Neumann's Law Research Articles

Simply solvable model capturing the approach to statistical self-similarity for the diffusive coarsening of bubbles, droplets, and grains.

Aqueous foams and a wide range of related systems are believed to coarsen by diffusion between neighboring domains into a statistically self-similar scaling state, after the decay of initial transients, such that dimensionless domain size and shape distributions become time independent and the average grows as a power law. Partial integrodifferential equationsfor the time evolution of the size distribution for such phase separating systems can be formulated for arbitrary initial conditions, but these are cumbersome for analyzing data on nonscaling state preparations. Here we show that essential features of the approach to the scaling state are captured by an exactly-solvable ordinary differential equationfor the evolution of the average bubble size. The key ingredient is to characterize the bubble size distribution approximately, using the average size of all bubbles and the average size of the critical bubbles, which instantaneously neither grow nor shrink. The difference between these two averages serves as a proxy for the width of the size distribution. Solution of our model shows that behavior is controlled by a signed length δ that is proportional to the width of the initial distribution relative to that in the scaling state. In particular, δ is negative if the initial preparation is too monodisperse, and is positive if it is too polydisperse. To test our approach, we compare with data for quasi-two dimensional dry foams created with three different initial amounts of polydispersity. This allows us to readily identify the critical radius from the average area of six-sided bubbles, whose growth rate is zero by the von Neumann law. The growth of the average and critical radii agree quite well with exact solution, though the most monodisperse sample crosses over to the scaling state faster than expected. A simpler approximate solution of our model performs equally well. Our approach is applicable to 3d foams, which we demonstrate by re-analyzing prior data, as well as to froths of dilute droplets and to phase separation kinetics for more general systems such as emulsions, binary mixtures, and alloys.

The influence exerted by active cooling on the permanent connection formed as a result of electric arc welding

The problem of temperature stresses is solved in the work. The process of electric arc welding with guaranteed penetration of steel plates of various thicknesses is simulated. The possibility of reducing residual stresses formed as a result of welding by applying heat removal from the surface of the near-weld region of a steel plate is being investigated. The temperature distribution is determined by solving a non-linear heat equation, in which the specific heat and thermal conductivity depend on temperature. The heat source is modeled using the double ellipsoid method proposed by John A. Goldak. The horseshoe-shaped heat sink follows immediately after the anode spot and is set by the heat flux in such a way that the main temperature of the plate does not decrease below its initial temperature. This limitation reduces the level of the temperature gradient, thereby reducing the stresses in the metal plate. To illustrate the effect of active heat sink, the temperature distribution in the top layer at the moment of termination of welding is compared with and without active heat sink. The material is assumed to be elastoviscoplastic, the deformations are small and consist of reversible and irreversible ones. Reversible deformations are related to stresses by the Duhamel - Neumann law. Irreversible strains increase when the von Mises plastic flow condition is satisfied, in which there is a plastic strain rate component responsible for the plastic flow viscosity. The elastic moduli (Young's modulus, Poisson's ratio) and the yield strength are assumed to be temperature dependent. The solution of the mechanical part of the problem is found by the method of simple iterations. For plates with different thicknesses, diagrams of residual stresses are given, located in the center of the plate from the seam to the periphery. As a result of the work, according to the obtained distributions of residual stresses in the plate material, it is concluded that the use of an active horseshoe-shaped heat sink from the near-weld zone when welding thin steel plates reduces residual stresses, therefore it is recommended to use in the welding process. The use of an active heat sink on the reverse side of the plate leads to an increase in the stress level, therefore it is not recommended for use.

Average evolution and size-topology relations for coarsening 2d dry foams

Two-dimensional dry foams coarsen according to the von Neumann law as dA/dt ∝ (n − 6) where n is the number of sides of a bubble with area A. Such foams reach a self-similar scaling state where area and side-number distributions are stationary. Combining self-similarity with the von Neumann law, we derive time derivatives of moments of the bubble area distribution and a relation connecting area moments with averages of the side-number distribution that are weighted by powers of bubble area. To test these predictions, we collect and analyze high precision image data for a large number of bubbles squashed between parallel acrylic plates and allowed to coarsen into the self-similar scaling state. We find good agreement for moments ranging from 2–20.

Experimentally testing a generalized coarsening model for individual bubbles in quasi-two-dimensional wet foams.

We present high-precision data for the time evolution of bubble area A(t) and circularity shape parameter C(t) for several bubbles in a quasi-two-dimensional foams consisting of bubbles squashed between parallel plates. In order to fully compare with earlier predictions, foam wetness is systematically varied by controlling the height of the sample above a liquid reservoir which in turn controls the radius r of the inflation of the Plateau borders. For very dry foams, where the borders are very small, classic von Neumann behavior is observed where a bubble's growth rate depends only on its number n of sides. For wet foams, the inflated borders impede gas exchange and cause deviations from von Neumann's law that are found to be in accord with the generalized coarsening equation. In particular, the overall growth rate varies linearly with the film height, which decrease as surface Plateau borders inflate. More interestingly, the deviation from dA/dt∝(n-6) von Neumann behavior grows in proportion to nCr/sqrt[A]. This is highlighted definitively by data for six-sided bubbles, which are forbidden to grow or shrink except for the existence of this term. It is tested quantitatively by variation of all four relevant quantities: n, C, r, and A.

Static interface profiles for contact lines on an elastic membrane with the Willmore energy.

We consider a fluid interface in contact with an elastic membrane and study the static profiles of the interface and the membrane. Equilibrium conditions are derived by minimizing the total energy of the system with volume constraints. The total energy consists of surface energies and the Willmore energy; the latter penalizes the bending of the membrane. It is found that, while the membrane is locally flat at the contact line with the contact angle satisfying the Young-Dupré equation, the gradient of the mean curvature of the membrane exhibits a jump across the contact line. This jump balances the surface tension of the fluid interface in the normal direction of the membrane. Asymptotic solutions are obtained for two-dimensional systems in the limits as the reduced bending modulus ν tends to +∞ and 0, respectively. In the stiff limit as ν→+∞, the leading-order solution is given by that of a droplet sitting on a rigid substrate with the contact angle satisfying the Young-Dupré equation; in contrast, in the soft limit as ν→0, a transition layer appears near the contact line and the interfaces have constant curvatures in the outer region with apparent contact angles obeying Neumann's law. These solutions are validated by numerical experiments.

Singular Nature of the Elastocapillary Ridge

The functionality of soft interfaces is crucial to many applications in biology and surface science. Recent studies have used liquid drops to probe the surface mechanics of elastomeric networks. Experiments suggest an intricate surface elasticity, also known as the Shuttleworth effect, where surface tension is not constant but depends on substrate deformation. However, interpretations have remained controversial due to singular elastic deformations, induced exactly at the point where the droplet pulls the network. Here we reveal the nature of the elastocapillary singularity on a hyperelastic substrate with various constitutive relations for the interfacial energy. First, we finely resolve the vicinity of the singularity using goal-adaptive finite element simulations. This confirms the universal validity, also at large elastic deformations, of the previously disputed Neumann's law for the contact angles. Subsequently, we derive exact solutions of nonlinear elasticity that describe the singularity analytically. These solutions are in perfect agreement with numerics, and show that the stretch at the contact line, as previously measured experimentally, consistently points to a strong Shuttleworth effect. Finally, using Noether's theorem we provide a quantitative link between wetting hysteresis and Eshelby-like forces, and thereby offer a complete framework for soft wetting in the presence of the Shuttleworth effect.

Beyond modified mean field: a case for a stochastic grain growth model

ABSTRACTWe address the basic mechanism of grain growth in two and three dimensions in single phase metallic systems. It is argued, a comprehensive grain growth model is best based on a stochastic process described by a Fokker Planck equation. This concept is sufficient to determine almost all features of grain growth in general terms. The existence of the von Neumann law and its counterpart in three dimensions enables us to provide an explicit form of the Fokker Planck equation and thus determine many features of grain growth in great detail. Finally, we illustrate this model by determining the salient features of grain growth in three dimensions in the long time limit, and comparing them with experimental and computational results.

Spreading on viscoelastic solids: are contact angles selected by Neumann's law?

The spreading of liquid drops on soft substrates is extremely slow, owing to strong viscoelastic dissipation inside the solid. A detailed understanding of the spreading dynamics has remained elusive, partly owing to the difficulty in quantifying the strong viscoelastic deformations below the contact line that determine the shape of moving wetting ridges. Here we present direct experimental visualisations of the dynamic wetting ridge using shadowgraphic imaging, complemented with measurements of the liquid contact angle. It is observed that the wetting ridge exhibits a rotation that follows exactly the dynamic liquid contact angle - as was previously hypothesized [Karpitschka et al., Nat. Commun., 2015, 6, 7891]. This experimentally proves that, despite the contact line motion, the wetting ridge is still governed by Neumann's law. Furthermore, our experiments suggest that moving contact lines lead to a variable surface tension of the substrate. We therefore set up a new theory that incorporates the influence of surface strain, for the first time including the so-called Shuttleworth effect into the dynamical theory for soft wetting. It includes a detailed analysis of the boundary conditions at the contact line, complemented by a dissipation analysis, which shows, again, the validity of Neumann's balance.

Modal analysis of thin parabolic shells

The modal analysis of parabolic shells of revolution is based on using the finite-element model of inhomogeneous shell. The shells can have complex-shaped midsurface, geometrical features throughout the thickness, or multilayer structure. To develop the finite-element shell model we approximate a thin shell by one spatial finite element throughout the thickness. The structural elements of an inhomogeneous shell require the finite element to be universal: it should be eccentrically arranged relative to the mid-surfaces of the casing, it should be possible to vary the thickness of the lateral edges of the finite element and ets. The universal finite element is based on an isoparametric spatial finite element with polylinear shape functions for coordinates and displacements. Additional variable parameters are introduced to enhance the capabilities of the modified finite element. Two hypotheses are used to describe the features of the stress–strain state of a thin inhomogeneous shell. The static hypothesis assumes that the compressive stresses in the fibers throughout the thickness are constant. The nonclassical kinematic hypothesis of deformed straight line is used: a straight segment along the thickness remains straight though stretched or shortened during deformation. This segment is not necessarily normal to the mid-surface of the shell.The stress–strain state of a shell and its structural elements is determined using the geometrically nonlinear equations of the three-dimensional theory of thermoelasticity. A linear elastic continuous medium with large displacements and small strains is used as a model whose properties correspond to the generalized Duhamel–Neumann law. To derive the governing finite-element equations for displacements the moment finite-element scheme is used. The moment finite-element scheme approximations of displacements and strains guarantee a correct description of the rigid-body displacements of finite elements, which enhances the convergence and accuracy of solutions on coarse meshes.The natural vibrations of parabolic shells with various heights have been investigated. The convergence of solutions has been studied and compared with the results obtained by other authors.During operation, realistic shell structures often undergo various changes in the temperature field. This can significantly affect their dynamic characteristics. Extension of this work to modal analysis of parabolic shells considering heating is currently being pursued.

Size-dependent non-linear dynamics of curvilinear flexible beams in a temperature field

• Dynamical stability loss of a curvilinear size-dependent beam is studied. • A combination of the couple stress theory and the Euler–Bernoulli and Duhamel–Neumann hypotheses is proposed. • Reliability of solution to non-linear thermomechanical PDEs is validated. • Numerous engineering-oriented problems are investigated. A mathematical model of the loss of dynamic stability of curvilinear size-dependent MEMS and NEMS elements embedded in a temperature field and subjected to large deflections was derived and studied. The fundamental governing dynamical equations of MEMS/NEMS members were yielded by Hamilton's principle. The investigations were based on combining the modified couple stress theory, the first-order approximation kinematic (Euler–Bernoulli) model, the von Kármán geometric non-linearity, and the Duhamel–Neumann law regarding the temperature input (the beam material is elastic, isotropic and there are no constraints imposed on the temperature distribution). The temperature field was defined by solving a heat transfer equation. The computational algorithm was based on the finite difference method and the Runge–Kutta method. The numerical methods were validated by estimating the temporal and spatial convergence and reliability of the obtained solution was validated with the Lyapunov exponents obtained by qualitatively different methods. A few case studies related to the loss of stability, the magnitude of the size-dependent parameter, the type and intensity of the temperature input, and the parameters of uniformly distributed transverse load were investigated.

Border-crossing model for the diffusive coarsening of two-dimensional and quasi-two-dimensional wet foams.

For dry foams, the transport of gas from small high-pressure bubbles to large low-pressure bubbles is dominated by diffusion across the thin soap films separating neighboring bubbles. For wetter foams, the film areas become smaller as the Plateau borders and vertices inflate with liquid. So-called "border-blocking" models can explain some features of wet-foam coarsening based on the presumption that the inflated borders totally block the gas flux; however, this approximation dramatically fails in the wet or unjamming limit where the bubbles become close-packed spheres and coarsening proceeds even though there are no films. Here, we account for the ever-present border-crossing flux by a new length scale defined by the average gradient of gas concentration inside the borders. We compute that it is proportional to the geometric average of film and border thicknesses, and we verify this scaling by numerical solution of the diffusion equation. We similarly consider transport across inflated vertices and surface Plateau borders in quasi-two-dimensional foams. And we show how the dA/dt=K_{0}(n-6) von Neumann law is modified by the appearance of terms that depend on bubble size and shape as well as the concentration gradient length scales. Finally, we use the modified von Neumann law to compute the growth rate of the average bubble area, which is not constant.

Heat propagation in a nonuniform rod of variable cross section

An integral formula is used to average a coupled problem of thermoelasticity for a nonuniform rod of variable cross section. Effective characteristics are found. It is shown that, in addition to the expected effective coefficients, there appear five independent coefficients characterizing the temperature change rate effect on the stresses in the rod, on the longitudinal heat flux, and on the entropy distribution along the length of the rod. A feature of these new coefficients is that they become equal to zero in the case of a uniform rod. The homogenization of the thermoelasticity equations for nonuniform rods allows one to propose a new theory of heat conduction in rods. This new theory differs from the classical one by the fact that some new terms are added to the Duhamel–Neumann law, to the Fourier heat conduction law, and to the entropy expression. These new terms are proportional to the temperature change rate with time. It is also shown that, in the new theory of heat conduction, the propagation velocity of harmonic heat perturbations is dependent on the oscillation frequency and is finite when the frequency tends to infinity.

Elasto-electro-capillarity: drop equilibrium on a charged, elastic solid.

We present a study here on elasto-electro-capillarity - for the first time, the matter of drop equilibrium on a soft (elastic and incompressible) and charged solid has been studied. Charges on the elastic solid induce an electric double layer or EDL at the solid-drop interface. Our analysis accounts for the electrostatic wetting contribution of the EDL in the overall energy balance. Our results reveal that (a) with an increase in "softness", the equilibrium solid-liquid contact angles show transition from the EDL-modified Young's law (rigid limit) to the EDL-modified Neumann's law (soft limit); (b) the EDL effects invariably enhance solid deformation and lower the apparent contact angle made by the drop with the undeformed solid; (c) the solid contact angles increase and the cusp made by the deformed solid undergoes enhanced rotation due to the EDL effects; and (d) the EDL effects are more prominent for the case where the solid-vapor surface energy exceeds the solid-liquid surface energy. The fact that the EDL effects invariably decrease the overall wetting energy of the system, thereby supporting a larger increase in the elastic energy associated with a larger solid deformation, explains all these findings and establishes that surface charges enhance the "softness" of a soft surface in the context of elastocapillarity.

Extended Use of Coulomb’s Law in Relation to Established Laws within Electromagnetism

The competence of the basically electrostatic Coulomb's Law has historically been assumed to be restricted to pure electrostatics. As soon as electric charges were studied in motion, new sets of laws were introduced to explain the electromagnetic forces that are impelled by the motion. Among these new laws are Neumann's law of induction, Grassmann's force law, Lorentz' force law, and Ampere's force law. Furthermore, the difficulties in explaining the nature of light have given rise to the so-called wave-particle paradox. 1997 was the first year of the public circulation of research results that succeeded in showing that the basic force behind cases involving electricity or, more precisely, electromagnetism, can be derived from electrostatics. The aim of this article is to unify the results of existing theoretical research that discusses problems inherent within the prevailing standpoint on electromagnetism. In this paper, the conflict between the Lorentz Force Law and Ampere's Law is explored. Simultaneously, an alternative based strictly on electrostatics is closely examined. The very limited interest in this field of research, however, makes the amount of existing papers rather limited. The present study's new intervention is as follows: The so-called Ampe re forces between collinear currents, as in Ampere's bridge and in exploding wires, have been explained to be due to electrostatics, provided that the propagation delay dependant on the motion of charges is correctly taken into account. The Lorentz Force Law fails in this case. Additionally, electromagnetic induction can be explained by applying electrostatics, whereas the induction law fails. Light on the orbit electrons in the atoms involved in excitation and de-excitation of states can be explained using Coulomb's law (this has been widely disputed within science). The appearance of light at an atom hit by electromagnetic radiation can be shown to constitute a case of electromagnetic induction. The present study's conclusion is therefore that Coulomb's law is the only necessary force law within electromagnetism.

The Claim that Neumann’s Induction Law Is Consistent with Ampère’s Law Rejected

There have been made several efforts to link Amperes law to different parts of electromagnetic field theory In this paper Neumann's effort to make his induction law to appear to be consistent with Ampere's law will be studied thoroughly., Since there exists a concept, named Ampere-Neumann electrodynamics, it has been regarded as necessary to analyze how Neumann derives the connection between induction and Ampere's force law. One conspicuous thing is that he by the electromotive force of the secondary loop is meaning a physical, mechanical force, contrary to what has usually has been understood as an induced voltage. This makes it possible for him to claim his ideas to be consistent with Ampere's law. On the contrary, recent papers have convincingly shown that the Continuity Equation of Electricity is able to explain, how a current is being induced in a secondary circuit, due to an alternate current in the primary circuit. Earlier discoveries that Coulomb's law is able to account for electromagnetic forces, without involving magnetic fields, provides the conceptual background, which makes the use of magnetic fields unnecessary also in connection with induction. Index Terms—Neumann's induction law, ampere's law, faraday's law of induction, grassmann's force law, coulomb's law.

Structure and coarsening at the surface of a dry three-dimensional aqueous foam

We utilize total-internal reflection to isolate the two-dimensional surface foam formed at the planar boundary of a three-dimensional sample. The resulting images of surface Plateau borders are consistent with Plateau's laws for a truly two-dimensional foam. Samples are allowed to coarsen into a self-similar scaling state where statistical distributions appear independent of time, except for an overall scale factor. There we find that statistical measures of side number distributions, size-topology correlations, and bubble shapes are all very similar to those for two-dimensional foams. However, the size number distribution is slightly broader, and the shapes are slightly more elongated. A more obvious difference is that T2 processes now include the creation of surface bubbles, due to rearrangement in the bulk, and von Neumann's law is dramatically violated for individual bubbles. But nevertheless, our most striking finding is that von Neumann's law appears to holds on average, namely, the average rate of area change for surface bubbles appears to be proportional to the number of sides minus six, but with individual bubbles showing a wide distribution of deviations from this average behavior.

Coarsening dynamics of three-dimensional levitated foams: From wet to dry

We study diamagnetically levitated foams with widely different liquid fractions. Due to the levitation, drainage is effectively suppressed and the dynamics is driven by the coarsening of the foam bubbles. For dry foams, the bubble size is found to increases as the square root of foam age, as expected from a generalized von Neumann law. At higher liquid content the behavior changes to that of Ostwald ripening where the bubbles grow with the 1/3 power of the age. Using Diffusing Wave Spectroscopy we study the local dynamics in the different regimes and find diffusive behavior for dry foams and kinetic behavior for wet foams.

Bubble statistics and coarsening dynamics for quasi-two-dimensional foams with increasing liquid content

We report on the statistics of bubble size, topology, and shape and on their role in the coarsening dynamics for foams consisting of bubbles compressed between two parallel plates. The design of the sample cell permits control of the liquid content, through a constant pressure condition set by the height of the foam above a liquid reservoir. We find that in the scaling regime, all bubble distributions are independent not only of time, but also of liquid content. For coarsening, the average rate decreases with liquid content due to the blocking of gas diffusion by Plateau borders inflated with liquid; we achieve a factor of 4 reduction from the dry limit. By observing the growth rate of individual bubbles, we find that von Neumann's law becomes progressively violated with increasing wetness and decreasing bubble size. We successfully model this behavior by explicitly incorporating the border-blocking effect into the von Neumann argument. Two dimensionless bubble shape parameters naturally arise, one of which is primarily responsible for the violation of von Neumann's law for foams that are not perfectly dry.

Contact Angles on a Soft Solid: From Young’s Law to Neumann’s Law

The contact angle that a liquid drop makes on a soft substrate does not obey the classical Young's relation, since the solid is deformed elastically by the action of the capillary forces. The finite elasticity of the solid also renders the contact angles differently from those predicted by Neumann's law, which applies when the drop is floating on another liquid. Here, we derive an elastocapillary model for contact angles on a soft solid by coupling a mean-field model for the molecular interactions to elasticity. We demonstrate that the limit of a vanishing elastic modulus yields Neumann's law or a variation thereof, depending on the force transmission in the solid surface layer. The change in contact angle from the rigid limit to the soft limit appears when the length scale defined by the ratio of surface tension to elastic modulus γ/E reaches the range of molecular interactions.

Coarsening of a two-dimensional foam on a dome

In this paper we report on bubble growth rates and on the statistics of bubble topology for the coarsening of a dry foam contained in the narrow gap between two hemispheres. By contrast with coarsening in flat space, where six-sided bubbles neither grow nor shrink, we observe that six-sided bubbles grow with time at a rate that depends on their size. This result agrees with the modification to von Neumann's law predicted by J. E. Avron and D. Levine [Phys. Rev. Lett. 69, 208 (1992)]. For bubbles with a different number of sides, except possibly seven, there is too much noise in the growth rate data to demonstrate a difference with coarsening in flat space. In terms of the statistics of bubble topology, we find fewer three-, four-, and five-sided bubbles, and more bubbles with six or more sides, in comparison with the stationary distribution for coarsening in flat space. We also find good general agreement with the Aboav-Weaire law for the average number of sides of the neighbors of an n-sided bubble.