Initial Inclination Angle Research Articles

An Enveloping, Centering, and Grabbing Mechanism for Harvesting Hydroponic Leafy Vegetables Cultivated in Pipeline

Manually harvesting hydroponic leafy vegetables from a cultivation pipeline is labor-intensive and expensive. Rapidly grabbing hydroponic leafy vegetables grown in different positions and orientations in the planting hole is the primary issue for efficient mechanical harvesting. Thus, a novel grabbing mechanism with double-pivot rotation cross fingers is proposed. The fingers’ inner surfaces could envelop the grabbable area of the leafy vegetable in the grasping process and position each leafy vegetable stalk to the center of the planting hole before taking it out. A grabbing mechanism for harvesting hydroponic Chinese kale was designed and optimized with less than 1 mm of grasping error, enough enveloping range, and no collision with the extended leaves. Laboratory experiments were conducted to investigate centering and grabbing at different initial positions and inclination angles of the hydroponic Chinese kale and varied finger deflection speeds. It was indicated that for the grasping inclination angle and grabbing success rate, the initial inclination angle was a significant factor, as was the position, whereas the finger deflection speed was insignificant. As the initial inclination angle of matured hydroponic Chinese kale in different initial positions is mostly larger than 60°, the best results were achieved with a finger deflection speed in the range of 40° s−1 to 60° s−1 and grasping inclination angles of 85° to 95°, with a grabbing success rate of more than 95%. This showed the promising applicability of the studied grabbing mechanism for harvesting hydroponic Chinese kale or other varieties of leafy hydroponic vegetables with similar growth characteristics.

Drop impact dynamic and directional transport on dragonfly wing surface

The ability of dragonflies to fly in the rain without being wetted by raindrops has motivated researchers to investigate the impact behavior of a drop on the superhydrophobic wings of dragonflies. This superhydrophobic surface is used as a reference for the design of directional surfaces and has attracted extensive attention owing to its wide applicability in microfluidics, self-cleaning, and other fields. In this study, the static contact angle and rebound process of a drop impacting a dragonfly wing surface are investigated experimentally, whereas the wetting pressure, Gibbs free energy, and Stokes number vs. coefficient of restitution are theoretically calculated to examine the dynamic and unidirectional transport behaviors of the drop. Results show that the initial inclination angle of the dragonfly wing is similar to the sliding angles along with the drop sliding. The water drop bounces from the bottom of the dragonfly wing to the distal position, demonstrating directional migration. The drop impacts the dragonfly wing surface, and the drop exhibits compression, recovery, and separation phases; in these three phases, the drop morphology evolves. As the Gibbs free energy and cross-sectional area evolve, the coefficient of restitution decreases as the drop continues to bounce, and the Stokes number increases.

Transverse penny-shaped hydraulic fracture propagation in naturally-layered rocks under stress boundaries: A 3D phase field modeling

The ground (tectonic) stress and layered structures of rocks are the among the main factors that influences the hydraulic fracturing behavior. The purpose of this study is to examine the suitability of the phase field model (PFM) in simulating transverse penny-shaped hydraulic fracture propagation and to investigate the phase field feature for the transverse penny-shaped hydraulic fracture in layered rocks under stress boundaries. A phase field model for 3D transverse penny-shaped hydraulic fracture propagation in naturally-layered rocks considering the effect of initial stress field is established. The mathematical model is based on Biot poroelasticity theory and the phase field fracture theory, while the governing equations are solved by using the finite element method in a staggered manner. The established PFM is validated experimentally and analytically by 2D and 3D examples. At last, the influences of the initial stress field, stiffness contrast and inclination angle of the layer interface on the penny-shaped fracture evolution in naturally-layered rocks are investigated by using the PFM. The study indicates that the phase field model has excellent feasibility and practicability in predicting penny-shaped hydraulic fractures. The layer inclination has nearly no effect on the penny-shaped fracture evolution in naturally-layered rocks. The stress ratio Sv/Sh on the boundaries has a significant effect on propagation of the penny-shaped hydraulic fracture. With the increase in Sv/Sh, the hydraulic fracture deflects and propagates along the direction of the maximum in-situ stress. For a high Sv/Sh, branching scenarios can be observed. The stiffness contrast of the rock layers determines whether the penny-shaped hydraulic fracture can penetrate into the adjacent layer. The predictions on the effects of the initial stress field, stiffness contrast and inclination angle of the layer interface provide new understanding of penny-shaped hydraulic fracture propagation in underground geological environment.

The study of mixed mode fatigue crack growth mechanism of 25CrNiMo compact tensile specimens by experiment and finite element simulation

Fatigue cracking is the most common failure mode that endangers the safety and service life of parts. Initial manufacturing flaws are a critical factor affecting the growth of fatigue cracks. Currently, mixed-mode crack growth research frequently applies a simplistic two-dimensional finite element model, which cannot adequately represent the crack growth mechanism. An improved finite element method is proposed in this paper to overcome this drawback. Fatigue fracture propagation investigations on 25CrNiMo compact tensile specimens are conducted in this study, and the crack propagation rate is calculated using the Paris formula. Moreover, the special element modeling method and second-order Runge-Kutta integration method are utilized to perform finite element calculations on the 3D model. The results are highly consistent with the experimental test results. The propagation mechanism of cracks with varied initial inclination angles is explored depending on the finite element method. The relationship between the stress intensity measures KI, KII, and KIII is discovered to be competitive. The results have significant application value for the crack growth path prediction and crack life in parts with initial flaws.

Heat transfer enhancement by a pair of asymmetric flexible vortex generators and thermal performance prediction using machine learning algorithms

• Heat transfer enhancement system including two flexible vortex generators is analyzed by using an immersed boundary method. • Effects of the inclination angle, bending rigidity, and the gap distance on the thermal performance are analyzed in detail. • The thermal performance of the present heat transfer system is enhanced up to 122% as compared to baseline flow without flexible vortex generators. Heat transfer should be considered in the design process of various industrial fields such as electric-electronics, plants and refrigeration and air conditioning, etc. Many researchers have introduced vortex generators to promote mixing of fluids, thereby improving heat transfer performance. Convective heat transfer can be enhanced by using vortex generators where, however, the disadvantage of increased mechanical loss due to increased pressure drop is not avoidable. In order to compensate for the penalty, the present study utilizes the characteristics of self-sustained flapping motions of the flexible vortex generator to improve heat transfer without much increasing the pressure drop. The two flexible vortex generators (FVGs) fixed on the upper and lower channel walls, respectively, are introduced in the present study and they take the form of asymmetric configurations by adjusting the inclination angles. The heat transfer performance is observed to depend on the parametric conditions of FVGs such as the inclination angle, bending rigidity, and the gap distance between them. The present system including the asymmetric FVGs shows the improvement of thermal performance by 122% as compared to the baseline flow when the bending rigidity is 0.06, the initial inclination angle is 75°, and the gap distance is 2.4 times the length of FVGs. The attracting Lagrangian coherent structures (LCS) are visualized by calculating the finite-time Lyapunov exponent (FTLE) field to analyze the effects of the vortex structures near the heated channel walls on the fluid mixing. The net energy loss and average heat transfer are predicted by the three machine learning algorithms, i.e ., artificial neural network (ANN), support vector regression (SVR), and random forest (RF), where the inclination angle, bending rigidity, and gap distance of FVGs are selected as input data. The SVR and ANN algorithms show the best performance in predicting the mean energy loss and the heat transfer, respectively, with the R 2 value of above 0.99 and 0.95.

The Design of Cone and Pendulum Scanning Mode Using Dual-Camera with Multi-Dimensional Motion Imaging Micro-Nanosatellite

This paper focuses on the design of a new optical cone and pendulum scanning imaging mode for micro-nanosatellites. This kind of satellite uses a high-resolution camera with a small imaging plane to achieve high-resolution and ultra-wide coverage imaging through the three-dimensional motion of the camera’s wobble, satellite spin, and satellite orbital motion. First, this paper designs a single-camera constant speed OCPSI (optical cone and pendulum scanning imaging) mode. On the premise of ensuring coverage, the motion parameters and imaging parameters are derived. Then, in order to improve the performance and imaging quality of the system, a dual-camera variable speed OCPSI mode is designed. In this method, in order to reduce the overlap ratio, the camera is oscillated at a variable speed. Turn on the cameras in turn at the same time to minimize the overlap. This paper details these working modes. The simulation experiment is carried out using the satellite orbit of 500 km, the focal length of 360 mm, the pixel size of 2.5 μm, the resolution of [5120 × 5120], the number of imaging frames in the pendulum scanning hoop of 10, and the initial camera inclination angle of 30°. The single-camera constant speed OCPSI mode has an effective swath of 1060 km at a ground sampling distance of 5.3 m. The dual-camera variable speed OCPSI mode has an effective width of 966 km under the same conditions. Finally, the ground experiment prototype of OCPSI imaging theory is designed. We choose a camera with a pixel size of 3.45 μm, a resolution of [1440 × 1080], and a focal length of 25 mm. The ground experiment was carried out at the initial camera inclination angle of 10°, the number of imaging frames in the pendulum scanning hoop of 3, and the orbit height of 11 m. The experimental result is that the effective width of OCPSI imaging mode reaches 10.8 m. Compared with the traditional push-broom mode using the same camera, the effective width of 1.64 m is increased by seven times, and the effective width of 3.83 m is increased by three times compared to the traditional whisk-broom imaging mode. This study innovatively integrates three-dimensional motion imaging into aerospace remote sensing and provides a reference for the research on the realization of high-resolution and ultra-wide coverage of micro-nano remote sensing satellites.

Pressure distribution on a flat plate in the context of the phenomenon of the Coanda effect hysteresis

As a result of the Coanda effect, a symmetrical free jet will flow as an asymmetrical wall jet. At the same time, at the obstacle along which the flow is observed, the wall jet generates pressure distribution. In this study, the obstacle located at the diffuser outlet is a flat plate with a variable inclination angle. The article presents results of the study on pressure distributions on a flat plate with a variable angle of inclination. In the experiment, the Reynolds number ranged from 16,192 to 42,240. A fixed geometry diffuser (Witoszyński nozzle) with a height of 0.60 m, width of 0.02 m and outlet velocity of 11.33–29.57 m/s was used. A plate with a length of 1.00 m and a variable inclination angle was installed at the diffuser outlet. What is new, however, is that the presented results of the experimental research include the influence of the Coanda effect hysteresis on the pressure distribution on the plate. The article shows how pressure distributions change on the plate depending on whether the initial angle of inclination was 0° and was increased gradually in the course of the experiment until a detachment of the jet flowing from the plate was observed, or the initial angle of inclination was close to 90° in the primal state and as the angle of the plate inclination was decreased, the jet flowing towards the plate reached the state of attachment to the plate surface. The study demonstrated that for a turbulent jet, pressure distribution on a flat plate is determined not only by the plate’s inclination angle, but also by the direction of its rotation.

A Lateral-Response Analyzed Model for Piles in Sloping Ground Considering Vertical Resistance Based on Transfer Matrix Method

In order to analyze the lateral response of piles subjected to combinational loads located at sloping ground, a simplified physical model is proposed and developed based on the Timoshenko beam theory and tri-parameter elastic foundation beam model. In this model, the effects of vertical resistance, shear deformation and initial inclination of pile shafts are taken into account, and the semi-analytical solution to the lateral behavior of pile is obtained by utilizing the transfer matrix approach. Specifically, the relative transfer matrix coefficients in the free, passive, and active segment of the pile are analytically determined through Laplace transformation. In addition, the validation of this model is conducted by comparing with field observations and predictions obtained from other existing methods. Moreover, the influences of initial inclination angle, vertical resistance in passive segments and the vertical load on the lateral response of the pile are also discussed in depth. The results indicate that, although there is a significant reduction effect emerged in both the lateral displacement and bending moment with the increase in the vertical resistance, a growth trend for those responses can be obtained with an increase in the initial inclination angle and vertical load.

Experimental Study on Gas–Liquid Interface Evolution during Liquid Displaced by Gas of Mobile Pipeline

The mobile pipeline is the most effective and reliable means for the emergency oil transfer task, which is temporarily laid on the field. After completing the task, the oil in the pipeline should be emptied before the mobile pipeline is removed. The oil in pipeline gradually displaced by the air is a main method of pipeline evacuation, which is the only choice of jet fuel pipeline. Due to the convection between gas and oil phases, the length of the oil–gas mixing section and the evolution of the oil–gas interface change with time. The evolution of the gas–liquid interface directly determines the different flow patterns of oil–gas two phases, which are very important for the mobile pipeline evacuation. In this paper, the characteristics of the gas–liquid interface evolution during the water displaced by air are studied, using the multi-phase pipeline experiment. Through the change of the gas–liquid interface in the initial stage of gas cap evacuation, it is found that the gas–liquid interface can be divided into smooth, wavy, and dispersed forms under different initial gas flow rates and different inclination angles. Slug phenomenon may occur in the process of gas-carrying liquid flow in the pipeline. The emergence of slug is mainly affected by hydraulic conditions and pipeline operating conditions. The liquid plug body is complex and unstable, and there will be a mutual transition between different liquid slug shapes. The increase of the inclination angle and gas phase velocity will accelerate the occurrence of liquid slug frequency.

The study of the deviations of the payload moved along the curved trajectory by the crawler crane

Reducing the uncontrolled deviations of the payload moved by a crawler crane with a constant boom length is a relevant objective, since it improves the operational efficiency and safety, as well as the positional accuracy of the payload at the target point. In this regard, the results of the research aimed at developing a trajectory synthesis method of two controlled coordinates of the crane, such as the swing angles of the rotary platform and boom raising which ensure a predefined trajectory of the moving payload, have been described. Furthermore, the influence of the payload movement parameters, such as the time of the prescribed payload displacement, the initial angle of the boom lift, the hoist rope length, the overall dimensions of the required payload displacement trajectory on the error of the predefined payload trajectory execution by the lifting crane has been studied. For example, we examined a predefined arc curved trajectory of the moving payload. The horizontal coordinates of the trajectory of the upper suspension point located on the boom head and providing the predefined payload trajectory were generated using the Matlab mathematical simulation model developed in the Simulink package for solving a linearized differential equation. This equation describes the deviations of a pendulum suspended payload with a horizontally moving suspension point in a separate vertically located plane of the space. The superposition of two horizontal trajectories obtained by solving the given equation ensures the payload movement along a curved trajectory predefined in the horizontal plane. The trajectory of the crane’s controlled coordinates was calculated from the horizontal coordinates of the suspension point. When executing the trajectory, the examined additional errors occur. It was revealed that when the time of moving along a predefined payload trajectory exceeds 40 seconds, the residual oscillations become insignificant and payload positional accuracy at the target point increases. It suggests the limited applicability of the developed method when implemented on the rotary lifting cranes with a constant boom length. The range of the local minima errors of the predefined trajectory implementation depending on the initial inclination angle of the boom, which varies in the average values range of the boom inclination angle, has been defined.

Numerical study of hydraulic fracturing near a wellbore using dual boundary element method

The pattern and path of hydraulic fracture (HF) from a wellbore circumference have a great impact on subsequent formation of complex fracture networks highly desired for permeability enhancement of unconventional reservoirs. In this paper, a new fully coupled hydraulic fracturing model is developed in the framework of the dual boundary element method (DBEM) that describes the deformation of rock matrix and the finite volume method that simulates the fluid flow in the fracture and the distribution of injection rate among fractures. The DBEM applies displacement boundary integral equation to one crack surface and traction boundary integral equation to the other one. The model presented is first validated against the existing semi-analytical and numerical solutions for single fracture in an infinite domain, and shows excellent agreements with a relative error less than 1% for injection pressure and half fracture length. Then the problem of multiple HFs including natural fractures (NFs) cutting the wellbore is investigated. It is shown that a larger product of fluid viscosity and injection rate or a small in-situ stress difference can smooth the fracture path and enhance the fracture opening. In the presence of multiple fractures, the HF and NF competes in growth, and the results show that the initial permeability and inclination angle of NF can alter the propagation preference from HF to NF. In particular, a large fluid viscosity and a more isotropic stress condition will propagate both NFs and HFs simultaneously.

Numerical investigation on the water entry of a 3D circular cylinder based on a GPU-accelerated SPH method

As a typical fluid–structure interaction (FSI) problem, water entry involves violent fluid flows and changing free surfaces, which presents great challenges for numerical modeling. Smoothed Particle Hydrodynamics (SPH) is a Lagrangian particle method that has natural advantages in modeling free surfaces and moving interfaces. However, SPH is computationally expensive due to the search of particle–particle interactions, and it causes great difficulties for performing large-scale simulations of 3D FSI problems. In this work, we present an accelerated SPH framework based on the Graphics Processing Unit (GPU) techniques to study water entry problems. The multi-threading programmed by Compute Unified Device Architecture (CUDA) is applied to enhance the computational performance in terms of efficiency and scale. Compared to the Single-CPU-based strategy, the newly presented GPU-accelerated SPH method is computationally more efficient with a speedup over hundreds times and enables a larger memory available for large-scale simulations of around ten million particles for three-dimensional cases. With the GPU-accelerated SPH method, the 3D water entry of a circular cylinder is investigated with some kinematic and dynamic characteristics explained. The results demonstrate that the rotational characteristic of a 3D cylinder in water entry is related to the dimensionless number γ defined as the ratio of the initial inclination angle to the initial velocity angle. The rotation of a cylinder changes from anticlockwise to clockwise with the increase in γ. A transition value of γ exists between the anticlockwise to clockwise rotation, which focuses on the range from 1.0 to 6.0. Meanwhile, the water entry of a 3D circular cylinder leads to a violent impact on the bottom of the cylinder, which causes a peak value of pressure being a maximum value at the early stage of the water entry. It is also indicated that the selection of the initial inclination angle has a great effect on the maximum pressure.

Peristaltic transport of elliptic particles: A numerical study

Peristaltic transport of elliptic particles suspended in Newtonian fluids is numerically investigated in a planar channel formed between two flexible membranes. Numerical results were obtained under creeping-flow conditions for centered and off-center particles using the lattice Boltzmann method. The results demonstrate the importance of aspect ratio and initial inclination angle on peristaltic transport of solid particles. For a domain comprising just one wave, it was shown that, in free-pumping mode, circular particles move faster than elliptic particles and experience less shear stress. They also resist a larger adverse pressure gradient before they are finally brought to rest. Above a critical Reynolds number, however, elliptic particles are predicted to move faster than circular particles. The effect was attributed to the vulnerability of circular particles to hydrodynamic instability, which is exhibited by the particle detaching itself from the centerline, thereby adopting a longer trajectory. This is the first time that peristaltic transport of elliptic particles is being numerically studied, and the results can be used for designing peristalsis-based micro-swimmers or microfluidic systems deemed for single-cell studies.

A macroscopic viscoelastic model of magnetorheological elastomer with different initial particle chain orientation angles based on fractional viscoelasticity

In this paper, a macroscopic viscoelastic modeling method for magnetorheological elastomer (MRE) based on fractional derivative model is presented to describe the dynamic viscoelastic properties of MRE with different initial particle chain orientation angles. The angle between the particle chain and the applied magnetic field is used as an indicator to describe the directionality of the particle chain. MRE samples with different initial inclination angles have been designed and fabricated. The dynamic viscoelastic properties of different MRE samples under shear working mode were measured using a parallel plate rheometer. The dynamic viscoelastic properties of MRE with different initial inclination angles are analyzed under the test conditions of different strain amplitude, frequency and magnetic flux density. The test results show that the initial inclination angle of the particle chain in the MRE has a significant effect on the dynamic viscoelastic properties of the MRE. A polynomial function is used to describe the relationship between the initial particle chain orientation angle and the magneto-induced modulus of MRE. A phenomenological model of magneto-induced modulus is established based on the fractional derivative model. The model parameters are identified using the nonlinear least square method. The predicted values of the model are in good agreement with the experimental results, indicating that the model can well describe the dynamic viscoelastic properties of MRE.

Global and local fracture behavior in a brittle solid with a set of pre-existing small-scale cracks

For understanding the mechanics of complicated seismic activities, not only global observations of large-scale material behavior but also more local tracing of smaller-scale fractures is needed. In this study, therefore, by using the technique of dynamic photoelasticity, the evolution of fractures and waves is experimentally traced in a two-dimensional linear elastic brittle specimen that has a set of pre-existing small-scale parallel fracture regions. The small-scale fracture regions or cracks, modeling a large-scale geological normal fault plane, are prepared by a digital laser cutter. Our experimental observations with a high-speed video camera, together with finite difference numerical simulations, indicate that the global and local fracture behavior significantly depends on the initial inclination angle and distribution pattern of the set of parallel cracks. In the case of pre-existing cracks that are more vertically inclined with respect to the quasi-static external load applied by a tensile testing machine, the main fracture develops dynamically and then the secondary fractures appear at spatiotemporally remote distances from the main fracture. The secondary fractures extend also dynamically, into the direction opposite to the main fracture. On the other hand, in the case of smaller inclination angles, the main fracture tends to evolve quasi-statically. Nevertheless, in both cases, it is found that fracture can be arrested and jump without difficulties in a brittle solid having a number of small-scale cracks, which implies that such small-scale cracks may play an important part in the generation of a cluster of seismic events.

Effect of Weibull parameters and crack distribution on the failure probability of multi-layered SiC cladding

In this paper, an improved Weibull failure probability model which considers initial cracks distributed in the cladding has been proposed. Considering the coupling of force, heat and irradiation acting on the multi-layered SiC cladding, failure probabilities of the cladding under different conditions have been obtained. The results indicate that compared to linear heat rate, the cladding is more sensitive to changes in inner pressure of fuel elements. The cladding can be optimized by adjusting the proportion of different layers. Further investigations indicate that effect of different Weibull parameters of CVD SiC on the failure probability is small and can be ignored in real fuel elements. Meanwhile, cracks in SiC composite layer and crack length have little effect on the failure probability. However, the initial inclination angle can affect the failure probability greatly. Results indicate that the most dangerous inclination angle in the cladding is about 73°.

Mechanical performance of inclined Dougong bracket sets under vertical load: Experimental tests and finite element calculation

This paper presents an investigation on the mechanical performance of inclined Dougong bracket sets under the vertical load. Initial inclination angles of Dougongs are determined based on material test results and embedment mechanism theories. Four full-scale Dougongs with different initial inclination angles are tested under the vertical load. The failure patterns, load-displacement curves, load distribution and displacement analyses are disclosed. Moreover, finite element calculation is conducted, model validation, stress and strain analyses are illustrated. Parametric analyses considering initial inclination angles of Dougongs, compressive strength and elastic moduli of wood, and friction coefficients between embedded surfaces are also carried out. The results indicate that inclined Dougongs under the vertical load behave similarly to a timber block under compression in the perpendicular-to-grain direction. Increasing the initial inclination angles can enhance vertical stiffness to some extent, but the bearing capacities of the inclined Dougongs are not promoted. For inclined Dougongs, part of the load transferred in the width direction is gradually carried by other components in the overhanging direction. The inclined Dougong rotates under the vertical load, and as the load increases, the overall rotation becomes more obvious. Parametric studies show that as initial inclination angles increases, plastic stiffness decreases to some extent. With the increment of compressive strength, elastic moduli, and friction coefficients, ultimate displacement decreases but plastic stiffness obviously increases.

Improving stability of movement of machine section for soil preparation and seeding

The subject research is operation of a combined machine for soil preparation and seeding of sunflower and corn seeds. Technological process of machine operation with installed guides, passive rotating flat discs with flanges (which properly ensure movement of soil along the plough share to loosening-separating device), a sowing device, a seed tube, a furrow former, a rotor, a separating grid, a parallelogram mechanism, a spring, a share is described. Dynamic prerequisites for increasing uniformity of depth of groove formation and seed placement in depth in soil are considered. Values of length of links of parallelogram mechanism, initial angle of their installation and stiffness of spring, values of deviations of combined machine section from given depth of movement of plough share are determined. It is proved that with an increase in length of levers of parallelogram mechanism, maximum deviations of section increase. An increase in initial angle of inclination of levers of parallelogram mechanism causes an increase in maximum deflections. As spring stiffness increases, maximum deflections decrease. Relevance of study lies in ensuring stability of copying soil surface by working bodies of combined machine while depth of seeding remains unchanged along entire length of movement, which will make it possible to increase movement speed and unit width. Target group of consumers of information in the article - designers, specialists involved in development of tillage machines.

A numerical study on free-fall of a torus with initial inclination angle at low Reynolds numbers

In this study, the sedimentation of a torus is studied numerically over a range of Reynolds number spanning 10 ≤ ReT≤ 90. The three-dimensional governing equations are solved using the fluid–structure interaction version of the SVD-GFD method on a hybrid Cartesian-cum-meshless grid. The torus has a fixed aspect ratio AR = 2, and is set to fall along the negative x direction at an initial inclination angle θ0, in the range of 0 ≤θ0≤ 80°. The flow behavior induced by the falling torus is characterized by the far field vortex sheet and the near field recirculation zone. At a given ReT, the flow reaches the same terminal state and the torus maintains the same final state, orientation and terminal velocity, regardless of θ0. The torus initially experiences a zigzag motion (along the x and z directions) in the global frame and the pitch motion in its body frame, which are more obvious at larger ReT and θ0. The offset of the torus centroid in the z direction between the initial and terminal states increases with increasing ReT and θ0 because of the θ0 effect. The falling time (Tf) for the torus traveling along the same distance in the x direction decreases with increasing θ0 when ReT≤ 50. However, when ReT > 50, Tf initially increases and then decreases with increasing θ0, with the maximum Tf occurring at θ0≈ 60°. A lift-like force is generated in the positive x direction due to the translational torus motion in the z direction, which originates from the pressure. This force is most pronounced at θ0≈ 60° for ReT > 50, which thus significantly decelerates the falling speed of the torus and increases Tf.

Analysis of an artificial satellite orbit around the Earth under an influence of a rotating gravitational field

In this article, the orbit of a satellite placed in a rotating gravitational (Kerr new metric) field is derived by using symmetries of the metric method. It reveals that the orbit is an ellipse depending on mass, angular momentum of the Earth and the initial inclination angle. A mathematical treatment analyzes the orbit of a satellite around the Earth at an inclination of the equatorial plane. The advance of perigee of the orbital motion is administered. Additionally, the periodic time and the advance of perigee for some artificial satellites are calculated. It is shown that the advance of perigee of the orbital motion of the satellite is inversely proportional to its periodic time. The perturbed differential equation, describing the orbit of the satellite, is utilized to govern stability criterion. Furthermore, the effects of mass, angular momentum of the Earth, and the semi-latus rectum of the elliptical orbit on the satellite motion are discussed. As a result of examining perturbed geodesics, the orbit stability problem is inspected. A new theoretical finding related to the stable satellite model, consistent with those of the previous works, is obtained. Moreover, a formulation of the precession of the nodes has been accomplished.