Bodies Of Finite Size Research Articles

Global weak solutions to a time-periodic body-liquid interaction problem

We prove existence of time-periodic weak solutions to the coupled liquid-structure problem constituted by an incompressible Navier–Stokes fluid interacting with a rigid body of finite size, subject to an undamped linear restoring force. The fluid flow is generated by a uniform, time-periodic velocity field {\boldsymbol V} far from the body. We emphasize that our result is global, in the sense that no restriction is imposed on the magnitude of {\boldsymbol V} and, rather remarkably, the frequency of {\boldsymbol V} is entirely arbitrary. Thus, in particular, it can coincide with any multiple of a natural frequency of vibration of the body so that, with this model, resonance cannot occur. Although based on the classical “invading domains” technique, our approach requires several new ideas. Indeed, due to the lack of sufficient dissipation, it appears quite unfeasible to show the existence of a fixed point of the Poincaré map at the finite-dimensional level along the Galerkin approximant. Therefore, unlike the usual strategy, such a result must be proven directly in a class of weak solutions, and therefore in the infinite-dimensional framework.

Analysis of a diffusion-reaction heat transfer problem in a finite thickness layer adjoined by a semi-infinite medium

Multilayer diffusion-reaction problems are of much interest for both heat and mass transfer. While past work in this direction has mainly addressed multilayer bodies of finite size, practical problems such as immersion cooling of Li-ion cells and thermal runaway in semiconductor devices necessitate considering one of the layers to be semi-infinite. This work presents theoretical analysis of a diffusion-reaction problem in a finite layer surrounded by an infinite medium on both sides, where heat generation proportional to the local temperature occurs in the finite layer. Transient temperature distributions in the two bodies are determined using Laplace transformation technique. Through analysis of the poles of the solution in the Laplace domain, it is proved that this problem is unconditionally unstable, in that temperature in the finite thickness layer is predicted to always diverge at large times. However, the time taken to reach the unstable regime is shown to depend strongly on the key non-dimensional parameters of the problem. Temperature in the finite layer is shown to exhibit non-monotonic behavior at small times, particularly for large values of the heat generation coefficient, which is explained on the basis of the balance between temperature-dependent heat generation, heat dissipation into the semi-infinite medium and eventual slowdown of heat dissipation due to temperature rise. The impact of thermal properties on thermal behavior of the system is examined. A practical problem related to thermal safety design of a Li-ion cell is solved. Results from this work expand the state-of-the-art in theoretical analysis of diffusion-reaction problems, and also offer practical tools for thermal design of engineering problems including Li-ion cells and semiconductor devices.

The role of thermal effusivity in heat exchange between finite-sized bodies

Thermal effusivity is a thermophysical property of materials that combines thermal conductivity and volumetric heat capacity. Thermal effusivity is often interpreted in terms of a body's ability to exchange heat when brought in contact with another body at a different temperature, and is relevant in thermal property measurements, such as the transient plane source method as well as in energy harvesting and thermal management. Well-known theoretical models show that thermal effusivity is the only thermophysical property that governs heat exchange between two semi-infinite bodies. In contrast, there is a lack of work on heat exchange between bodies of finite thickness, which may be a relevant consideration in several practical scenarios. This work presents an analytical solution for heat exchange between two finite bodies initially at different temperatures. The effect of thermal contact resistance is accounted for. Results from this work are shown to approach semi-infinite results when the layer thickness becomes large. It is shown that while the heat flux for finite thickness in general depends on both thermal effusivity and thermal diffusivity, exclusive dependence on effusivity may occur under certain conditions, particularly for poorly conducting materials. The sensitivity of interfacial heat flux on these thermal properties is examined in the regimes in which most materials of interest lie. The limits in which the semi-infinite approximation is reasonable are determined. A practical problem is solved as an illustration. This work improves the fundamental understanding of thermal effusivity and thermal interaction between finite-sized bodies, and may find applications in a variety of engineering problems such as thermal property measurement, energy harvesting and thermal management.

Thermal equilibrium spin torque: Near-field radiative angular momentum transfer in magneto-optical media

Spin and orbital angular momentum of light plays a central role in quantum nanophotonics as well as topological electrodynamics. Here, we show that the thermal radiation from finite-sized bodies comprising of nonreciprocal magneto-optical materials can exert a spin torque even in global thermal equilibrium. Moving beyond the paradigm of near-field heat transfer, we calculate near-field radiative angular momentum transfer between finite-sized nonreciprocal objects by combining Rytov's fluctuational electrodynamics with the theory of optical angular momentum. We prove that a single magneto-optical cubic particle in non-equilibrium with its surroundings experiences a torque in the presence of an applied magnetic field (T-symmetry breaking). Furthermore, even in global thermal equilibrium, two particles with misaligned gyrotropic axes experience equal magnitude torques with opposite signs which tend to align their gyrotropic axes parallel to each other. Our results are universally applicable to semiconductors like InSb (magneto-plasmas) as well as Weyl semi-metals which exhibit the anomalous Hall effect (gyrotropy) at infrared frequencies. Our work paves the way towards near-field angular momentum transfer mediated by thermal fluctuations for nanoscale devices.

Confinement Enhances the Diversity of Microbial Flow Fields.

Despite their importance in many biological, ecological, and physical processes, microorganismal fluid flows under tight confinement have not been investigated experimentally. Strong screening of Stokelets in this geometry suggests that the flow fields of different microorganisms should be universally dominated by the 2D source dipole from the swimmer's finite-size body. Confinement therefore is poised to collapse differences across microorganisms, which are instead well established in bulk. We combine experiments and theoretical modeling to show that, in general, this is not correct. Our results demonstrate that potentially minute details like microswimmer spinning and the physical arrangement of the propulsion appendages have in fact a leading role in setting qualitative topological properties of the hydrodynamic flow fields of microswimmers under confinement. This is well captured by an effective 2D model, even under relatively weak confinement. These results imply that active confined hydrodynamics is much richer than in bulk and depends in a subtle manner on the size, shape, and propulsion mechanisms of the active components.

Heat source identification of some parabolic equations based on the method of fundamental solutions

In this paper, we consider inverse problems of the heat conduction process in one and two-dimensional homogeneous bodies of finite size, subject to the given initial and boundary conditions. The considered inverse problems are severely ill-posed since their solutions; if they exist, they do not depend continuously on the input data. We obtain stable solutions for several inverse problems by proposing a meshless regularization technique based on the combination of the method of fundamental solutions and the Tikhonov’s regularization method. In particular, we use the given information at the terminal state to estimate the space-dependent heat source in the one-dimensional case and the space- and time-dependent heat source in the two-dimensional case. Numerical results demonstrate high accuracy and low computational cost.

Charged particle in a constant electric field: force on a parallel plate charged capacitor

We analyze the problem about the motion of a classical charged particle with negligible radiation losses inside a parallel plate charged capacitor and determine mutual forces acting between the charge and insulating capacitor plates in the rest frame of the capacitor and in the rest frame of the charge, correspondingly. In the latter case, the force on the capacitor’s plates additionally contains a component associated with the flow of energy of mechanical stresses in the plates. We show that the misbalance of forces of action and reaction in this system is compensated by the time variation of momentum of the interactional electromagnetic field created by the capacitor and charge, which ensures the conservation of total momentum. The results obtained are helpful for better understanding the interaction of point-like charge with a charged body of finite size.

Light Emission by Nonequilibrium Bodies: Local Kirchhoff Law

The goal of this paper is to introduce a local form of Kirchhoff law to model light emission by nonequilibrium bodies. While absorption by a finite-size body is usually described using the absorption cross section, we introduce a local absorption rate per unit volume and also a local thermal emission rate per unit volume. Their equality is a local form of Kirchhoff law. We revisit the derivation of this equality and extend it to situations with subsystems in local thermodynamic equilibrium but not in equilibrium between them, such as hot electrons in a metal or electrons with different Fermi levels in the conduction band and in the valence band of a semiconductor. This form of Kirchhoff law can be used to model (i) thermal emission by nonisothermal finite-size bodies, (ii) thermal emission by bodies with carriers at different temperatures, and (iii) spontaneous emission by semiconductors under optical (photoluminescence) or electrical pumping (electroluminescence). Finally, we show that the reciprocity relation connecting light-emitting diodes and photovoltaic cells derived by Rau is a particular case of the local Kirchhoff law.Received 21 December 2017DOI:https://doi.org/10.1103/PhysRevX.8.021008Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasElectroluminescenceNanoantennasPhotovoltaic absorbersSpontaneous emissionPhysical SystemsOptical microcavitiesTechniquesPhotoluminescenceCondensed Matter, Materials & Applied Physics

Generalized Knudsen Number for Unsteady Fluid Flow.

We explore the scaling behavior of an unsteady flow that is generated by an oscillating body of finite size in a gas. If the gas is gradually rarefied, the Navier-Stokes equations begin to fail and a kinetic description of the flow becomes more appropriate. The failure of the Navier-Stokes equations can be thought to take place via two different physical mechanisms: either the continuum hypothesis breaks down as a result of a finite size effect or local equilibrium is violated due to the high rate of strain. By independently tuning the relevant linear dimension and the frequency of the oscillating body, we can experimentally observe these two different physical mechanisms. All the experimental data, however, can be collapsed using a single dimensionless scaling parameter that combines the relevant linear dimension and the frequency of the body. This proposed Knudsen number for an unsteady flow is rooted in a fundamental symmetry principle, namely, Galilean invariance.

Application of the Microstructural Finite Element Alternating Method to assess the impact of specimen size and distributions of contact/residual stress fields on fatigue strength

This paper shows an implementation of the Microstructural Finite Element Alternating Method (MFEAM) to calculate fatigue strength of engineering components subjected to complex loading conditions. These conditions include combination of primary cyclic loading and residual stress fields or localised stress distributions due to the contact of two solids. The alternating method uses the finite element method (FEM) to solve the boundary value problem of the un-cracked body, allowing treatment of arbitrary shaped components, in conjunction with a short crack model to account for the interaction of the crack with the microstructural barriers, implemented within the distributed dislocation technique (DDT) to assess the crack problem in an infinite medium. An iterative scheme between the FEM and the DDT solutions is proposed in order to predict the required applied load to propagate the crack in the finite size body. Comparisons of results with reported data in the literature show that the MFEAM can be used effectively to analyse finite size components and that it is capable of capturing the effect of localised stress concentration features on cyclic behaviour.

A numerical study of super-resolution through fast 3D wideband algorithm for scattering in highly-heterogeneous media

We present a wideband fast algorithm capable of accurately computing the full numerical solution of the problem of acoustic scattering of waves by multiple finite-sized bodies such as spherical scatterers in three dimensions. By full solution, we mean that no assumption (e.g. Rayleigh scattering, geometrical optics, weak scattering, Born single scattering, etc.) is necessary regarding the properties of the scatterers, their distribution or the background medium. The algorithm is also fast in the sense that it scales linearly with the number of unknowns. We use this algorithm to study the phenomenon of super-resolution in time-reversal refocusing in highly-scattering media recently observed experimentally (Lemoult et al., 2011), and provide numerical arguments towards the fact that such a phenomenon can be explained through a homogenization theory.

Movement of a finite body in channel flow.

A body of finite size is moving freely inside, and interacting with, a channel flow. The description of this unsteady interaction for a comparatively dense thin body moving slowly relative to flow at medium-to-high Reynolds number shows that an inviscid core problem with vorticity determines much, but not all, of the dominant response. It is found that the lift induced on a body of length comparable to the channel width leads to differences in flow direction upstream and downstream on the body scale which are smoothed out axially over a longer viscous length scale; the latter directly affects the change in flow directions. The change is such that in any symmetric incident flow the ratio of slopes is found to be [Formula: see text], i.e. approximately 0.900969, independently of Reynolds number, wall shear stresses and velocity profile. The two axial scales determine the evolution of the body and the flow, always yielding instability. This unusual evolution and linear or nonlinear instability mechanism arise outside the conventional range of flow instability and are influenced substantially by the lateral positioning, length and axial velocity of the body.

Modeling Sliding Contact Temperatures, Including Effects of Surface Roughness and Convection

The ability to predict contact surface temperatures in rolling/sliding contacting bodies is important if failure of tribological components is to be avoided. Many works on surface temperature analysis and prediction have been published over the past 75 years or so, but most of the analytical solutions that are readily available do not include such important factors as finite body geometry, surface roughness, or convective cooling. Approaches for addressing these deficiencies are presented in this paper. This paper builds on previous analytical work by the authors and others, and presents models that are based on experimental observations of contact temperatures and factors that affect them. It is shown that the total surface temperature rise above ambient temperature is the sum of nominal temperature rise and flash temperature rise. Models are developed for calculating nominal surface temperature rise for sliding bodies of finite size, including effects of both convection and conduction. Flash temperature models are developed for both single and multiple contacts, as would be found with rough surfaces. Methods are presented that are valid for a variety of geometries and kinematic operating conditions, and techniques are also presented for partitioning the frictional heat between the two contacting surfaces. Examples of the use of the methodology are presented, along with experimental verification of the predictions.

The force function of two rigid celestial bodies in Delaunay–Andoyer variables

Two new expansions of the force function of two rigid celestial bodies of finite size and arbitrary shape are obtained in Delaunay–Andoyer variables with any degree of accuracy, in the form of a partial sum of an eight dimensional Fourier series. These expansions of the force function contain products of expressions for the momenta and Stokes constants in terms of sines and cosines, whose arguments are linear combinations of the Delaunay and Andoyer angular variables. These representations of the force function are compact and convenient for applications in various problems in celestial mechanics and astrodynamics.

Calculation of flexoelectric deformations of finite-size bodies

The previously developed approximate theory of flexoelectric deformations of finite-size bodies has been considered as applied to three special cases: a uniformly polarized ball, a uniformly polarized circular rod, and a uniformly polarized thin circular plate of an isotropic material. For these cases simple algebraic formulas have been derived. In the case of the ball, the solution is compared with the previously obtained exact solution.

Generalized Three-body Problem and the Instability of the Core-halo Objects in Binary Systems

Goal of the presented research is to construct simplified model of the core-halo structures in binary systems. Examples are provided by Thorne-Zytkov objects, hot Jupiters, protoplanets with large moons, red supergiants in binaries and globular clusters with central black hole. Instability criteria due to resonance between internal and orbital frequencies in such a systems has been derived. To achieve assumed goals, generalized planar circular restricted three body problem is investigated with one of the point masses, $M$, replaced with spherical body of finite size. Mechanical system under consideration includes two large masses $m$ and $M$ and the test body with small mass $\mu$. Only gravitational interactions are considered. Equations of motion are presented, and linear instability criteria are derived using quantifier elimination. Motion of the test mass $\mu$ is shown to be unstable due to resonance between orbital and internal frequencies if $\frac{M}{d^3} < \frac{4}{3} \pi \rho < \frac{ M + 3 m \left( 1+\mu/M \right)^{-1}}{d^3}$, where $\rho$ is the central density of mass $M$, and $d$ distance between masses $m$ and $M$ (circular orbit diameter). The above result is important for core-collapse supernova theory, with mass $\mu$ identified with helium core of the exploding massive star. The instability cause off-center supernova "ignition" relative to the center-of-mass of the hydrogen envelope. The instability is also inevitable during protoplanet growth, with hypothetical ejection of the rocky core from gas giants and formation of the "puffy planets" due to resonance with orbital frequency. Hypothetical central intermediate black holes of the globular clusters are also in unstable position with respect to perturbations caused by the Galaxy.

Modeling of pre-fracture zones for limited permeable crack in piezoelectric materials

A plane-strain problem for a limited permeable crack in an adhesive thin interlayer between two semi-infinite piezoelectric spaces is considered. The tensile mechanical stress and the electric displacement are applied at infinity. The interlayer is assumed to be softer than the connected materials; therefore, the zones of mechanical yielding and electric saturations can arise at the crack tips on the continuations of the crack. These zones are considered in this work. It was assumed that the length of electric saturation zones is larger than the length of mechanical yield zones. The zones of mechanical yielding are modeled by the crack continuations with normal compressive stresses applied at its faces. The electric saturation zones are modeled by segments at the crack continuations with prescribed saturated electric displacements. These electric displacements can linearly vary along the mechanical yielding zones. The problem is reduced to the Hilbert–Riemann problem of linear relationship, which is solved exactly. The equation for the determination of the yielding zones length, the expressions for the crack-opening displacement jump, electric potential jump, and J-integral is obtained in an analytical form. In case of finite size body, the finite elements method is used and the variation in the fracture mechanical parameters with respect to this size is demonstrated.

On the flexoelectric deformations of finite size bodies

Exact equations describing flexoelectric deformation in solids, derived previously within the framework of a continuum media theory, are partial differential equations of the fourth order. They are too complex to be used in the cases interesting for applications. In this paper, using the fact of smallness of the elastic moduli of a higher order, simplified equations are proposed. Solution of the exact equations is approximately represented as a sum of two parts: the first part obeys one-dimensional differential equations and exponentially decays near surface and the second part satisfies the equations of classical theory of elasticity. The first part can be constructed in an explicit form. For the second part, boundary conditions are obtained. They have a form of the classical boundary conditions for the body under external forces on surface.

Vortex motion on surfaces of small curvature

We consider a single Abelian Higgs vortex on a surface Σ whose Gaussian curvature K is small relative to the size of the vortex, and analyse vortex motion by using geodesics on the moduli space of static solutions. The moduli space is Σ with a modified metric, and we propose that this metric has a universal expansion, in terms of K and its derivatives, around the initial metric on Σ. Using an integral expression for the Kähler potential on the moduli space, we calculate the leading coefficients of this expansion numerically, and find some evidence for their universality. The expansion agrees to first order with the metric resulting from the Ricci flow starting from the initial metric on Σ, but differs at higher order. We compare the vortex motion with the motion of a point particle along geodesics of Σ. Relative to a particle geodesic, the vortex experiences an additional force, which to leading order is proportional to the gradient of K. This force is analogous to the self-force on bodies of finite size that occurs in gravitational motion.

The stability in a strict non-linear sense of a trivial relative equilibrium position in the classical and generalized versions of Sitnikov's problem

Stability, in a strict non-linear sense, of a trivial relative equilibrium position is investigated in the classical and generalized versions of Sitnikov's problem in the case of small eccentricities of the orbits of bodies of finite dimensions. In the classical version (n=2) of the problem, it is proved that there are no second-, third- and fourth-order resonances and a degenerate case. In the generalized version (2<n≤5·105), it is proved that there are no second- and third-order resonances and a degenerate case. A fourth-order resonance occurs in versions of the problem in which the number of finite size bodies satisfies the inequality 45000≤n≤62597 and the orbital eccentricities e < 0.25. Use of the Arnold–Moser and Markeyev theorems enables one to establish the Lyapunov stability of the trivial positions of relative equilibrium in the above-mentioned versions of Sitnikov's problem.