General Scaling Laws Research Articles

Use of IR Thermography To Investigate Heated Droplet Evaporation and Contact Line Dynamics

In this paper we present the results of an experimental study investigating interfacial properties during the evaporation of sessile water droplets on a heated substrate. This study uses infrared thermography to map the droplet interfacial temperature. The measurements evidence nonuniform temperature and gradients that evolve in time during the evaporation process. A general scaling law for the interfacial temperature is deduced from the experimental observations. A theoretical analysis is performed to predict the local evaporation rates and their evolution in time. The use of energy conservation laws enabled us to deduce a general expression for the interfacial temperature. The comparison between the theory and experiments shows good agreement and allows us to rationalize the experimental observations. The thermography analysis also enabled the detection of the three-phase contact line location and its dynamics. To our knowledge, such measurements are performed for the first time using thermography.

Scaling laws for beach and dune erosion processes

This paper is focussed on the derivation of a set of general scaling laws valid for both beach and dune erosion volumes based on scaling law analysis, existing and new experimental results. This latter experiments concern beach profile changes in three different laboratory flumes using identical wave conditions based on Froude scaling. The experiments with planar sloping beaches have been done at three scales: large-scale Hannover wave flume experiment (beach slope of 1 to 15), medium scale Barcelona wave flume experiment (beach slope of 1 to 15) and small-scale Delft wave flume experiments (beach slopes of 1 to 10, 15 and 20) using an identical wave train of irregular waves (single topped spectrum). The available data sets have been used to derive a general set of scaling laws which is valid for both beach and dune erosion under storm conditions. The morphological time scale depends on the depth scale and the length scale, and only indirectly on the sediment size scale. The analysis of the scaling laws shows that laboratory scale models can be operated at distorted scales, if necessary. Finally, two applications of the scaling laws are presented.

Nonlinear quenches of power-law confining traps in quantum critical systems

We describe the coherent quantum evolution of a quantum many-body system with a time-dependent power-law confining potential. The amplitude of the inhomogeneous potential is driven in time along a nonlinear ramp which crosses a critical point. Using Kibble-Zurek-like scaling arguments we derive general scaling laws for the density of excitations and energy excess generated during the nonlinear sweep of the confining potential. It is shown that, with respect to the sweeping rate, the densities follow algebraic laws with exponents that depend on the space-time properties of the potential and on the scaling dimensions of the densities. We support our scaling predictions with both analytical and numerical results on the Ising quantum chain with an inhomogeneous transverse field varying in time.

Optimality of the Münch mechanism for translocation of sugars in plants.

Plants require effective vascular systems for the transport of water and dissolved molecules between distal regions. Their survival depends on the ability to transport sugars from the leaves where they are produced to sites of active growth; a flow driven, according to the Münch hypothesis, by osmotic gradients generated by differences in sugar concentration. The length scales over which sugars are produced (Lleaf) and over which they are transported (L(stem)), as well as the radius r of the cylindrical phloem cells through which the transport takes place, vary among species over several orders of magnitude; a major unsettled question is whether the Münch transport mechanism is effective over this wide range of sizes. Optimization of translocation speed predicts a scaling relation between radius r and the characteristic lengths as r∼(Lleaf Lstem)1/3. Direct measurements using novel in vivo techniques and biomimicking microfluidic devices support this scaling relation and provide the first quantitative support for a unified mechanism of sugar translocation in plants spanning several orders of magnitude in size. The existence of a general scaling law for phloem dimensions provides a new framework for investigating the physical principles governing the morphological diversity of plants.

Size-dependent elastic properties of Au nanowires under bending and tension—Surfaces versus core nonlinearity

The present work investigates contributions from surfaces and core nonlinearity to the size-dependent elastic properties of nanowires under bending and tension-compression. When a nanowire is formed by removing it from its parent bulk material, relaxation occurs inevitably because of high energy of newly created surfaces or born high surface eigenstress. Relaxation-induced initial strain could be large and nonlinear, which causes the size-dependent elastic properties of nanowires. If relaxation-induced initial strain is small and linear, the size-dependent elastic properties of nanowires are caused by surface Young’s modulus. The eigenstress model for surface stress of solids {Zhang et al. [Phys. Rev. B 81, 195427 (2010)]} is further developed here for nanowires under bending and tension-compression. The developed eigenstress model leads to general scaling laws for nanowires under bending and tension-compression. In the scaling laws, there are the surface and nonlinearity factors, which measure quantitatively the contributions of surfaces and core nonlinearity, respectively, to the nominal Young’s modulus of nanowires. Atomistic simulations on Au [001] nanowires verify the theoretical predictions.

A systematic numerical study of the tidal instability in a rotating triaxial ellipsoid

The full non-linear evolution of the tidal instability is studied numerically in an ellipsoidal fluid domain relevant for planetary cores applications. Our numerical model, based on a finite element method, is first validated by reproducing some known analytical results. This model is then used to address open questions that were up to now inaccessible using theoretical and experimental approaches. Growth rates and mode selection of the instability are systematically studied as a function of the aspect ratio of the ellipsoid and as a function of the inclination of the rotation axis compared to the deformation plane. We also quantify the saturation amplitude of the flow driven by the instability and calculate the viscous dissipation that it causes. This tidal dissipation can be of major importance for some geophysical situations and we thus derive general scaling laws which are applied to typical planetary cores.

Bimodal Analysis Reveals a General Scaling Law Governing Nondirected and Chemotactic Cell Motility

Cell motility is a fundamental process with relevance to embryonic development, immune response, and metastasis. Cells move either spontaneously, in a nondirected fashion, or in response to chemotactic signals, in a directed fashion. Even though they are often studied separately, both forms of motility share many complex processes at the molecular and subcellular scale, e.g., orchestrated cytoskeletal rearrangements and polarization. In addition, at the cellular level both types of motility include persistent runs interspersed with reorientation pauses. Because there is a great range of variability in motility among different cell types, a key challenge in the field is to integrate these multiscale processes into a coherent framework. We analyzed the motility of Dictyostelium cells with bimodal analysis, a method that compares time spent in persistent versus reorientation mode. Unexpectedly, we found that reorientation time is coupled with persistent time in an inverse correlation and, surprisingly, the inverse correlation holds for both nondirected and chemotactic motility, so that the full range of Dictyostelium motility can be described by a single scaling relationship. Additionally, we found an identical scaling relationship for three human cell lines, indicating that the coupling of reorientation and persistence holds across species and making it possible to describe the complexity of cell motility in a surprisingly general and simple manner. With this new perspective, we analyzed the motility of Dictyostelium mutants, and found four in which the coupling between two modes was altered. Our results point to a fundamental underlying principle, described by a simple scaling law, unifying mechanisms of eukaryotic cell motility at several scales.

Quench dynamics near a quantum critical point: Application to the sine-Gordon model

We discuss the quench dynamics near a quantum critical point focusing on the sine-Gordon model as a primary example. We suggest a unified approach to sudden and slow quenches, where the tuning parameter $\lambda(t)$ changes in time as $\lambda(t)\sim \upsilon t^r$, based on the adiabatic expansion of the excitation probability in powers of $\upsilon$. We show that the universal scaling of the excitation probability can be understood through the singularity of the generalized adiabatic susceptibility $\chi_{2r+2}(\lambda)$, which for sudden quenches ($r=0$) reduces to the fidelity susceptibility. In turn this class of susceptibilities is expressed through the moments of the connected correlation function of the quench operator. We analyze the excitations created after a sudden quench of the cosine potential using a combined approach of form-factors expansion and conformal perturbation theory for the low-energy and high-energy sector respectively. We find the general scaling laws for the probability of exciting the system, the density of excited quasiparticles, the entropy and the heat generated after the quench. In the two limits where the sine-Gordon model maps to hard core bosons and free massive fermions we provide the exact solutions for the quench dynamics and discuss the finite temperature generalizations.

Experimental Determination of Zonal Winds Driven by Tides

We describe a new phenomenon of zonal wind generation by tidal forcing. Following a recent theoretical and numerical analysis [A. Tilgner, Phys. Rev. Lett. 99, 194501 (2007)], we present the first experimental evidence that the nonlinear self-interaction of a tidally forced inertial mode can drive an intense axisymmetric flow in a rotating deformed sphere. Systematic measurements of zonal flows are carried out by an embarked system of particle image velocimetry, allowing the determination of general scaling laws. These results are fully relevant for zonal winds generation in planets and stars, and illustrate a generic mechanism of geostrophic flow generation by periodic forcing.

Critical Quench Dynamics in Confined Systems

We analyze the coherent quantum evolution of a many-particle system after slowly sweeping a power-law confining potential. The amplitude of the confining potential is varied in time along a power-law ramp such that the many-particle system finally reaches or crosses a critical point. Under this protocol we derive general scaling laws for the density of excitations created during the nonadiabatic sweep of the confining potential. It is found that the mean excitation density follows an algebraic law as a function of the sweeping rate with an exponent that depends on the space-time properties of the potential. We confirm our scaling laws by first order adiabatic calculation and exact results on the Ising quantum chain with a varying transverse field.

A scaling approach for the prediction of high-frequency mean responses of vibrating systems

This analysis presents a scaling approach to predict high-frequency mean responses of vibrating systems. The basis of the approach lies in the dynamic similitude between the original systems and the scaled models. A general scaling law is formulated using Skudrzyk's mean-value theorem and its specific form is derived for the case of a flexural plate. Modal density is scaled down to reduce the computational cost in the high-frequency mean response prediction. Different scaling procedures are numerically experimented and some insights are given about the accuracy of the scaling approach as compared with a dense finite element analysis.

Scaling for the ensemble statistics prediction of a random system subjected to harmonic excitations

This paper is concerned with the ensemble statistics of the dynamic responses of a random system subjected to harmonic excitations. Random point process theory is employed to derive general scaling laws with the Gaussian orthogonal ensemble assumption about the system natural frequencies. A scaled model is built to simulate the high-frequency vibrations of the original system. Specific forms of the scaling laws are presented for a mass-loaded plate regarding the scaling factors for the structural parameters. The ensemble statistics predicted from the scaled model are compared favorably with those obtained from the original system.

Generalized Rosenfeld scalings for tracer diffusivities in not-so-simple fluids: Mixtures and soft particles

Rosenfeld [Phys. Rev. A 15, 2545 (1977)] originally noticed that casting the transport coefficients of simple monatomic equilibrium fluids in a specific dimensionless form makes them approximately single-valued functions of excess entropy. This observation has predictive value because, while the transport coefficients of dense fluids can be difficult to estimate from first principles, the excess entropy can often be accurately predicted from liquid-state theory. In this work, we use molecular simulations to investigate whether Rosenfeld's observation is a special case of a more general scaling law relating the tracer diffusivities of particles in mixtures to the excess entropy. Specifically, we study the tracer diffusivities, static structure, and thermodynamic properties of a variety of one- and two-component model fluid systems with either additive or nonadditive interactions of the hard-sphere or Gaussian-core form. The results of the simulations demonstrate that the effects of mixture concentration and composition, particle-size asymmetry and additivity, and strength of the interparticle interactions in these fluids are consistent with an empirical scaling law relating the excess entropy to a dimensionless (generalized Rosenfeld) form of tracer diffusivity, which we introduce here. The dimensionless form of the tracer diffusivity follows from knowledge of the intermolecular potential and the transport/thermodynamic behavior of fluids in the dilute limit. The generalized Rosenfeld scaling requires less information and provides more accurate predictions than either Enskog theory or scalings based on the pair-correlation contribution to the excess entropy. As we show, however, it also suffers from some limitations especially for systems that exhibit significant decoupling of individual component tracer diffusivities.

Scaling stellar jets to the laboratory: The power of simulations

AbstractAdvances in laser and Z-pinch technology, coupled with the development of plasma diagnostics, and the availability of high-performance computers, have recently stimulated the growth of high-energy density laboratory astrophysics. In particular, a number of experiments have been designed to study radiative shocks and jets with the aim of shedding new light on physical processes linked to the ejection and accretion of mass by newly born stars. Although general scaling laws are powerful tools to link laboratory experiments with astrophysical plasmas, the phenomena modeled are often too complicated for simple scaling to remain relevant. Nevertheless, the experiments can still give important insights into the physics of astrophysical systems and can be used to provide the basic experimental validation of numerical simulations in regimes of interest to astrophysics. We will illustrate the possible links between laboratory experiments, numerical simulations, and astrophysics in the context of stellar jets. First we will discuss the propagation of stellar jets in a cross-moving interstellar medium and the scaling to Z-pinch produced jets. Our second example focuses on slab-jets produced at the Prague Asterix Laser System laser installation and their practical applications to astrophysics. Finally, we illustrate the limitations of scaling for radiative shocks, which are found at the head of the most rapid stellar jets.

A universal Hamiltonian for motion and merging of Dirac points in a two-dimensional crystal

We propose a simple Hamiltonian to describe the motion and the merging of Dirac points in the electronic spectrum of two-dimensional electrons. This merging is a topological transition which separates a semi-metallic phase with two Dirac cones from an insulating phase with a gap. We calculate the density of states and the specific heat. The spectrum in a magnetic field B is related to the resolution of a Schrodinger equation in a double well potential. They obey the general scaling law e_n \propto B^{2/3} f_n(Delta /B^{2/3}. They evolve continuously from a sqrt{n B} to a linear (n+1/2)B dependence, with a [(n+1/2)B]^{2/3} dependence at the transition. The spectrum in the vicinity of the topological transition is very well described by a semiclassical quantization rule. This model describes continuously the coupling between valleys associated with the two Dirac points, when approaching the transition. It is applied to the tight-binding model of graphene and its generalization when one hopping parameter is varied. It remarkably reproduces the low field part of the Rammal-Hofstadter spectrum for the honeycomb lattice.

Hyperfine quenching of the 3s3p 3P0 level in Mg-like ions

Hyperfine quenching rates of the 3s3p3P0 level in Mg-like ions were calculated using the GRASP2K package based on the multi-configuration Dirac–Hartree–Fock method. Valence and core-valence correlation effects were accounted for in a systematic way. Breit interactions and QED effects were included in the subsequent relativistic CI calculations. Calculated rates were compared with other theoretical values and with experiment, and a good agreement with the latest experimental value for the ion (Rosenband et al 2007 Phys. Rev. Lett. 98 220801) was found. Furthermore, we showed in detail the contributions from Breit interaction and QED effects to concerned physical properties. Finally, electronic data were presented in terms of a general scaling law in Z that, given isotopic nuclear spin and magnetic moment, allows hyperfine-induced decay rates to be estimated for any isotope along the isoelectronic sequence.

Numerical modeling of 2‐D granular step collapse on erodible and nonerodible surface

The study of the collapse of a granular step is of great interest for understanding transient dense granular flow conditions and for modeling geophysical flows in granular materials. We present the results of a series of finite elements simulations considering variable column aspect ratios and properties for an elastoplastic material with a Mohr‐Coulomb yield rule and nonassociate flow rule. The adopted approach does not suffer limitations of typical shallow water equation methods, being able to consider strong vertical motion components. Transition from initial instability to complete flow development is simulated for columns with different aspect ratios (a ≤ 20). Simulation results are compared to original tests and available well‐documented experimental data, in terms of flow development, duration, profile geometry, velocity distribution, erosion and deposition, and evolution of the interface between static and moving material. Tests involving a thick erodible layer have been performed and numerical simulation results are compared also with a real case study. Numerical results support both those of qualitative and theoretical models and the proposed general scaling laws and clarify the dependence on frictional properties. Power laws describe the normalized runout versus aspect ratio (a > 4) relationship with constants of proportionality dependent on internal friction angle and exponents ranging between 0.68 and 0.77, in good agreement with experimental results. Total duration and evolution in three successive phases agree with observations. Time for the flow front to cease motion with respect to aspect ratio is best represented by the 3.68a0.448 relationships for a 30° internal friction angle material.

Adiabatic dynamics of a quantum critical system coupled to an environment: Scaling and kinetic equation approaches

We study the dynamics of open quantum many-body systems driven across a critical point by quenching a parameter of the Hamiltonian at a certain velocity. General scaling laws are derived for the density of excitations and of energy produced during the quench as a function of the quench velocity and the bath temperature. The scaling laws and their regimes of validity are verified for the $XY$ spin chain locally coupled to bosonic baths. A detailed derivation and analysis of the kinetic equations for the problem is presented.

Revision of capillary cone-jet physics: Electrospray and flow focusing

Capillary cone jets are natural microfluidic structures arising in steady capillary tip streaming, whose paradigms are electrospray and flow focusing phenomena. In this work, we make a profound revision of the basic underlying physics of generic cone jets from thousands of experimental measurements, most of them reported in the literature. First, the boundaries of the stability region of steady jetting are calculated. We describe these limitations by instability mechanisms associated with the local flow structure in the tip and the issuing jet and with the global behavior of the meniscus. Second, to undertake a general physical treatment of cone jets in steady regime, we analyze the energy balance taking place in the tips of both flow focusing and electrospray. This analysis yields a fundamental result: if the electrospray data are expressed in terms of an effective pressure drop, both phenomena satisfy the same scaling law for the droplet size, which exhibits nearly complete similarity in the parameter window where quasimonodisperse sprays are produced. That effective pressure drop is a function of the liquid properties exclusively, i.e., it does not depend on the operational parameters (flow rate and applied voltage). Moreover, the stability limits of the operational regimes are analyzed in detail, finding fundamental coincidences between flow focusing and electrospray as well. These results provide most useful general description and predictive scaling laws for nearly monodisperse microspraying or nanospraying based on steady cone jets, of immediate applicability in analytical chemistry, chemical engineering, biochemistry, pharmaceutical and food technologies, painting, and many other technological fields.

Stochastic analysis of spatial and opportunistic aloha

Spatial Aloha is probably the simplest medium access protocol to be used in a large mobile ad hoc network: each station tosses a coin independently of everything else and accesses the channel if it gets heads. In a network where stations are randomly and homogeneously located in the Euclidean plane, there is a way to tune the bias of the coin so as to obtain the best possible compromise between spatial reuse and per transmitter throughput. This paper shows how to address this questions using stochastic geometry and more precisely Poisson shot noise field theory. The theory that is developed is fully computational and leads to new closed form expressions for various kinds of spatial averages (like e.g. outage, throughput or transport). It also allows one to derive general scaling laws that hold for general fading assumptions. We exemplify its flexibility by analyzing a natural variant of Spatial Aloha that we call Opportunistic Aloha and that consists in replacing the coin tossing by an evaluation of the quality of the channel of each station to its receiver and a selection of the stations with good channels (e.g. fading) conditions. We show how to adapt the general machinery to this variant and how to optimize and implement it. We show that when properly tuned, Opportunistic Aloha very significantly outperforms Spatial Aloha, with e.g. a mean throughput per unit area twice higher for Rayleigh fading scenarios with typical parameters.