dimensional velocity fields that are automatically invariant under three-dimensional Lorentz transformations. An interesting special case gives rise to families of particle paths for which the magnitude of the velocity is the speed of light. This is indicative of the abundant possibilities existing in the “fast lane”.

On the existence of Rayleigh waves with full impedance boundary condition

The existence of Rayleigh waves (propagating in isotropic elastic half-spaces) with the tangential and normal impedance boundary conditions was investigated. It has been shown that for the tangential impedance boundary condition (TIBC), there always exists a unique Rayleigh wave, while for the normal impedance boundary condition (NIBC), there exists a domain (of impedance and material parameters) in which exactly one Rayleigh wave is possible and outside this domain a Rayleigh wave is impossible. In this paper, we consider the existence of Rayleigh waves with the full impedance boundary condition (FIBC) that contains both TIBC and NIBC. It is shown that the existence picture of Rayleigh waves for this general case is more complicated. It contains domain for which exactly one Rayleigh wave exists, domain where a Rayleigh wave is impossible, and domain for which all three possibilities may occur: two Rayleigh waves exist, one Rayleigh wave exists, and no Rayleigh wave exists at all. The obtained existence results recover the existence results established previously for the cases of TIBC and NIBC. The formulas for the Rayleigh wave velocity are derived. As these formulas are totally explicit, they are very useful in various practical applications, especially in the non-destructive evaluation of the mechanical properties of structures. In order to establish the existence results and derive formulas for the Rayleigh wave velocity, the complex function method, which is based on the Cauchy-type integrals, is employed.

Effect of physicochemical properties on critical sinking and attachment of respirable coal mine dust impacting on a water surface

Respirable coal mine dust (RCMD) inhalation is identified as the main cause of the resurgence of coal worker’s pneumoconiosis (CWP) since the mid-1990s. At present, the predominant dust control technology is the water spray system. However, in practice, the capture efficiency of RCMD by this technology is relatively low. To understand the capturing mechanism and develop improvement strategies, this research is focused on the surface chemistry study of RCMD and its impact on a water surface using a dynamic model. Proximate analysis, chemical, and mineral composition of a run-of-mine (ROM) coal sample from Appalachian region were analyzed using a proximate analyzer, Inductively Coupled Plasma Mass Spectrometry (ICP-MS), and X-ray Diffraction (XRD), respectively. Contact angles were measured by capillary rise test using the Washburn equation. Based on the dynamic model, the effects of particle size, density, contact angle, and surface tension on the critical sinking were investigated. It was pointed out in this work that reducing surface tension, in turn, decreases contact angle, which has been neglected in the literature. Regime maps for different minerals were created and showed that organic matter has the highest critical velocity due to its low density and high contact angle. Reducing water surface tension to the critical solid surface tension of coal around 30 mN/m could maximize the attachment efficiency. Scaling laws, constructed by force balance, led to the criteria of critical sinking: Ucr∼61-cosθ2D+1γlvρlR, i.e., Wecr∼121-cosθ2D+1. A semi-empirical formula for critical velocity was obtained by fitting the simulation data, Ucr=1.0961-cosθ2D+1γlvρlR. Attachment efficiency was defined and formulated as Pa=U02Ucr2=We0Wecr, establishing relationships between attachment efficiency and the physicochemical properties of RCMD and water droplets.

Prediction of Pier Scour Depth under Extreme Typhoon Storm Tide

The Western Pacific region is highly vulnerable to typhoon storm surge disasters, with localized erosion posing a particularly prominent issue for coastal marine structures. The prevalence of extreme typhoon storm surges poses a significant threat to the safety of engineering projects in these areas. In this study, a parameterized wind field model with precise calculation of wind speed was employed to establish a numerical model for typhoon storm tides. Based on the Western Pacific typhoon data from 1949 to 2023, hydraulic simulations were conducted for Hangzhou Bay, Xiangshan Port, and Yueqing Bay, revealing maximum flow velocities of 4.5 m/s, 1.95 m/s, and 2.09 m/s, respectively. These velocities exceeded the maximum possible tidal flow by 0.47–1.17 m/s. Additionally, using Sun’s velocity formula, the initiation flow velocities were calculated to be 1.85 m/s, 1.81 m/s, and 2.06 m/s for the aforementioned locations. Through localized erosion tests conducted around typical bridge piers and the subsequent application of similarity criteria, the maximum depth of localized erosion in the study area was determined to range from 2.16 m to 16.1 m, which corresponds to 1.1–2.3 times the scour caused by the maximum tidal flow scenario. A comparison of the erosion test results with calculations based on several formulas demonstrated that the scour prediction formula proposed by Sun exhibited the highest accuracy. This study supplements the understanding of the impact of typhoon storm surges on bridge pier erosion and provides a scientific basis for the design of bridge foundations.

The Influence of Mallet Mass and Velocity on the Fracture Patterns in Osteotomies.

Osteotomies are routinely incorporated in rhinoplasty, however, the influence of mass, velocity, kinetic energy (KE), and momentum (p) of the mallet on fracture patterns has not been studied. An experimental sledge guillotine setup was designed simulating a mallet strike with adjustable height and mass and 2 mm-thick Sawbone blocks. KE and p were calculated using KE = ½ mass × velocity2 and p = mass × velocity formulas. Fracture lengths and angles were measured. Ten groups with varying mallet masses and drop heights were tested with 10 bones per group. Fracture length positively correlated with KE (R = 0.542, p < 0.001) and p (R = 0.508, p < 0.001). Fracture angle also positively correlated with KE (R = 0.367, p < 0.001) and p (R = 0.329, p < 0.001). In groups with similar KE, osteotomies with higher p (heavier mallet with slower velocity) had greater fracture lengths (29.31 ± 0.68 vs. 27.68 ± 2.12 mm, p = 0.013) but similar fracture angles (p = 0.189). In groups with similar p, osteotomies with higher KE (lighter hammer with faster velocity) had significantly greater fracture lengths (28.28 ± 1.28 vs. 20.45 ± 12.20 mm, p = 0.041) and greater divergent fracture angles (3.13 ± 1.97° vs. 1.40 ± 1.36°, p = 0.031). Regression modeling of the relationship between KE and fracture lengths and angles demonstrated that cubic followed by logarithmic regression models had the best fits. Osteotomy fracture patterns positively correlated with the mallet's KE more so than its p, suggesting that the mallet's velocity has an increased impact effect than its mass. Clinically, a heavier mallet with a lower velocity will likely generate a smaller fracture length and fracture angle, indicating a more controlled and ideal fracture. NA Laryngoscope, 2024.

Model and new imbalance thrust force method mechanical model for thrust-type soil landslides

The slice method used in traditional slope stability analysis has been improved, but the development of monitoring technology means that its assumption of the rigid contact condition has become inapplicable. This study considered the case of the Ya'an Baoxing landslide, which is a thrust-type soil landslide in Sichuan Province (China). First, on the basis of the Imbalanced Thrust Force Method, a mechanical model with a Kelvin body added between the slice blocks was established, and the slice block velocity formula was derived using the dynamic differential equation. Notably, this model is currently at the stage of qualitative research and more effective simplification methods are required; however, this study does promote both a new concept for the slope stability method and a new approach to landslide mechanism analysis. Comparison of model test monitoring data and the analytical results revealed that the mechanical model could effectively reflect the internal mechanical transmission law of the thrust-type soil landslide. It verified that the landslide evolution process conformed to the Saito model, and revealed that the unbalanced force of the non-disintegrated landslide, which followed the law of internal force transmission from the top to the foot of the slope, initially rose and then fell rapidly. Finally, the interaction mechanism between the ideal elastoplastic anchor cable and the slope was examined based on engineering experience. This study innovatively reinterpreted the traditional slice methods, explored and proved to a certain extent the transmission mode of internal stress in a thrust-type soil landslide, and verified the possibility of achieving high-precision monitoring and early warning of slope instability using a mechanical monitoring method.

Modeling the longitudinal wave in a nanorod based on a novel theory of elastic waves with surface effects

This paper investigates the impact of surface effects on the propagation behavior of longitudinal waves in a nanorod. A theoretical model has been established on the basis of a newly proposed theory of elastic waves with surface effects. The surface effects comprise two components: the effect of surface energy and the effect of surface inertia. An analytical formula for the longitudinal wave velocity of a nanorod has been derived. Two inherent lengths at nanoscale have been deduced to characterize these two types of surface effects. The results indicate that the longitudinal wave in a nanorod is still nondispersive. However, an attractive phenomenon uncovered is that when the size of a rod reduces to the inherent lengths at nanoscale, the longitudinal wave velocity becomes size-dependent due to the effects of surface energy and surface inertia. The former increases the longitudinal wave velocity, whereas the latter decreases it. This can be understood as the former equivalently increasing the stiffness of the nanorod, whereas the latter enhancing its effective density. On the other hand, when the rod is at the macroscale, the longitudinal wave velocity degenerates to the classical velocity for a macroscopic rod without any surface effects. The current findings not only enhance our understanding of the size-dependent wave velocity of longitudinal waves in nanorods but also facilitate precisely designing the elastic wave nanodevices.

Sedimentation of a Charged Soft Sphere within a Charged Spherical Cavity.

The sedimentation of a soft particle composed of an uncharged hard sphere core and a charged porous surface layer inside a concentric charged spherical cavity full of a symmetric electrolyte solution is analyzed in a quasi-steady state. By using a regular perturbation method with small fixed charge densities of the soft sphere and cavity wall, a set of linearized electrokinetic equations relevant to the fluid velocity field, electrical potential profile, and ionic electrochemical potential energy distributions are solved. A closed-form formula for the sedimentation velocity of the soft sphere is obtained as a function of the ratios of core-to-particle radii, particle-to-cavity radii, particle radius-to-Debye screening length, and particle radius-to-porous layer permeation length. The existence of the surface charge on the cavity wall increases the settling velocity of the charged soft sphere, principally because of the electroosmotic enhancement of fluid recirculation within the cavity induced by the sedimentation potential gradient. When the porous layer space charge and cavity wall surface charge have the same sign, the particle velocity is generally enhanced by the presence of the cavity. When these fixed charges have opposite signs, the particle velocity will be enhanced/reduced by the presence of the cavity if the wall surface charge density is sufficiently large/small relative to the porous layer space charge density in magnitude. The effect of the wall surface charge on the sedimentation of the soft sphere increases with decreases in the ratios of core-to-particle radii, particle-to-cavity radii, and particle radius-to-porous layer permeation length but is not a monotonic function of the ratio of particle radius-to-Debye length.

Comparative analysis of viscous dissipation effects on Prandtl fluid including contraction and relaxation phenomena

AbstractThis paper investigates the effects of magnetohydrodynamics and heat transfer on the peristaltic transport of Prandtl fluid in a vertical endoscopic tube. The reduction of the complexity of the equations governing the flow of Prandtl fluid entails the use of long wavelength and low Reynolds number approximations. These complex equations for the pressure gradient and velocity profile are handled analytically using the perturbation technique with convective boundary conditions, and the temperature and concentration profiles are carefully solved for the exact solution. The frictional forces and pressure rise are also simulated with numerical integration. The resulting formulas for velocity, temperature, concentration, pressure rise, and pressure gradient are graphed using the MATLAB and MATHMATICA software, and the effects of all the different physical parameters are investigated and assessed. The streamlines with five distinct wave types are sketched at the conclusion to show the phenomenon of trapping. It is investigated that the velocity profile rises due to buoyancy forces and falls due to the influence of magnetic forces.

DISEASED BILE FLOW UNDER THE EFFECTS OF HEAT TRANSFER AND CHEMICAL REACTIONS

The movement of bile inside ducts in a diseased state is investigated using a mathematical model that takes into account heat transport and chemical interactions. Bile is considered as a viscous fluid, geometry of ducts is assumed as finite length asymmetric channel and the flow is induced by peristaltic wave along the length of channel walls. Under the presumption of a long wave length and a low Reynolds number, the formulas for axial velocity, pressure gradient, volume flow rate, stream function, pressure increase, shear stress, and heat transfer coefficient are produced. In the end, a plot and discussion are made of how different emergent factors affect the relevant physical quantities.

Functional Forms for Lorentz Invariant Velocities

Lorentz invariance lies at the very heart of Einstein’s special relativity, and both the energy formula and the relative velocity formula are well-known to be invariant under a Lorentz transformation. Here, we investigate the spatial and temporal dependence of the velocity field itself u(x,t) and we pose the problem of the determination of the functional form of those velocity fields u(x,t) which are automatically invariant under a Lorentz transformation. For a single spatial dimension, we determine a first-order partial differential equation for the velocity u(x,t), which appears to be unknown in the literature, and we investigate its main consequences, including demonstrating that it is entirely consistent with many of the familiar outcomes of special relativity and deriving two new partial differential relations connecting energy and momentum that are fully compatible with the Lorentz invariant energy–momentum relations.

Settling velocity of atmospheric particles in seawater: Based on hydrostatic sedimentation method using video imaging techniques

When atmospheric particles deposit to the ocean, their settling velocities and residence times associated are critical for their effects on oceanic ecosystems. We developed a hydrostatic sedimentation method using video imaging techniques to track particles of 5–20 μm in diameter falling into seawater and determine the particle settling velocities in relation to their diameter, shape, organic matter contained, and seawater salinity. The measured settling velocities varied from 0.025 to 0.41 mm/s. Irregular particle shape and organic matter contained in particles also, however, reduced the values. The settling velocities were decelerated by the dissolution process of particle in seawater. Combined with the experimental results, a formula for calculating the settling velocity formulae for atmospheric particles was estimated. Using this equation, the residence time of particles is estimated to be less than one month in continental shelf sea and more than 100 days in the oceans.

The Aeolian harp sound from the Madelung model of quantum mechanics

According to Madelung, Bohm and Vigier, Wilhelm, Rosen and others, the original Schroedinger equation can be transformed into the hydrodynamical system of equations by using the so called Madelung ansatz. We derive in such quantum hydrodynamics, the non-relativistic and relativistic Strouhal number from the so called von Karman's vortex street. The relativistic derivation of this formula follows from the addition formula for velocities. The Strouhal friction tones are generated also during the motion of cosmic rays in relic photon sea, during the motion of bolids in atmosphere, during the Saturn rings motion in the relic black-body sea, during the motion of bodies in superfluid helium and so on.

ANALYTICAL SOLUTION FOR POTENTIAL FLOW AROUND A ROTATING SPHERE AND COMPARISON WITH CFD

A three-dimensional analytical solution was derived for an incompressible steady potential at an initially uniform velocity flow around a sphere, which translates forward, backward, rotates longitudinally or transversally. The concept of relative velocity was used to analyze the flow around the transitionally moving sphere. To analyze the flow around the longitudinally rotating sphere, A formula of the circumferential velocity of the fluid is found at the equatorial plane of the sphere and then generalized to the whole sphere as an approximation. The superposition principle of velocities was used to analyze the flow around the transversely rotating sphere, the stagnation points were detected and analyzed, the pressure distribution at the equator was calculated and compared with the experimental and CFD results, and the lift coefficient was calculated and compared with the experimental results and a good agreement for C_p and C_L was found at low spin factors, In contrast, at high spin factors the results begin to diverge due to the viscous effects and eddy formation.

Approximate theory of armour perforation by ductile hole expansion

The present study suggests a complete analytical model for armour perforation by ductile hole enlargement process. The new approximate model is applicable for entry, tunnelling and exit phases of a rigid nose-pointed projectile with an arbitrary nose-shape. The theory is based on dividing the target into infinitesimal thin layers, in which a cylindrical hole expansion is assumed. The elastoplastic work that is consumed by the target for infinitesimal projectile advancement is based on the specific cavitation energy and leads to reduction of the projectile kinetic energy. Several equivalent formulae for ballistic limit velocity, and a formula for the ballistic limit thickness are found to be independent of projectile nose-shape and nose length. The approximate model is compared with numerical simulations for AA6061-T651 targets that are impacted by two different ogive nose-shape projectiles at several striking velocities. A small nose-shape effect on the ballistic limit velocity is observed in the numerical simulations and is attributed mainly to the target axial effect and the bulge formation at the target rear surface, which are not considered in the approximate model.