Numerical simulations of a freely falling rigid sphere in bounded and unbounded water domains

Numerical studies were conducted on the hydrodynamics of a freely falling rigid sphere in bounded and unbounded water domains to investigate the drag coefficient, normalized velocity, pressure coefficient, and skin friction coefficient as a function of dimensionless time. The bounded domain was simulated by bringing the cylindrical water container's wall closer to the impacting rigid sphere and linking it to the blockage ratio (BR), defined as the ratio of the projection area of a freely falling sphere to that of the cross-section area of the cylindrical water container. Six cases of bounded domains (BR= 1%, 25%, 45%, 55%, 65%, and 75%) were studied. However, the unbounded domain was considered with a BR of 0.01%. In addition, the k–ω shear stress transport (SST) turbulence model was employed, and the computed results of the bounded domain were compared with those of other studies on unbounded domains. In the case of the bounded domain, which has a higher value of BR, a substantial reduction in normalized velocity and an increase in the drag coefficient were found. Moreover, the bounded domain yielded a significant increase in the pressure coefficient when the sphere was half-submerged; however, an insignificant effect was found on the skin friction coefficient. In the case of the unbounded domain, a significant reduction in the normalized velocity occurred with a decrease in the Reynold number (Re) whereas the drag coefficient increases with a decrease in Reynolds number.

The Contact Mechanics for Indentation of Single Asperity and Rough Surfaces

A Finite Element Analysis of a rigid sphere contact with a deformable elastic-plastic plat called indentation model is studied. The numerical results are applied on the rough surfaces contact of the GW model. A series of the relationships of the rough surfaces contact parameters are obtained. The contact parameters of the indentation model and the flattening model are compared in detail and the reasons for their differences are analyzed. In the case of single asperity contact, for ω/ωc > 1, the Indentation model reaches the initial plastic yield while the flattening model is ω/ωc = 1. In ω/ωc = 10, the plastic yield reaches the contact surface for the first time, and the corresponding point of ψ = 0.5 the flattening model is relatively earlier in . The contact parameters of rough surface in different plasticity indexes are compared again. On the point of ω/ωc = 6, the contact parameters of the flattening model and the indentation model coincide perfectly. For 0.5 < ψ < 4, the difference between the parameters curves become larger and larger. To the point of ψ = 4, when the distance difference reaches the maximum, it begins to decrease until the two curves are close to coincide again. The dimensionless elastic-plastic contact hardness is introduced. The relation between real contact area and the contact pressure of the indentation model can be acquired quickly. The results show that the geometric shape of deformable contact parts has an important effect on the contact parameters, especially for the extension of plastic deformation region within a specific range of plasticity index.

Lubricated Soft Normal Elastic Contact of a Sphere: A New Numerical Method and Experiment

An important problem in lubrication is the squeezing of a thin liquid film between a rigid sphere and an elastic substrate under normal contact. Numerical solution of this problem typically...

Soft-lubrication interactions between a rigid sphere and an elastic wall

The motion of an object within a viscous fluid and in the vicinity of a soft surface induces a hydrodynamic stress field that deforms the latter, thus modifying the boundary conditions of the flow. This results in elastohydrodynamic interactions experienced by the particle. Here, we derive a soft-lubrication model, in order to compute all the forces and torque applied on a rigid sphere that is free to translate and rotate near an elastic wall. We focus on the limit of small deformations of the surface with respect to the fluid-gap thickness, and perform a perturbation analysis in dimensionless compliance. The response is computed in the framework of linear elasticity, for planar elastic substrates in the limiting cases of thick and thin layers. The EHD forces are also obtained analytically using the Lorentz reciprocal theorem.

Nonlinear oscillations in a constant gravitational field

Exploring non-linear oscillations is a challenging task since the related differential equations cannot be directly solved in terms of elementary functions. Thus, complicated mathematical or numerical methods are usually employed to find accurate or approximate expressions that describe the behavior of the system with respect to time. In this paper, the vertical oscillations of an object under the influence of its weight and an opposite force with magnitude F=cyn, where n>0 are being explored. Accurate and approximate simple solutions regarding the object’s position with respect to time are presented and the dependence of the oscillation’s period from the oscillation’s range of displacements and the exponent n is revealed. In addition, the special case in which n=3/2 (which describes the oscillation of a rigid sphere on an elastic half space) is also highlighted. Lastly, it is shown that similar cases (such as the case of a force with magnitude F=kx+λx2) can be also treated using the same approach.

The Symmetry and Stability of the Flow Separation around a Sphere at Low and Moderate Reynolds Numbers

The flow separation state reflects the symmetry and stability of flow around spheres. The three-dimensional structures of flow around a rigid sphere at moderate Reynolds number (Re) between 20 and 400 by using finite volume method with adaptive mesh refinement are presented, and the process of separation angles changing from stable to oscillating state with increasing of Re is analyzed. The results show that the flow is steady, and the separation angles are stable and axisymmetric at Re in less than 200. The flow is unsteady and time-periodic, and the flow separation becomes regular fluctuations and asymmetric at Re = 300, which leads to the nonzero value of lateral force and the phase difference between lift and lateral force. At Re = 400, the flow is unsteady, non-periodic, and asymmetric, as is the flow separation. It’s concluded that the flow separation angle increases when Re increases within a range between 40 and 200. With Re continues to increase, the flow separation state changes from stable to periodically regular until quasi-periodically irregular. The vortex structure changes from no shedding to asymmetric periodic shedding, and finally to asymmetric and intermittently periodic vortex shedding. These results have important implications for the stability of flow around spheres.

Calculation and AFM Experimental Research on Slip Friction for Unlubricated Spherical Contact with Roughness Effect

Previous research on friction calculation models has mainly focused on static friction, whereas sliding friction calculation models are rarely reported. In this paper, a novel sliding friction model for realizing a dry spherical flat contact with a roughness effect at the micro/nano scale is proposed. This model yields the sliding friction by the change in the periodic substrate potential, adopts the basic assumptions of the Greenwood–Williamson random contact model about asperities, and assumes that the contact area between a rigid sphere and a nominal rough flat satisfies the condition of interfacial friction. It subsequently employs a statistical method to determine the total sliding friction force, and finally, the feasibility of this model presented is verified by atomic force microscopy friction experiments. The comparison results show that the deviations of the sliding friction force and coefficient between the theoretical calculated values and the experimental values are in a relatively acceptable range for the samples with a small plasticity index (Ψ ≤ 1).

Theoretical Modeling of Conformal Criterion for Flexible Electronics Attached onto Complex Surface

Transferring completed electronic devices onto curvilinear surfaces is popular for fabricating three-dimensional curvilinear electronics with high performance, while the problem of conformality between the unstretchable devices and the surfaces needs to be considered. Prior conformability design based on conformal mechanics model is a feasible way to reduce the non-conformal contact. Former studies mainly focused on stretchable film electronics conforming onto soft bio-tissue with a sinusoidal form microscopic morphology or unstretchable film conforming onto rigid sphere substrate, which limits its applicability in the aspect of shape and modulus of the substrate. Here, a conformal mechanics model with general geometric shape and material is introduced by choosing a bicurvature surface as the target surface, and the conformal contact behavior of film electronics is analyzed. All eight fundamental local surface features is obtained by adjusting two principal curvatures of the bicurvature surface, and the conformal performance is simulated. A dimensionless conformal criterion is given by minimizing the total energy as a function of seven dimensionless parameters, including four in geometric and three in material. The model and analysis results are verified by the finite element analysis, and it can provide a guidance for prior conformability design of the curvilinear electronic devices during the planar manufacturing process.

Elastic Scattering of Slow Electrons by Noble Gases—The Effective Range Theory and the Rigid Sphere Model

We report on an extensive semi-empirical analysis of scattering cross-sections for electron elastic collision with noble gases via the Markov Chain Monte Carlo-Modified Effective Range Theory (MCMC−MERT). In this approach, the contribution of the long-range polarization potential (∼r−4) to the scattering phase shifts is precisely expressed, while the effect of the complex short-range interaction is modeled by simple quadratic expression (the so-called effective range expansion with several adjustable parameters). Additionally, we test a simple potential model of a rigid sphere combined with r−4 interaction. Both models, the MERT and the rigid sphere are based on the analytical properties of Mathieu functions, i.e., the solutions of radial Schrödinger equation with pure polarization potential. However, in contrast to MERT, the rigid sphere model depends entirely upon one adjustable parameter—the radius of a hard-core. The model’s validity is assessed by a comparative study against numerous experimental cross-sections and theoretical phase shifts. We show that this simple approach can successfully describe the electron elastic collisions with helium and neon for energies below 1 eV. The purpose of the present analysis is to give insight into the relations between the parameters of both models (that translate into the cross-sections in the very low energy range) and some “macroscopic” features of atoms such as the polarizability and atomic “radii”.