Planar Free Surface Research Articles

Rayleigh-type wave in non-planar pre-stressed nearly incompressible elastic structure

The primary objective of the current work is to establish the impacts of planar boundary and non-planar boundary on the displacement components (DCs) of Rayleigh-type wave propagation in an elastic structure. The elastic structure, precisely, incorporates a pre-stressed nearly incompressible elastic material in the half-space along with the non-planar boundary at the free surface. The methodology applied features variable separable technique and Taylor’s series expansion along with substitution method to establish the expressions for DCs and frequency relation. The frequency relation of Rayleigh-type wave is obtained for the situation considering planar free surface, which is further reduced as a special case of the study and validated with the classical result present in the literature. The DCs of Rayleigh-type wave have been numerically computed, and their reliance on wave number have been exhibited graphically. In addition to this, the influences of affecting parameters viz., hydrostatic stress, principal Cauchy stress and curvature parameter on DCs have been traced out. The computational result also manifests the domain of existence of Rayleigh-type wave propagating in the considered model in the presence of planar free surface.

Surface instability of static liquid metal in magnetized fusion plasma

Understanding surface instability in magnetized fusion plasma supports the appropriate implementation and handling of liquid metal as plasma facing components (PFCs) in future fusion reactors. A Lagrange equation describing a viscous liquid surface deformation in a magnetized plasma is derived using Rayleigh’s method. Its solution justifies the general instability criterion and helps in characterizing the key interactions driving such instability under fusion conditions. Surface tension and gravity, especially with the poloidal angles of the lower part of a plasma chamber, mainly stabilize the liquid surface at small and large disturbance wavelengths, respectively. The sheath electric field and the external tangential magnetic field cause the liquid surface to disintegrate at an intermediate wavelength. Practically, a magnetic confinement fusion (MCF) device requires a strong magnetic field for confinement. The study suggests that such a strong field dominates the rest and governs instability. In addition, this implies that the configuration of a static planar free liquid surface is difficult to adopt as a candidate for handling the liquid metal as PFCs in next step MCF devices.

Coupling characteristics of bubbles with a free surface initially disturbed by water waves

The interactions between bubbles and water waves have important applications in ocean engineering, and their coupling characteristics are strongly associated with the wave phase angle, wavelength, and wave amplitude. Based on the assumption that the liquid is inviscid and incompressible, the coupling characteristics between bubbles and water waves are solved by the Euler equations with the finite volume method, and the bubble surface and water wave surface are tracked by the front tracking method. The accuracy of the numerical method is verified by comparison with a spark-generated bubble experiment. Compared with the bubble near the initially plane free surface, the rising height of the water spike is reduced by water waves in the crest state, where a concave shape forms on the falling water wave during bubble contraction when the wavelength λ≤ 4.00 and the wave amplitude h≥ 0.364. The rising height of the water spike is significantly strengthened by water waves in the trough state with smaller wavelengths and larger wave amplitudes, which produce a thinner and higher water spike. The bubble cycle is shortened by water waves in the crest state with smaller wavelengths and prolonged by water waves in the trough state with smaller wavelengths and larger wave amplitudes. The results presented in this paper provide guidance for the study of underwater explosions in complex water wave environments.

A holed-plate with material dislocations: Formulation and verification

Standard text-book dislocation theory for the elastic fields of a dislocation is provided in an infinite medium. For problems involving free surfaces, or image stresses, the solution for such elastic fields is more complex. The current article extends the state of knowledge for problems with free surfaces to address the stress field of a screw dislocation in a plate (two planar free surfaces) containing a hole/void. Such surfaces interact and thus affect the stress field of a dislocation in deviation of the infinite medium solution. The numerical solutions provided herein, based on reflective image stresses and the collocation-point method, have been verified to the extent available and such solutions allow for further applied problems. One main finding of the paper is that the equilibrium location of a screw dislocation is always closer to the hole/void surface than the planar free surface.

Exact solutions for the formation of stagnant caps of insoluble surfactant on a planar free surface

A class of exact solutions is presented describing the time evolution of insoluble surfactant to a stagnant cap equilibrium on the surface of deep water in the Stokes flow regime at zero capillary number and infinite surface Péclet number. This is done by demonstrating, in a two-dimensional model setting, the relevance of the forced complex Burgers equation to this problem when a linear equation of state relates the surface tension to the surfactant concentration. A complex-variable version of the method of characteristics can then be deployed to find an implicit representation of the general solution. A special class of initial conditions is considered for which the associated solutions can be given explicitly. The new exact solutions, which include both spreading and compactifying scenarios, provide analytical insight into the unsteady formation of stagnant caps of insoluble surfactant. It is also shown that first-order reaction kinetics modelling sublimation or evaporation of the insoluble surfactant to the upper gas phase can be incorporated into the framework; this leads to a forced complex Burgers equation with linear damping. Generalized exact solutions to the latter equation at infinite surface Péclet number are also found and used to study how reaction effects destroy the surfactant cap equilibrium.

The role of pre-existing defects in shock-generated ejecta in copper

The interaction of shock waves with non-planar free surfaces can cause atoms to eject from the surface, leading to the formation of ejecta. These non-planarities in the free surface can occur due to machining of the free surface or can be induced in the shock wave itself due to the presence of heterogeneities in the material. Both cases lead to the formation of ejecta. While the effect of machining on ejecta has been well studied, the latter has not been a topic of significant investigations. In this work, molecular dynamics simulations are used to systematically investigate the effect of size and concentration of He bubbles in Cu with planar free surfaces on ejecta production. It is shown that the presence of defects leads to the formation of non-planarity in the shock wave itself producing ejecta as the front reaches the flat free surface. The cluster size and velocity of ejected particles greatly exceeds that of pure Cu; the radius, density, and nature of the helium-filled voids alter the mass, velocity, and size distribution of the ejected matter.

The planar spread of a liquid jet and hydraulic jump on a porous layer

The flow of a planar liquid free surface jet impinging on a porous layer is theoretically examined, with particular emphasis on the influence of porosity ϕ, stress jump coefficient χ, and depth of the porous layer on the super- and sub-critical regions. Despite the numerous studies in the literature on the flow over a porous medium, the jet impingement on a porous layer has not been studied. An averaging integral approach is adopted to capture the flow in the developing boundary-layer and fully viscous regions upstream of the hydraulic jump. Asymptotic analyses for small distance from impingement, small porosity, and small porous layer depth are also conducted, elucidating the various mechanisms behind the behavior predicted numerically. We find a domain of validity for the stress jump coefficient χ in which numerical and experimental values of χ from the literature seem to fall. The transition point, where the outer edge of the boundary layer intersects the film surface, moves downstream with increasing porosity and stress jump coefficient accompanied by a drop in the film thickness. While the height of the hydraulic jump generally decreases with increasing ϕ for any permeability, the jump location decreases for small χ and increases for large χ.

Incompressible SPH simulation of flow past horizontal cylinder between plane wall and free surface

Flow past a circular cylinder located between free-surface and wall boundaries is modelled numerically using non-uniform particle size incompressible smoothed particle hydrodynamics (SPH). Several enhancements including Rhie and Chow interpolation, colour function-based free surface tracking, and modified particle shifting are introduced to increase the accuracy and efficiency of the method. A parameter study at constant Reynolds number, Re=150, examines the influence of Froude number (Fr=0.2−0.6), submergence ratio (H∕D=0.5−1.0, where H is the distance between the apex of the cylinder and undisturbed free surface, and D is the cylinder diameter), and bottom gap ratio (G∕D=1.0−5.0, where G is the distance between the base of the cylinder and the bed) on the ambient free-surface flow pattern and hydrodynamic force on the cylinder. It is found that the vortex shedding pattern depends on all three parameters. As the Froude number increases for fixed submergence ratio and bottom gap ratio, vortex shedding is eventually suppressed. As the submergence and bottom gap ratios increase, the threshold of vortex shedding suppression shifts to higher values of Froude number. The mean drag coefficient depends on the submergence ratio and bottom gap ratio but is independent of Froude number. Meanwhile, the lift coefficient depends on submergence ratio and Froude number but is independent of bottom gap ratio. Spectral analysis of force and free-surface elevation time signals shows that the free-surface deformation and lift force are closely related during the vortex shedding regime.

FD3D_TSN: A Fast and Simple Code for Dynamic Rupture Simulations with GPU Acceleration

AbstractWe introduce FD3D_TSN—an open-source Fortran code for 3D dynamic earthquake rupture modeling based on the staggered grid fourth-order finite-difference method employing a regular cubical spatial discretization. Slip-weakening and fast-velocity-weakening rate-and-state fault friction laws are combined with vertical planar fault geometry, orthogonal to a planar free surface. FD3D_TSN demonstrates good agreement with other methods in a range of benchmark exercises of the Southern California Earthquake Center and U.S. Geological Survey dynamic rupture code verification project. Efficient graphic processing units (GPU) acceleration using the OpenACC framework yields a factor of 10 speed-up in terms of time to solution compared to a single-core solution for current hardware (Intel i9-9900K and Nvidia RTX 2070). The software is fast and easy-to-use and suitable explicitly for data-driven applications requiring a large number of forward simulations such as dynamic source inversion or probabilistic ground-motion modeling. The code is freely available for the scientific community and may be incorporated in physics-based earthquake source imaging and seismic hazard assessment, or for teaching purposes.

Pressure Responsive Multi-Layered Composite Nanoparticles for through Silicon Via (TSV) Chemical Mechanical Planarization (CMP) Applications

Through silicon vias (TSV) have gained much attention due to the emergence of the vertical alignment necessary to produce 3D integrated circuit devices. Due to the large overburden caused by the electroplating process in TSV, high removal rate of copper (Cu) is essential for producing highly planar and defect free substrates. Therefore, focus has shifted to the development of processes and chemistries that produce highly planar and defect free Cu surfaces. In order to obtain this planarity, a process known as chemical mechanical planarization (CMP) is implemented in which a silica nanoparticle dispersion coupled with a mechanical force synergistically removes excess topography. The demands on TSV CMP have become stringent in order to meet the following criteria: very high removal rates, low dishing, high selectivity to nitride after barrier clear, low defectivity, and uniform surface topography. This work focuses on the development of a pressure responsive, multi-layer composite nanoparticle to provide a process driven synergy with the slurry formulation. These composites can be held together either covalently or non-covalently to the nanoparticle core using additives with amine, amino acid, and surfactant functionalities. The additives respond to an applied force and shear at the slurry-pad interface, resulting in controlled release of the slurry chemistry. Initial results show that at downforces ranging between 2.5-5 psi removal rates are at an excess of 1 micron per minute, while at below 1 psi removal rates are below 0.2 micron per minute and produce uniform surface quality at low surface roughness.

Дія радіаційної сили в звуковому полі на сферичну краплю в околі вільної поверхні ідеальної рідини

Acoustic radiation force effect on a liquid spherical drop placed in the vicinity of an ideal liquid free surface is studied. The problem of determination of the radiation forces acting on an obstacle in ideal liquid is formulated with respect to the Lagrange coordinate system. Thus, the radiation pressure is defined as time-averaged value of the acoustic pressure over the obstacle surface. This approach is adequate if, at determining of the acoustic pressure in a fluid, the deviation of the pressure from the harmonic law in time domain is taken into account in the obstacle vicinity. An action of the acoustic radiation force on a spherical drop of ideal liquid placed in turn in a liquid by its free plane surface is studied here for the case of the incident plane sound wave propagating perpendicularly to the liquid boundary. As a result, the liquid sphere is appeared to be located in the standing sound wave of pressure which has its displacement node at the free surface. Problem solution is obtained as a three step procedure. Initially, solution of the problem of an incident wave scattering at the drop is derived. With making use of the results obtained, the second step encompasses determining of hydrodynamic forces acting on the liquid spherical drop with their subsequent averaging over the suitable time interval at the third step. It is found there frequencies of the incident wave exist that provide zero radiation force acting on the drop which is immobile in this case. These equilibrium positions of the spherical drop could be stable or unstable with respect to the incident wave frequency variation.

Stokes velocity generated by a point force in various geometries

.In this short review, we visualize the fluid velocity generated by a point force close to a plane free surface or a plane rigid wall. We present separately contributions from all the multipoles which form the corresponding classical systems of images. Such graphical images might be useful in the theoretical and numerical modeling of the dynamics of micro-objects moving close to an interface.Graphical abstract

Time-domain full-waveform inversion of Rayleigh and Love waves in presence of free-surface topography

Correct estimation of near-surface seismic-wave velocity when encountering lateral heterogeneity and free surface topography is one of the challenges to current shallow seismic. We propose to use time-domain full-waveform inversion (FWI) of surface waves, including both Rayleigh and Love waves, to solve this problem. We adopt a 2D time-domain finite-difference method with an improved vacuum formulation (IVF) to simulate shallow-seismic Rayleigh wave in presence of free-surface topography. We modify the IVF for SH-wave equation for the simulation of Love wave in presence of topographic free surface and prove its accuracy by benchmark tests. Checkboard model tests are performed in both cases when free-surface topography is included or neglected in FWI. Synthetic model containing a dipping planar free surface and lateral heterogeneity was then tested, in both cases of considering and neglecting free-surface topography. Both checkerboard and synthetic models show that Rayleigh- and Love-wave FWI have similar ability of reconstructing near-surface structures when free-surface topography is considered, while Love-wave FWI could reconstruct near-surface structures better than Rayleigh-wave when free-surface topography is neglected.

Two-layer film flow on a rough rotating disk in the presence of air shear

The development of a two-layer thin film over a rough non-uniformly rotating disk with constant air shear is analyzed under the consideration of planar interface and free surface. von Karman’s similarity variables are applied to transform the guiding Navier–Stokes equations into a set of coupled unsteady nonlinear partial differential equations. These equations with moving boundary conditions are solved using the finite difference method. Here, it is found that the azimuthal roughness slows down the film thinning rate and the radial roughness enhances the thinning rate slightly. Also, the effect of air shear on film thinning is discussed. For different types of rotation, effects on the flow due to azimuthal roughness, radial roughness, and air shear remain the same.

The three-dimensional O(n) ϕ 4 model on a strip with free boundary conditions: Exact results for a nontrivial dimensional crossover in the limit n→∞

We briefly review recent results of exact calculations of critical Casimir forces of the O(n) ϕ4 model as n→∞ on a three-dimensional strip bounded by two planar free surfaces at a distance L. This model has long-range order below the critical temperature Tc of the bulk phase transition only in the limit L→∞ but remains disordered for all T > 0 for an arbitrary finite strip width L /< 0, and the low-temperature asymptotic behavior of the residual free energy and the Casimir force) by a combination of boundary-operator and short-distance expansions, proper extensions of the inverse scattering theory, new trace formulas, and semiclassical expansions.

Трехмерная $O(n)$-$\phi^4$-модель в плоско-параллельном слое со свободными граничными условиями: точные результаты для нетривиального размерного кроссовера в пределе $n\to\infty$

Дан краткий обзор недавних результатов точных расчетов критических сил Казимира в $O(n)$-$\phi^4$-модели при $n\to\infty$ для трехмерного слоя ширины $L$, ограниченного двумя параллельными плоскостями со свободными граничными условиями. В этой модели дальний порядок имеет место при температурах ниже критической точки $T_{\mathrm c}$ объемного фазового перехода только в пределе $L\to\infty$, однако дальний порядок в модели отсутствует при всех температурах $T&gt;0$ при произвольной конечной ширине слоя $L&lt;\infty$. Последовательное описание скейлингового поведения системы вблизи $T_{\mathrm c}$ оказывается весьма сложной задачей, так как наряду с объемным и поверхностным скейлингом, а также конечно-размерным скейлингом необходимо учитывать нетривиальный размерный кроссовер. Модель допускает точное решение в пределе $n\to\infty$ в терминах собственных значений и собственных функций самосогласованного уравнения Шредингера. Это решение содержит потенциал $v(z)$, который имеет при $z\to +0$ сингулярное поведение $v(z)\approx-1/(4z^2)+4m/(\pi^2z)$ в приграничном слое, где $m=1/\xi_+(|t|)$ - обратная корреляционная длина в объемной системе, $t\sim(T-T_{\mathrm c})/T_{\mathrm c}$, и аналогичная сингулярность имеет место вблизи второй граничной плоскости. В недавней работе нами совместно с соавторами численными методами с высокой точностью были вычислены потенциал $v(z)$, избыточная свободная энергия и сила Казимира. Дано объяснение того, как эти численные результаты могут быть дополнены точными аналитическими результатами для целого ряда величин (коэффициенты разложения потенциала $v(z)$, данные рассеяния для потенциала $v(z)$ в полубесконечной геометрии $L=\infty$ при всех значениях $m\gtreqless 0$, низкотемпературная асимптотика избыточной свободной энергии и силы Казимира). Эти аналитические результаты были получены сочетанием таких методов, как операторные разложения, обобщенная теория обратной задачи рассеяния, новая формула следов и квазиклассические (ВКБ) разложения.

Modeling study on the surface morphology evolution during removing the optics surface/subsurface damage using atmospheric pressure plasma processing

Plasma processing has been widely reported as an effective tool in relieving or removing surface/subsurface damage induced by previous mechanical machining process. However, the surface morphology evolution during removing the damage using plasma processing is rarely reported. In this research, this procedure is studied based on experiments and robust numerical models developed on the basis of Level Set Method (LSM). Even if some unique properties of plasma etching are observed, such as particle redistribution, the dominant role of isotropic etching of plasma processing is verified based on experiments and 2D LSM simulations. With 2D LSM models, the damage removal process under various damage characteristics is explored in detail. Corresponding peak-to-valley roughness evolution is investigated as well. Study on morphology evolution is also conducted through the comparison between experiments and 3D LSM computations. The modeling results and experiments show good agreement with each other. The trends of simulated roughness evolution agree with the experiments as well. It is revealed that the plasma processing may end up with a planar surface depending on the damage characteristics. The planarization procedure can be divided into four parts: crack opening and pit formation; pit coalescing and shallow pits subsumed by deep ones; morphology duplicate etching; and finally a planar and damage free surface.

Generation of a dipole of misfit dislocations in an axisymmetrical precipitate embedded in a semi-infinite matrix

The strain relaxation of an axisymmetrical precipitate embedded in a semi-infinite matrix with a planar free-surface has been theoretically investigated through the generation in the structure of a dislocation dipole. An energy variation calculation has been performed and the formation of the dipole composed of two misfit edge dislocations has been characterised in the interface. It is found that, contrary to the case where the inclusion is embedded in an infinite-size matrix, there exists near the free-surface a preferential plane selected for the dipole formation in the interface which corresponds to the plane perpendicular to the free surface and containing the inclusion centre. A stability diagram of the structure with respect to the dipole formation in the interface has been provided as a function of the misfit strain and the distance of the inclusion from the free-surface, for different values of the inclusion radius.

Inverse-scattering-theory approach to the exact n→∞ solutions of O(n) ϕ⁴ models on films and semi-infinite systems bounded by free surfaces.

The O(n) ϕ(4) model on a strip bounded by a pair of planar free surfaces at separation L can be solved exactly in the large-n limit in terms of the eigenvalues and eigenfunctions of a self-consistent one-dimensional Schrödinger equation. The scaling limit of a continuum version of this model is considered. It is shown that the self-consistent potential can be eliminated in favor of scattering data by means of appropriately extended methods of inverse scattering theory. The scattering data (Jost function) associated with the self-consistent potential are determined for the L=∞ semi-infinite case in the scaling regime for all values of the temperature scaling field t=(T-T(c))/T(c) above and below the bulk critical temperature T(c). These results are used in conjunction with semiclassical and boundary-operator expansions and a trace formula to derive exact analytical results for a number of quantities such as two-point functions, universal amplitudes of two excess surface quantities, the universal amplitude difference associated with the thermal singularity of the surface free energy, and potential coefficients. The asymptotic behaviors of the scaled eigenenergies and eigenfunctions of the self-consistent Schrödinger equation as function of x=t(L/ξ(+))(1/ν) are determined for x→-∞. In addition, the asymptotic x→-∞ forms of the universal finite-size scaling functions Θ(x) and ϑ(x) of the residual free energy and the Casimir force are computed exactly to order 1/x, including their x(-1)ln|x| anomalies.

An accuracy analysis of a Hamiltonian particle method with the staggered particles for seismic-wave modeling including surface topography

A Hamiltonian particle method (HPM), which is one of the mesh-free methods, can simulate seismic wavefields for models including surface topography in a simple manner. Numerical error caused by a curved free surface or by particles not aligned with the surface is not obvious in HPM. In general, the accommodation of irregular free surfaces requires more grids or particles in a minimum wavelength for achieving sufficient accuracy in the simulation. We tested the accuracy of HPM with staggered particles for simulating seismic-wave propagation including the surface topography, and we established the relationship between desired accuracy and spatial resolution. We conducted numerical simulations for models with a planar free surface aligned with the regular particle alignment and a dipping free surface. Our accuracy tests revealed that the numerical error strongly depends on the dipping angle of the slope. We concluded that about 25 particles in a minimum wavelength are required to calculate Rayleigh waves propagating along the irregular topography with good accuracy. Finally, we simulated Rayleigh wave propagation along irregular topography using a layered model with a hill. HPM can reproduce not only surface-wave propagation but also the reflected and refracted waves. Our numerical results were in good agreement with those from a finite-element method. Our investigations indicated that HPM could be a solution to simulate Rayleigh waves in the presence of complex surface topography.