Fluid Mechanics: Landau and Lifshitz: Course of Theoretical Physics
Abstract
Fluid Mechanics, Second Edition deals with fluid mechanics, that is, the theory of the motion of liquids and gases. Topics covered range from ideal fluids and viscous fluids to turbulence, boundary layers, thermal conduction, and diffusion. Surface phenomena, sound, and shock waves are also discussed, along with gas flow, combustion, superfluids, and relativistic fluid dynamics. This book is comprised of 16 chapters and begins with an overview of the fundamental equations of fluid dynamics, including Euler's equation and Bernoulli's equation. The reader is then introduced to the equations of motion of a viscous fluid; energy dissipation in an incompressible fluid; damping of gravity waves; and the mechanism whereby turbulence occurs. The following chapters explore the laminar boundary layer; thermal conduction in fluids; dynamics of diffusion of a mixture of fluids; and the phenomena that occur near the surface separating two continuous media. The energy and momentum of sound waves; the direction of variation of quantities in a shock wave; one- and two-dimensional gas flow; and the intersection of surfaces of discontinuity are also also considered. This monograph will be of interest to theoretical physicists.
... Combining the continuity of mass flux density, M := ρu, and momentum flux density, ρu 2 + P, over the front [the corresponding expressions can easily be read off from the terms on the left-hand side of Eqs. (1) and (2)], we arrive at the Rayleigh criterion (see Landau and Lifshitz, 1987) relating the unburnt and burnt states (denoted with subscripts "u" and "b", respectively): ...
... Here, we introduce the specific internal energy e int := e tot − u 2 /2. This results in a thermodynamic condition for the "burnt" state, the so-called Hugoniot adiabat (see Landau and Lifshitz, 1987): ...
... For common equations of state, the Hugoniot adiabat is a parabola in the P-V -plane connecting the unburnt state to all thermodynamically allowed "burnt" states. It is important to note that the slope of the tangent to the Hugoniot adiabat at a certain point measures the speed of sound in the corresponding state, while the slope of the Rayleigh line through that same point represents the velocity of the front with respect to that state (see Landau and Lifshitz, 1987). The curve's intersection with the Rayleigh line marks the physically realized (i.e. both mechanically and thermodynamically admissible) "burnt" state (V b , P b ). ...
Type Ia supernovae are associated with thermonuclear explosions of white dwarf stars. Combustion processes convert material in nuclear reactions and release the energy required to explode the stars. At the same time, they produce the radioactive species that power radiation and give rise to the formation of the observables. Therefore, the physical mechanism of the combustion processes, as reviewed here, is the key to understand these astrophysical events. Theory establishes two distinct modes of propagation for combustion fronts: subsonic deflagrations and supersonic detonations. Both are assumed to play an important role in thermonuclear supernovae. The physical nature and theoretical models of deflagrations and detonations are discussed together with numerical implementations. A particular challenge arises due to the wide range of spatial scales involved in these phenomena. Neither the combustion waves nor their interaction with fluid flow and instabilities can be directly resolved in simulations. Substantial modeling effort is required to consistently capture such effects and the corresponding techniques are discussed in detail. They form the basis of modern multidimensional hydrodynamical simulations of thermonuclear supernova explosions. The problem of deflagration-to-detonation transitions in thermonuclear supernova explosions is briefly mentioned.
... von Karman) is given in the handbook on theoretical physics of Landau and Lifshitz [1] and in the handbook of Schlichting [2]. ...
... In this regard, a natural question arises as to why in the handbook of Landau and Lifshitz [1] and in the handbook of Schlichting [2] the radial velocity depends on the coordinate r according to linear law: ...
... We can see from these graphs that the pressure profi le in the boundary layer of incompressible fl uid for our problem is convex. In the thin boundary layer (very small z) the pressure signifi cantly arises which corresponds to the presence of a large tangentialvelocity gradient, resulting in a large viscous dissipation of energy [1]. ...
The flow in the neighborhood of a rotating disk is of great practical importance, particularly in connection with rotary machines. It becomes turbulent at larger Reynolds numbers, in the same way as the flow about a plate. In this article, we consider a motion of incompressible fluid that is always turbulent in azimuthal direction (Reynolds number based on azimuthal velocity) and is of both kinds in a radial direction, i.e. laminar (Reynolds number based on radial velocity) and turbulent. The equations of analyticity of functions of a spatial complex variable (shortly, the equations of tunnel mathematics) afford a possibility to seek the solutions of steady Navier-Stokes equation in view of elementary functions. All vector fields, including those obeying the Navier-Stokes equation, satisfy the equations of tunnel mathematics. The Navier-Stokes equations themselves are afterward applied for verification of obtained solutions and calculation of the pressure. Obtained formulae for pressure allow us to visualize the presence of the boundary layer and estimate its thickness for laminar and turbulent flows. We use Prandtl`s concept of considering fluids with small viscosities, i. e. we suppose that the Reynolds number is enough large and the viscosity has an important effect on the motion of fluid only in a very small region near the disk (boundary layer). We also suppose that the fluid and the disk had at the beginning the same temperatures and the energy dissipation occurs only by means of internal friction.
... Here we consider how such a thin film at a liquid surface modify hydrodynamic motion. It is well-known that the presence of a film increases the damping of surface waves [3,4]. The history of this phenomenon dates back to antiquity, when ancient Greeks used oil to calm rough seas. ...
... We consider the bulk motion of a liquid, which obeys the Navier-Stokes equation [3,4] ...
... Therefore the boundary conditions at a liquid surface z = h are modified in comparison with the free surface, see e.g. [4], they are ...
Recently a theoretical scheme explaining the vorticity generation by surface waves in liquids was developed [S. Filatov et al., Phys. Rev. Lett. 116, 054501 (2016)]. Here we study how a thin (monomolecular) film presented at the surface of liquid affects the generated vorticity. We demonstrate that the vorticity becomes parametrically larger than for the case with a clean surface and now it depends on viscosity of the liquid. We also discuss the motion of particles passively advected by the generated surface flow. The results can be used in different applications: from the analysis of pollutants' diffusion on the ocean surface till the reconstruction of vorticity based on the particle image velocimetry (PIV) measurements.
... DPD typically has a softer potential between particles than that of MD, therefore it allows for a larger time step. In the hydrodynamic limit with large spatial-temporal scales, it may be considered as a Lagrangian discrete counterpart of the fluctuating hydrodynamics described by the Landau-Lifshitz-Navier-Stokes equations [42,44]. At small wave-length and high frequency, it may be considered as the representation of the generalized hydrodynamics [43,45], especially when the pairwise forces of DPD are obtained via the coarse-graining and non-Markovian effects are nonnegligible [46][47][48]. ...
... Each of the components is according to the reversible, irreversible, and stochastic process, respectively. Assuming that the pressure is isotropic and the viscous stress depends only on the first derivatives of velocity, Π C and Π D read as [44] Π C µσ = −pδ µσ , (2.6) ...
... For fluids at mesoscale, there are fluctuations in the state variables governed by the framework of thermodynamics, therefore local spontaneous stress does occur. By assuming an underlying Gaussian-Makovian process for the unresolved degrees of freedom, the conditions for the random stress tensor satisfying the fluctuation-dissipation theorem read as [44] ...
We study correlations of hydrodynamic fluctuations in shear flow analytically and also by dissipative particle dynamics~(DPD) simulations. The hydrodynamic equations are linearized around the macroscopic velocity field and then solved by a perturbation method in Fourier-transformed space. The autocorrelation functions~(ACFs) from the analytical method are compared with results obtained from DPD simulations under the same shear-flow conditions. Upto a moderate shear rate, various ACFs from the two approaches agree with each other well. At large shear rates, discrepancies between the two methods are observed, hence revealing strong additional coupling between different fluctuating variables, which is not considered in the analytical approach. In addition, the results at low and moderate shear rates can serve as benchmarks for developing multiscale algorithms for coupling of heterogeneous solvers, such as a hybrid simulation of molecular dynamics and fluctuating hydrodynamics solver, where thermal fluctuations are indispensable.
... When ultrasound is used to stimulate a microbubble, the fluid around the microbubble follows the laws of mass and momentum. 32 The gas within the microbubble is assumed to be uniform, so it follows the ideal gas state equation. The pressure inside the microbubble depends on its volume and is defined as a function of the radius: ...
... The gas pressure and initial gas pressure inside the microbubble are represented by p g and p g0 , while V and V 0 stand for the volume and initial volume of the microbubble, respectively. The polytropic constant of the gas is represented by c. 25,32 The vapor pressure p v is assumed to be zero. Hydrostatic pressure in the fluid, equilibrium microbubble radius, and microbubble radius at a given time t are presented by P 0 , R 0 , and R, respectively. ...
... Main parameters used in this study.25,30,32 ...
This study examines the effect of acoustic driving parameters, both in single and dual-frequency sonication, on the pressure applied to blood vessel walls due to microbubble oscillations in blood. This study aims to derive a safe sonication protocol to open the blood–brain barrier. The finite element method was used to perform simulations of a microbubble. Activations were carried out at 1 MHz (1–3 W/cm²) and 150 kHz (0.1 and 0.2 W/cm²), with 0, π/2, π, and 3π/2 phase differences and different pulse modes. The safe protocols were acquired based on the experimental study. The pulse pressure average created on the vessel wall (PPA) for single-frequency (2.99 kPa, 1 MHz and 116 Pa, 150 kHz) was lower than dual-frequency (4.20 kPa). With increased intensity, PPA increased by 74%–80% for different pulse modes. The effect of duty factor on PPA at 1 MHz and 150 kHz was about 50% and less than 10%, respectively. The maximum change of PPA in phase difference was less than 10%. The order of influence of the studied parameters on the PPA is intensity > duty factor > phase difference. Safe protocols for animal models were reported to open the blood–brain barrier.
... where η S αβγδ = (η αβγδ + η γδαβ )/2 and η A αβγδ = (η αβγδ − η γδαβ )/2. The viscous energy dissipation in the fluid per unit time and unit volume can be calculated as [80]Ė kin = −σ vis αβ ∂ β v α , where σ vis αβ is the viscous stress tensor and ∂ β v α is the shear rate. Accordingly, only the symmetric contributions of the viscosity are associated with dissipation because the anti-symmetric parts cancel in the summation. ...
... Equation (4) can be closed by supplying a relation between the density and the pressure. A common approach is to assume incompressiblity [80] such that Equation (4) together with ∂ α v α = 0 fully determines the dynamics. Note that the compression term in Equation (4) to ζ then also vanishes. ...
... In systems in which density inhomogeneities play a crucial role, such as in systems featuring shock waves, an alternative route could be to explicitly allow for weak density inhomogeneities and close Equation (4) with the continuity equation [71]. If the Reynolds number (the ratio of inertial to viscous forces in the fluid) is sufficiently small as is typical for soft matter systems on the micrometer scale, the left-hand side of Equation (4) can be typically neglected [80]. In an incompressible chiral active fluid with sufficiently high rotational and odd viscosities, we thus arrive at the closed Stokes equation for chiral active fluids [34]: ...
In recent years, there has been growing interest in the study of chiral active materials, which consist of building blocks that show active dynamics featuring chiral symmetry breaking, e.g., particles that rotate in a common direction. These materials exhibit fascinating phenomena such as odd viscosity, odd diffusivity, active turbulence in fluids, vivid dislocation dynamics or odd elasticity in crystals or elastic materials, and hyperuniform states. The systematic study of soft chiral active matter systems is relatively new, starting around 2017, but has already shown promising applications in robust cargo transport, segregation and mixing dynamics, or manipulation of metamaterials. In this review, we summarize recent experimental and theoretical advances in this field, highlighting the emergence of anti-symmetric and odd stresses and ensuring effects such as odd viscosity or topologically protected edge modes. We further discuss the underlying mechanisms and provide insights into the potential of chiral active matter for various applications.
... It will be noted that this is only a model 2 , in the sense of formal logic, since it cannot correctly bring out any viscosity of the air, for example. However, we know that without it, no plane could fly (7). Nevertheless, this theory has the advantage of being able to shed some light on what is happening and lends itself, with a few approximations, to affordable calculations. ...
... Indeed, the set of natural numbers, by definition, is countable and we know that the set of its parts is of cardinal 1 . Calculations limited to steps will therefore lead us to a decidability slightly more powerful than that of a Turing machine in our world 7 . But if we can allow ourselves 1 steps , ...
... Maybe not! Indeed, a little less known than the conjecture above, there is a theorem that says that there are oracles such that in some cases we have A A P NP And in others we have 7 Let us nevertheless mention the question of the construction of the field of surreal numbers by John Conway who makes an induction up to and "the day We obtain the whole of the proper class No (16). 8 Recall that a non-standard model can be seen as to which an appendix has been added which is, typically, ...
We explore the possibility of making true physical oracles and propose a specific use in the field of cryptography.
... It will be noted that this is only a model 2 , in the sense of formal logic, since it cannot correctly bring out any viscosity of the air, for example. However, we know that without it, no plane could fly (7). Nevertheless, this theory has the advantage of being able to shed some light on what is happening and lends itself, with a few approximations, to affordable calculations. ...
... Indeed, the set of natural numbers, by definition, is countable and we know that its power set is of cardinal 1 . Calculations limited to steps will therefore lead us to a decidability slightly more powerful than that of a Turing machine in our world 7 . But if we can allow ourselves 1 steps , ...
... Maybe not! Indeed, a little less known than the conjecture above, there is a theorem that says that there are oracles such that in some cases we have A A P NP And in others we have 7 Let us nevertheless mention the question of the construction of the field of surreal numbers by John Conway who makes an induction up to and "the day We obtain the whole of the proper class No (16). 8 Recall that a non-standard model can be seen as to which an appendix has been added which is, typically, For an attacker to have a chance of hacking the communication, she must replicate the oracle A , which is possible in our non-Archimedean model, but she would also have to be able to know the common result obtained by Alice and Bob of the oracle A . ...
We explore the possibility of making and mastering truely infite Turing machines able to make infinite computation in an infinitesimal amount of time. That is, we explore, in some way, the possibility of making true oracles.
... For equilibrium reversible processes of a two-component fluid, the thermodynamic Gibbs identity has the form Landau and Lifshitz (1987) (1) ...
... where E is the internal energy per unit volume, is the absolute temperature, S is the entropy per unit volume, is the total density of the mixture, 0 is the chemical potential of the mixture, c is the mass concentration of the supplied component and is the chemical potential of this component. Since thermodynamic pressure is the derivative of energy with respect to volume, it follows from equation (1) that pressure satisfies the following representation formula Landau and Lifshitz (1987): ...
... The concentration flux j obeys Fick's law where is the mobility Landau and Lifshitz (1987). As for the velocity vector v , it is governed by the Navier-Stokes equation ...
We address the two-scale homogenization of the Navier–Stokes and Cahn–Hilliard equations in the case of a weak miscibility of a two-component fluid. To this end a notion of the miscibility strength is formulated on the basis of a correlation between the upscaling parameter and the surface tension. As a result, a two-scale model is derived. Macro-equations turn out to be a generalization of the Darcy law enjoying cross-coupling permeability tensors. It implies that the Darcy velocity of each phase depends on pressure gradients of both phases. Micro-equations serve for determination both of the permeability tensors and the capillary pressure. An example is constructed by analytical tools to describe capillary displacement of oil by mixture of water with carbon dioxide in a system of hydrophobic parallel channels.
... Inertial waves propagate in a globally rotating medium and capillary waves run at the interface between the media. The existence of a large group of hybrid waves is provided by the combined action of a number of factors [1,2]. ...
... Accordingly, when studying the waves of other types-acoustic [14] or gravitationalcapillary-at the interface between the atmosphere and the hydrosphere [1,15], the unperturbed density was assumed to be homogeneous. Here and further, general rotation effects and associated inertial waves [1,16] are not considered. ...
... Accordingly, when studying the waves of other types-acoustic [14] or gravitationalcapillary-at the interface between the atmosphere and the hydrosphere [1,15], the unperturbed density was assumed to be homogeneous. Here and further, general rotation effects and associated inertial waves [1,16] are not considered. ...
The density of a fluid or gas, which depends on the temperature, pressure and concentration of dissolved substances or suspended particles, changes under the influence of a large number of physical factors. We assume that an undisturbed liquid is heterogeneous. The propagation of periodic flows in viscous, uniformly stratified fluids is considered. The analysis is based on a system of fundamental equations for the transfer of energy, momentum and ma er in periodic flows. Taking into account the compatibility condition, dispersion relations are constructed for two-dimensional internal, acoustic and surface linear periodic flows with a positive definite frequency and complex wave number in a compressible viscous fluid exponentially stratified by density. The temperature conductivity and diffusion effects are neglected. The obtained regularly perturbed solutions of the dispersion equations describe the conventional weakly damped waves. The families of singular solutions, specific for every kind of periodic flow, characterize the before unknown thin ligaments that accompany each type of wave. In limited cases, the constructed regular solutions transform into well-known expressions for a viscous homogeneous and an ideal fluid. Singular solutions are degenerated in a viscous homogeneous fluid or disappear in an ideal fluid. The developing method of the fundamental equation system analysis is directed to describe the dynamics and spatial structure of periodic flows in heterogeneous fluids in linear and non-linear approximations.
... In practice, however, this type of microscopic modeling will be infeasible, even on proposed exascale architectures, for many mesoscopic problems of interest. A more efficient and tractable numerical approach for mesoscopic fluids is fluctuating hydrodynamics [33,49]. This theory extends conventional hydrodynamics by including a random component to the dissipative fluxes. ...
... Assuming that the viscous stress tensor is unaffected by the electric field then both τ and τ are the same as for neutral fluids. We use the formulation as given in [2,33,49] and ignore bulk viscosity effects, so τ = η∇v ≡ η[∇v + (∇v) T ], with viscosity η. The stochastic contribution to the viscous stress tensor is formally modeled as, ...
We formulate and study computationally the low Mach number fluctuating hydrodynamic equations for electrolyte solutions. We are interested in studying transport in mixtures of charged species at the mesoscale, down to scales below the Debye length, where thermal fluctuations have a significant impact on the dynamics. Continuing our previous work on fluctuating hydrodynamics of multicomponent mixtures of incompressible isothermal miscible liquids (A. Donev, et al., Physics of Fluids, 27, 3, 2015), we now include the effect of charged species using a quasielectrostatic approximation. Localized charges create an electric field, which in turn provides additional forcing in the mass and momentum equations. Our low Mach number formulation eliminates sound waves from the fully compressible formulation and leads to a more computationally efficient quasi-incompressible formulation. We demonstrate our ability to model saltwater (NaCl) solutions in both equilibrium and nonequilibrium settings. We show that our algorithm is second-order in the deterministic setting, and for length scales much greater than the Debye length gives results consistent with an electroneutral/ambipolar approximation. In the stochastic setting, our model captures the predicted dynamics of equilibrium and nonequilibrium fluctuations. We also identify and model an instability that appears when diffusive mixing occurs in the presence of an applied electric field.
... Now, let us give some notes about energy spectrum of turbulence regardless the method. In the model of the Kolmogorov turbulence the properties of turbulent motion are determined by a single specific parameter ǫ, which is a specific flux of the energy transmitted from large-scale pulsations to small-scale pulsations (Landau & Lifshitz 1959). If we assume this quantity to be constant over the turbulent cascade, then we should not have energy loss within the inertial interval. ...
... Thus, from dimension considerations one can obtain all the main relations for the Kolmogorov turbulence. We should remind the reader that in the frame of this model (Landau & Lifshitz 1959) the exponent of the spectrum of turbulence is α = 5/3 within the inertial interval, and is α = 3 within the dissipative interval. However, the model of turbulence may be self-similar if one accepts a more general relation ...
We present a method that can be used to recover the spectrum of turbulence from observations of optically thin emission lines formed in astrophysical disks. Within this method we analyze how line intensity fluctuations depend on the angular resolution of the instrument, used for the observations. The method allows us to restore the slope of the power spectrum of velocity turbulent pulsations and estimate the upper boundary of the turbulence scale.
... The leading vortex contributes to the entrainment by inducing the Biot-Savart-like flow field which forces the ambient toward the trailing tail [58][59][60][61][62][63][64]. The entrainment velocity vector at the TNTI is mostly perpendicular to it, and it's about two orders of magnitude lower than the jet centerline velocity in the corresponding cross-section [63][64][65]. ...
... The utilization of combined shadowgraphy and smoke-wire techniques allowed us to distinguish between the different jet zones and the phases inside it, to measure the ambient entrainment characteristics and to further investigate the jet collapsing phenomenon which affects the unsteady entrainment profile of the hollow-cone jet. Figure 6 the ambient entrainment velocity in [63][64][65], in the reported study we monitor the visual Lagrangian trajectory of the induced vapor flow. The round jet entrainment characteristics were compared in this study with the hollow-cone jet entrainment performance. ...
Recently published studies report on significant Particulate Matter (PM) formation, comprising organic carbon aerosols, in hydrogen- and hydrogen-rich reformate-fed Internal Combustion Engines. There is a lack of the knowledge about the hollow- cone jet entrainment characteristics at different stages of the jet development, jet interaction with a lubricated cylinder wall, and the entrainment characteristics comparison with the round-jet. The method of shadowgraphy imaging accompanied by the smoke-wire visualization of the ambient entrainment by the jet is suggested and employed for the first time for entrainment and jet structure investigation. We show that the hollow-cone jets exhibit different vapor front entrainment behavior, compared to the round jet. In contrast to the round jet. The phenomenon of a twofold impact of the leading vortex on the farfield ambient with repulsing the ambient downstream the leading vortex and pulling it toward the trail-jet region upstream the vortex is found in the hollow-cone jets after the collapsing. We demonstrate that in contrast to the round jet, the hollow-cone one entrains the ambient in the near field through the inner TNTI.
... On remarquera que cela n'est qu'un modèle 2 , au sens de la logique formelle, puisqu'il ne peut pas faire émerger correctement une quelconque viscosité de l'air par exemple. Or, on sait que sans elle, aucun avion ne pourrait voler (7). Néanmoins, cette théorie a l'avantage de pouvoir éclairer un peu sur ce qui se passe et se prête, moyennant quelques approximations, à des calculs abordables. ...
... Pour construire cela, il suffit de considérer l'axiomatique de Peano avec le schéma d'induction et de rajouter dans la signature du langage un terme c qui soit différent de tous les entiers standards. On montre facilement alors que toute théorie finie qui ampute les axiomes de Peano, mais qui conserve les propositions closes7 Evoquons quand même la question de la construction du corps des nombres surréels par John Conway qui fait une induction jusqu'à et « le jour » on obtient la totalité de la classe propre No(16). ...
Nous explorons le possibilité de machines de Turing composées d'une infinité de particules infinitésimales et capables d'opérer un nombre infini d'étapes de calcul en un temps infiniment bref.
... Tangential oscillations of two fluids near the interface accompanied by the discontinuity of the tangential velocity cause the excitation of the Kelvin-Helmholtz oscillatory instability [1][2][3][4], which manifests itself in the form of the quasi-stationary relief (frozen wave). It is noteworthy that the Kelvin-Helmholtz instability [5] is studied in detail for the case of stationary tangential velocity discontinuity [6,7] including the complicating factors, like: proximity of the wall [8], flows in Hele-Shaw cell [9], heat and mass transfer from viscoelastic liquid film [10]. The practical interest in the study of the Kelvin-Helmholtz instability, including the oscillatory type of the instability, is explained, in particular, by the influence of this effect on the film coating and fluid spraying [11][12][13][14]. ...
... When the thickness of the viscous film is small compared to the thickness of the layer, the pressure inside the film does not depend on the distance to the end wall and is determined by the pressure at the interface, that is, the pressure in a uniformly rotating low-viscosity fluid. This approach is standard when dealing with boundary layer flows [5]. The proposed model is the first attempt to describe film formation and requires further theoretical study. ...
... The idea of universality in turbulence relies on the assertion that the smallest scales of the flow are the result of its internal nonlinear dynamics and therefore become independent of the external forcing mechanism that generates the turbulence. Next to the Reynolds number, it was put forward by Landau (Landau & Lifshitz 1987) that sufficiently slow, large-scale variations will constrain universality. In order to ensure comparability across different turbulent flows, it is therefore important to identify and characterize the properties of the flow that affect small-scale universality. ...
Turbulent flows in three dimensions are characterized by the transport of energy from large to small scales through the energy cascade. Since the small scales are the result of the nonlinear dynamics across the scales, they are often thought of as universal and independent of the large scales. However, as famously remarked by Landau, sufficiently slow variations of the large scales should nonetheless be expected to impact small-scale statistics. Such variations, often termed large-scale intermittency, are pervasive in experiments and even in simulations, while differing from flow to flow. Here, we evaluate the impact of temporal large-scale fluctuations on velocity, vorticity and acceleration statistics by introducing controlled sinusoidal variations of the energy injection rate into direct numerical simulations of turbulence. We find that slow variations can have a strong impact on flow statistics, raising the flatness of the considered quantities. We discern three contributions to the increased flatness, which we model by superpositions of statistically stationary flows. Overall, our work demonstrates how large-scale intermittency needs to be taken into account in order to ensure comparability of statistical results in turbulence.
... Some theory of non-equilibrium thermodynamics is given in Bedeaux [4], Mauri [27] as well as DeGroot and Mazur [12]. An introduction to balance laws from a thermodynamic point of view can be found in Landau and Lifschitz [24], Müller [30] as well as Müller and Müller [31]. ...
This paper gives a concise but rigorous mathematical description of a material control volume that is separated into two parts by a singular surface at which physical states are discontinuous. The geometrical background material is summarized in a unified manner. Transport theorems for use in generic balance laws are given with proofs since they provide some insight into the results. Also the step from integral balances to differential equations is given in some detail.
... In the description of the shock conditions, quantities with subscript '+' denote the PS parameters, and those with subscript '-' indicate the pre-shock (PRS) parameters. Following Landau and Lifshitz (1959); Sarkar and Das (2016); ; Sarkar and Rao (2020), across the shock front, (a) the mass accretion rate has to be conserved (Ṁ + =Ṁ − ), (b) there has to be pressure balance across the shock (W + + Σ + u 2 + = W − + Σ − u 2 − ), (c) the specific energy has to be conserved across the shock front (ε + = ε − ), and also (d) the magnetic flux advection rate is conserved across the shock front (Φ + =Φ − ). Following Samadi et al. (2014); Das and Sarkar Fig. 1(a). ...
This paper investigates the effects of saturated thermal conduction (TC) and thermal-driven winds (TDWs) on magnetized advection-dominated accretion onto a rotating black hole (BH). We incorporate dissipative processes in the magnetized accretion flow and expect the accretion disk to be threaded by predominantly toroidal and turbulent magnetic fields. We solve the magnetohydrody-namics equations and construct a self-consistent steady model of the magnetized accretion flow surrounding a rotating BH, which includes TC and TDWs. We seek global accretion solutions spanning from the BH horizon to a large distance and analyze the so-lution's characteristics as a function of dissipation parameters. Accretion solutions with multiple critical points may exhibit shock waves if they meet the standing shock criteria. We found steady, global transonic, and shocked accretion solutions around the rotating BH. In particular, the wind parameter (m) and the saturated conduction parameter (Φ s) significantly influence the dynamical behavior of shocks. The shock location moves away from the BH horizon as Φ s and m increase, assuming fixed conditions at the disk's outer edge. Our formalism explains the declining phase of BH outbursts, characterized by a monotonic decrease in QPO frequency as the burst decays. Based on our findings, we conclude that the combined effect of Φ s and m parameters substantially alters the steady shock specific energy vs angular momentum parameter space and also modifies the corresponding post-shock luminosity vs QPO frequency parameter space. We propose, based on our theoretical model, that the Φ s and m parameters may significantly influence the evolution of the BH outbursts.
... Since the seminal works of Kraichnan [1] and Lighthill [2], it has been established that acoustic waves and turbulent flows interact in complex ways. This interaction is bidirectional: Sound propagation is influenced by turbulence and vortical flows [3,4], while acoustic waves can trigger instabilities in mixing layers or jets [5,6]. Therefore, the investigation of incipient instabilities in compressible flows and of their underlying physical mechanisms is an active field of research [7][8][9][10] with many engineering applications [11,12]. ...
The stability of time-dependent compressible linear flows, which are characterized by periodic variations in either their shape or their shear, is investigated. Two novel parametric instabilities are found: an instability that occurs for periodically wobbling elliptic vortices at a number of discrete oscillation frequencies that are proportional to the Mach number and an instability that occurs for all linear flows at various frequencies of the shear oscillation that depend on the Mach number. In addition, the physical mechanism underlying the instabilities is explained in terms of the linear interaction of three waves with time-varying wavevectors that describe the evolution of perturbations: a vorticity wave representing the evolution of incompressible perturbations and two counter-propagating acoustic waves. Elliptical instability occurs because the scale of the acoustic waves decreases exponentially and their wave action is conserved, leading to an exponential increase in the acoustic waves’ energies. The instability in shear-varying flows is driven by the interaction between vorticity and the acoustic waves, which couple through the shear and for specific frequencies resonate parametrically, leading to exponential or linear growth.
... For highly supersonic pulsar proper velocities Vns/cs ≡ Ms 1 (here Ms is the Mach number with respect to the external medium) the forward shock is perpendicular to the flow velocity only at the apex point (as a result the shocked flow there is always subsonic in the pulsar frame). Away from the apex point, where the forward shock front makes a sufficiently small angle with the flow velocity, φ < arcsin (γism + 1)/(2γism), (as can be obtained for the case Ms 1 from the shock polar equation, see Landau & Lifshitz 1959, here γism is the adiabatic index of the ISM) the shocked ISM flow remains subsonic (although the shock is still strong). ...
Bow-shock pulsar wind nebulae (PWNe) show a variety of morphological shapes. We attribute this diversity to the geometrical factors: relative orientations of the pulsar rotation axis, proper velocity, and the line of sight (magnetic inclination angle may also have a certain influence on the morphology). We identify three basic types of bow-shock nebulae: (i) a "Rifle Bullet" (pulsar spin axis and proper velocity are aligned); (ii) a "Frisbee" (pulsar spin axis and proper velocity are orthogonal with the spin axis lying in the plane of the sky), and (iii) a Cart Wheel" (like frisbee but the spin axis is perpendicular to the plane of the sky). Using 3D RMHD simulations, as well as analytical calculations, we reproduce the key morphological features of the bow-shock PWNe, as well as variations, are seen across different systems. magnetic stresses within the shocked pulsar wind affect the overall structure strongly, producing "whiskers", "tails", "filled-in" and "mushroom" shapes, as well as non-symmetric morphologies. On the other hand, the interstellar medium inhomogeneities and the anisotropy of the energy flux in the pulsar wind have only a mild impact of the PWN morphology. In a few cases, when we clearly identify specific morphological structures, our results do not favor alignment of the pulsar spin axis and proper velocity. Our calculations of the underlying emission processes explain the low synchrotron X-ray efficiency (in terms of the spin-down luminosity) and imply an energetical subdominant contribution of the inverse Compton process.
... water) caused by gravity in a V-shaped tube: they will attempt to fill a vacancy at a lower point, and the heights of both sides are the same in equilibrium. However, we cannot directly use fluid dynamics equations to describe the day-to-day arrival time shifting dynamics, since (i) the equilibrium surface of the fluid is not flat in Figure 4(b); (ii) the former can lead to oscillations at the equilibrium in a frictionless V-shaped tube (Landau and Lifshitz, 1987), and (iii) more importantly, the latter involves nonlocal behaviors, which cannot be captured in the former. ...
In this paper we present a stable day-to-day dynamical system for drivers' departure time choice at a single bottleneck. We first define within-day traffic dynamics with the point queue model, costs, the departure time user equilibrium (DTUE), and the arrival time user equilibrium (ATUE). We then identify three behavioral principles: (i) Drivers choose their departure and arrival times in a backward fashion (backward choice principle); (ii) After choosing the arrival times, they update their departure times to balance the total costs (cost balancing principle); (iii) They choose their arrival times to reduce their scheduling costs or gain their scheduling payoffs (scheduling cost reducing or scheduling payoff gaining principle). In this sense, drivers' departure and arrival time choices are driven by their scheduling payoff choice. With a single tube or imaginary road model, we convert the nonlocal day-to-day arrival time shifting problem to a local scheduling payoff shifting problem. After introducing a new variable for the imaginary density, we apply the Lighthill-Whitham-Richards (LWR) model to describe the day-to-day dynamics of scheduling payoff choice and present splitting and cost balancing schemes to determine arrival and departure flow-rates accordingly. We also develop the corresponding discrete models for numerical solutions. We theoretically prove that the day-to-day stationary state of the LWR model leads to the scheduling payoff user equilibrium (SPUE), which is equivalent to both DTUE and ATUE and stable. We use one numerical example to demonstrate the effectiveness and stability of the new day-to-day dynamical model.
... A drag law that follows a U 2 scaling is a reasonable approximation of the drag scaling for steady streamlined bodies operating at high Reynolds numbers, that is Re ≥ O(10 6 ) (Munson et al. 1998). A U 3/2 Blasius scaling for laminar boundary layers could also be used (Landau & Lifshitz 1987) to model swimming at lower Reynolds numbers. In fact, the transition between a high Reynolds number drag law and a Blasius drag law has been noted to occur at lower Re in fish swimming than in steady flow cases around Re = O(10 4 ) determined from a wide range of biological data (Gazzola et al. 2014). ...
Inviscid computational results are presented on a self-propelled virtual body combined with an airfoil undergoing pitch oscillations about its leading-edge. The scaling trends of the time-averaged thrust forces are shown to be predicted accurately by Garrick's theory. However, the scaling of the time-averaged power for finite amplitude motions is shown to deviate from the theory. Novel time-averaged power scalings are presented that account for a contribution from added-mass forces, from the large-amplitude separating shear layer at the trailing-edge, and from the proximity of the trailing-edge vortex. Scaling laws for the self-propelled speed, efficiency and cost of transport (CoT) are subsequently derived. Using these scaling relations the self-propelled metrics can be predicted to within 5% of their full-scale values by using parameters known a priori. The relations may be used to drastically speed-up the design phase of bio-inspired propulsion systems by offering a direct link between design parameters and the expected CoT. The scaling relations also offer one of the first mechanistic rationales for the scaling of the energetics of self-propelled swimming. Specifically, the cost of transport is shown to scale predominately with the added mass power. This suggests that the CoT of organisms or vehicles using unsteady propulsion will scale with their mass as , which is indeed shown to be consistent with existing biological data.
... However, a number of results on the fluctuating Boltzmann equation are available in the Math and Physics literature [8,24,37,43,41,42,44]. In particular, the articles of Bixon/Zwanzig [8] and Fox/Uhlenbeck [24] outline a formal derivation of Landau and Lifshitz's equations of fluctuating hydrodynamics [33], from the fluctuating linear Boltzmann equation. The connection with macroscopic fluid equations arises from studying the correlation structure of the fluctuations at the level of the kinetic description. ...
This article studies the Cauchy problem for the Boltzmann equation with stochastic kinetic transport. Under a cut-off assumption on the collision kernel and a coloring hypothesis for the noise coefficients, we prove the global existence of renormalized (in the sense of DiPerna/Lions) martingale solutions to the Boltzmann equation for large initial data with finite mass, energy, and entropy. Our analysis includes a detailed study of weak martingale solutions to a class of linear stochastic kinetic equations. This study includes a criterion for renormalization, the weak closedness of the solution set, and tightness of velocity averages in .
... As it is well known, radial stability is guaranteed once Rayleigh's criterion is satisfied (Landau and Lifshitz 2007;Letelier 2003). ...
We consider the three-dimensional bounded motion of a test particle around razor-thin disk configurations, by focusing on the adiabatic invariance of the vertical action associated with disk-crossing orbits. We find that it leads to an approximate third integral of motion predicting envelopes of the form , where R is the radial galactocentric coordinate, Z is the z-amplitude (vertical amplitude) of the orbit and represents the surface mass density of the thin disk. This third integral, which was previously formulated for the case of flattened 3D configurations, is tested for a variety of trajectories in different thin-disk models.
... For this task we choose to study the oscillation of a free droplet that has been perturbed from its spherical equilibrium. The linear frequency of oscillation of an inviscid droplet (in the eigenmode of interest) was found by Rayleigh (1879) (with a more succinct derivation given by Landau and Lifshitz (1987)) to be ...
The study of the shape of droplets on surfaces is an important problem in the physics of fluids and has applications in multiple industries, from agrichemical spraying to microfluidic devices. Motivated by these real-world applications, computational predictions for droplet shapes on complex substrates -- rough and chemically heterogeneous surfaces -- are desired. Grid-based discretisations in axisymmetric coordinates form the basis of well-established numerical solution methods in this area, but when the problem is not axisymmetric, the shape of the contact line and the distribution of the contact angle around it are unknown. Recently, particle methods, such as pairwise force smoothed particle hydrodynamics (PF-SPH), have been used to conveniently forego explicit enforcement of the contact angle. The pairwise force model, however, is far from mature, and there is no consensus in the literature on the choice of pairwise force profile. We propose a new pair of polynomial force profiles with a simple motivation and validate the PF-SPH model in both static and dynamic tests. We demonstrate its capabilities by computing droplet shapes on a physically structured surface, a surface with a hydrophilic stripe, and a virtual wheat leaf with both micro-scale roughness and variable wettability. We anticipate that this model can be extended to dynamic scenarios, such as droplet spreading or impaction, in the future.
... An alternative approach to quantify the vortex size can be obtained from the energy spectra, which quantify the kinetic energy stored at different length scales. The translation energy spectrum can be obtained from the Fourier transformation of the velocity correlation function 55 , yielding ...
Chiral active fluids show the emergence of a turbulent behaviour characterised by multiple dynamic vortices whose maximum size varies for each experimental system, depending on conditions not yet identified. We propose and develop an approach to model the effect of friction close to a surface in a particle based hydrodynamic simulation method in two dimensions, in which the friction coefficient can be related to the system parameters and to the emergence of a damping length. This length is system dependent, limits the size of the emergent vortices, and influences other relevant system properties such as the actuated velocity, rotational diffusion, or the cutoff of the energy spectra. Comparison of simulation and experimental results of a large ensemble of rotating colloids sedimented on a surface shows a good agreement, which demonstrates the predictive capabilities of the approach, which can be applied to a wider class of quasi-two-dimensional systems with friction.
... This section focuses on the initial mass flow rate of liquid released from the pipeline immediately after the pipeline is punctured or ruptured. The basis for this analysis is the phenomenological mass balancing equation, which itself emerges from a generalisation of a similar calculation for ideal gas cases given in Landau and Lifshitz (1987). The mass balancing equation is given by: ...
... For η = 0.5, the fourfold state bifurcates at Re ∼ 489 from the axisymmetric state, and a threefold state occurs at 491 [23], followed by the twofold state bifurcating around Re ∼ 600. All states originated in the axisymmetric state, which first lost stability at Re = 489 against a perturbation with a wavenumber of 4. The torque values exceeded the Stokes flow by a factor of 3 [36] in the range of Re > 500. The torque values of the m-fold states are smaller than those of the axisymmetric state, and the torque slightly decreases as the wavenumber m decreases. ...
A pioneering study conducted by Egbers and Rath [Acta Mech. 111 pp. 125–140 (1995)] experimentally captured spiral waves to elucidate the transition in the wide-gap spherical Couette flow. However, the physical field quantities of the spiral waves corresponding to light patterns of various intensities, as obtained in the experiment, remain unclear, and we have yet to move beyond the understanding that the reflected light from shear-sensitive flake tracers responds to a flow that appears at the transition. In this study, the experiment to visualize spiral waves using aluminum flakes, as performed by Egbers and Rath, was numerically reproduced by solving the translational and rotational motions of the particles in a spiral wave. First, the spiral wave in a spherical Couette flow with an aspect ratio was numerically calculated using the Newton–Raphson method. Subsequently, the image that was numerically reproduced from the spiral wave was compared with an experimentally visualized image. The torque acting on the inner sphere and the phase angular velocity of the spiral waves with various wavenumbers were provided. Attempts have been made to determine the instantaneous physical quantity that corresponds to the light and dark patterns observed in the flow visualization. From the attempts, we concluded the orientation motion of the flakes developed in the advective history of the flow is essential to yield these patterns. Exploring the correlation between flow visualization results and shear structures may provide a new avenue for quantitatively estimating spatial structures and time scales in complex and quickly time-varying flow fields, such as turbulence.
... This defines the Beltrami velocity Generalized Gromeka-Beltrami flow and the Beltrami vorticity. As usual, one may re-express the Navier-Stokes equation (38) in the form that includes only velocity [27]. For that one should simply apply the operator curl to equation (38). ...
In this note we focus on the backreaction effects due to the chiral anomaly. The chiral anomaly gives rise to certain modifications in the conserved currents. In the case of the Maxwell gauge fields there appears a new contribution to the electric current proportional to the background magnetic field. This phenomenon is known as the chiral magnetic effect and it is widely discussed in the current literature. In the case of the gravitational field, the anomaly, as we show here, induces a new contribution to the stress-energy tensor. We analyze the possible manifestations of these modifications in the gravitational field and in hydrodynamics in the chiral media. We also make some comments regarding the backreaction effects in the electrodynamics in the chiral media. In each of these directions we observe the systematical appearance of the Beltrami type fields. In electrodynamics or in hydrodynamics the Beltrami field (e.g., magnetic field or fluid velocity) is a vector that is parallel to its own curl. We suggest a generalization of the Beltrami fields for the tensorial gravitational perturbations. The respective solutions to the gravitational equations we call gravitational spheromaks by analogy with a similar phenomenon in electrodynamics. In the modified hydrodynamics in the chiral media the vorticity is asymptotically a Beltrami vector field in a generalized Gromeka-Beltrami flow.
... where ϵ˙ is the volumetric strain rate and U → the fluid acceleration at the microscopic unit cell scale. These quantities are related to the primary micro-scale unknown field p x → , t , i.e. the fluid pressure disturbance around the background equilibrium (see Landau & Lifshitz [42]), through the constitutive relations of a compressible inviscid fluid ...
A reduced-order homogenization framework is proposed, providing a macro-scale-enriched continuum model for locally resonant acoustic metamaterials operating in the subwavelength regime, for both time and frequency domain analyses. The homogenized continuum has a non-standard constitutive model, capturing a metamaterial behaviour such as negative effective bulk modulus, negative effective density and Willis coupling. A suitable reduced space is constructed based on the unit cell response in a steady-state regime and the local resonance regime. A frequency domain numerical example demonstrates the efficiency and suitability of the proposed framework.
This article is part of the theme issue ‘Current developments in elastic and acoustic metamaterials science (Part 2)’.
... But as stated in p.58 of Gunzburger [34] or p.45 of Landau and Lifshitz [40], when ν is not a constant, the viscosity ν can not be taken out of the differential operator and only ∇ · (νǫ(u)) is the meaningful choice since it is derived from the principle of conservation of the linear momentum ∇ · σ = −f and the Newton-Poisson constitutive equation ...
A stationary Stokes problem with a piecewise constant viscosity coefficient in multiple subdomains is considered in the paper. For standard finite element pairs, a robust inf-sup condition is required to show the robustness of the discretization error with respect to the discontinuous viscosity, which has only been proven for the two-subdomain case in the paper [Numer. Math. (2006) 103: 129--149]. To avoid the robust inf-sup condition of a discrete finite element pair for multiple subdomains, we propose an ultra-weak augmented mixed finite element formulation. By adopting a Galerkin-least-squares method, the augmented mixed formulation can achieve stability without relying on the inf-sup condition in both continuous and discrete settings. The key step to having a robust priori error estimate is to use two norms, one energy norm and one full norm, in robust continuity. The robust coercivity is proved for the energy norm. A robust a priori error estimate in the energy norm is then derived with the best approximation property in the full norm for the case of multiple subdomains. Additionally, the paper introduces a singular Kellogg-type example with exact solutions for the first time. Extensive numerical tests are conducted to validate the robust error estimate.
... In the description of the shock conditions, quantities with subscript '+' denote the PS parameters, and those with subscript '-' indicate the pre-shock (PRS) parameters. Following Landau and Lifshitz (1959); Sarkar and Das (2016); ; Sarkar and Rao (2020), across the shock front, (a) the mass accretion rate has to be conserved (Ṁ + =Ṁ − ), (b) there has to be pressure balance across the shock (W + + Σ + u 2 + = W − + Σ − u 2 − ), (c) the specific energy has to be conserved across the shock front (ε + = ε − ), and also (d) the magnetic flux advection rate is conserved across the shock front (Φ + =Φ − ). Following Samadi et al. (2014); Das and Sarkar Fig. 1(a). ...
This paper investigates the effects of saturated thermal conduction (TC) and thermal-driven winds (TDWs) on magnetized advection-dominated accretion onto a rotating black hole (BH). We incorporate dissipative processes in the magnetized accretion flow and expect the accretion disk to be threaded by predominantly toroidal and turbulent magnetic fields. We solve the magnetohydrodynamics equations and construct a self-consistent steady model of the magnetized accretion flow surrounding a rotating BH, which includes TC and TDWs. We seek global accretion solutions spanning from the BH horizon to a large distance and analyze the solution's characteristics as a function of dissipation parameters. Accretion solutions with multiple critical points may exhibit shock waves if they meet the standing shock criteria. We found steady, global transonic, and shocked accretion solutions around the rotating BH. In particular, the wind parameter (m) and the saturated conduction parameter () significantly influence the dynamical behavior of shocks. The shock location moves away from the BH horizon as and m increase, assuming fixed conditions at the disk's outer edge. Our formalism explains the declining phase of BH outbursts, characterized by a monotonic decrease in QPO frequency as the burst decays. Based on our findings, we conclude that the combined effect of and m parameters substantially alters the steady shock specific energy vs angular momentum parameter space and also modifies the corresponding post-shock luminosity vs QPO frequency parameter space. We propose, based on our theoretical model, that the and m parameters may significantly influence the evolution of the BH outbursts.
... Also relevant 25 is the equation of state (EoS), which relates pressure, temperature and density of a fluid [37]: ...
A new variational inference method, SPH-ParVI, based on smoothed particle hydrodynamics (SPH), is proposed for sampling partially known densities (e.g. up to a constant) or sampling using gradients. SPH-ParVI simulates the flow of a fluid under external effects driven by the target density; transient or steady state of the fluid approximates the target density. The continuum fluid is modelled as an interacting particle system (IPS) via SPH, where each particle carries smoothed properties, interacts and evolves as per the Navier-Stokes equations. This mesh-free, Lagrangian simulation method offers fast, flexible, scalable and deterministic sampling and inference for a class of probabilistic models such as those encountered in Bayesian inference and generative modelling.
... Flow velocity of creeping flow near the rock surface is probabilistically distributed over the porous space. Under the assumption of Stokes flow in the porous space, velocity distribution over the porous space is determined by the macroscale Darcy's velocity U, i.e., local microscale speeds are proportional to U. 80 The mechanical equilibrium failure conditions (32,33) indicate whether each attached to rock surface particle is broken or remains attached under a given velocity U. It makes attached concentration a function of velocity that is called the MRF (maximum retention function). ...
... Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. wake [1], which expands or 'spreads' behind a moving object such as a boat, and expands downstream (distance ∼x) with width w ∼ x 1/3 . Entrainment is a mixing process, where a turbulent region invades and mixes into a laminar region. ...
Predictions of heat load widths λq based on particle orbits alone are very pessimistic. This paper shows that pedestal peeling-ballooning (P-B) magnetohydrodynamic (MHD) turbulence broadens the stable scrape-off layer (SOL) by the transport, or spreading, of fluctuation energy from the pedestal. λq is seen to increase with Γε , the fluctuation energy density flux. We elucidate the fundamental physics of the spreading process. Γε increases with pressure fluctuation correlation length. P-B turbulence is seen to be especially effective at spreading, on account of its large effective mixing length. Spreading is shown to be a multiscale process, which is enhanced by the synergy of large and small-scale modes. Pressure fluctuation skewness correlates well with the spreading flux–with the zero crossing of skewness and Γε spatially coincident–suggesting the role of coherent fluctuation structures and the presence of intermittency in λq broadening. λq∼Bp−1 scaling persists for the broadened SOL. We show that the spreading flux increases for increasing pedestal pressure gradient ∇P0 and for decreasing pedestal collisionality υped∗ . This trend is due to the dominance of peeling modes for large ∇P0 and low υped∗ . Ultimately, we see that a state of weak MHD turbulence, as for small ELMs, is very attractive for heat load management. Our findings have transformative implications for future fusion reactor designs and call for experimental investigations to validate the observed trends.
The first principle of numerical modeling of die extrusion of energetic materials is carried out to reduce the needed pressure gradient along the die. The proposed new die design with a converging shape outlet appears to have a smaller pressure drop compared to the current U.S. Army Armament Research, Development and Engineering Center (ARDEC) die shape. The optimal shape was obtained by finite-volume fluid dynamics computations through a range of die designs. The presented computations have been performed for a 3D die equipped with different outflow pipes. The features of the flow field are obtained for the non-Newtonian fluid through the apparatus. The change of fluid model from Newtonian to non-Newtonian complying power law does not make a considerable change in velocity profile at outlets for the same mass flow rate. Nevertheless, there is a substantial increase in the pressure gradient needed to transport the fluid through the die. For the new proposed die design, apparent viscosity steadily drops along the centerline of the outlet. As the viscosity magnitude determines the needed pressure drop, the new die design with a converging shape outlet has a substantially smaller pressure drop compared to the current die.
Astrophysical shocks at all scales, from those in the heliosphere up to the cosmological shock waves, are typically "collisionless", because the thickness of their jump region is much shorter than the collisional mean free path. Across these jumps, electrons, protons, and ions are expected to be heated at different temperatures. Supernova remnants (SNRs) are ideal targets to study collisionless processes because of their bright post-shock emission and fast shocks. Although optical observations of Balmer-dominated shocks in young SNRs showed that the post-shock proton temperature is higher than the electron temperature, the actual dependence of the post-shock temperature on the particle mass is still widely debated. We tackle this longstanding issue through the analysis of deep multi-epoch and high-resolution observations of the youngest nearby supernova remnant, SN 1987A, made with the Chandra X-ray telescope. We introduce a novel data analysis method by studying the observed spectra in close comparison with a dedicated full 3-D hydrodynamic simulation. The simulation is able to reproduce self-consistently the whole broadening of the spectral lines of many ions altogether. We can therefore measure the post shock temperature of protons and selected ions through comparison of the model with observations. We have obtained information about the heating processes in collisional shocks by finding that the ion to proton temperature ratio is always significantly higher than one and increases linearly with the ion mass for a wide range of masses and shock parameters.
We describe the diffusion and random velocities of solid particles due to stochastic forcing by turbulent gas. We include the orbital dynamics of Keplerian disks, both in-plane epicycles and vertical oscillations. We obtain a new result for the diffusion of solids. The Schmidt number (ratio of gas to particle diffusivity) is Sc = 1 + (Omega t_stop)^2, in terms of the particle stopping time, t_stop, and the orbital frequency, Omega. The standard result, Sc = 1 + t_stop/t_eddy, in terms of the eddy turnover time, t_eddy, is shown to be incorrect. The main difference is that Sc rises quadratically, not linearly, with stopping time. Consequently, particles larger than ~ 10 cm in protoplanetary disks will suffer less radial diffusion and will settle closer to the midplane. Such a layer of boulders would be more prone to gravitational collapse. Our predictions of RMS speeds, vertical scale height and diffusion coefficients will help interpret numerical simulations. We confirm previous results for the vertical stirring of particles (scale heights and random velocities), and add a correction for arbitrary ratios of eddy to orbital times. The particle layer becomes thinner for t_eddy > 1/Omega, with the strength of turbulent diffusion held fixed. We use two analytic techniques -- the Hinze-Tchen formalism and the Fokker-Planck equation with velocity diffusion -- with identical results when the regimes of validity overlap. We include simple physical arguments for the scaling of our results.
Turbulence has associated chaotic features. In the past couple of decades, there has been growing interest in the study of these features as an alternative means of understanding turbulent systems. Our own input to this effort is in contributing to the initial studies of chaos in Eulerian flow using direct numerical simulation (DNS). In this review, we discuss the progress achieved in the turbulence community in understanding chaotic measures including our own work. A central relation between turbulence and chaos is one by Ruelle that connects the maximum Lyapunov exponent and the Reynolds number. The first DNS studies, ours amongst them, in obtaining this relation have shown the viability of chaotic simulation studies of Eulerian flow. Such chaotic measures and associated simulation methodology provides an alternative means to probe turbulent flow. Building on this, we analyze the finite-time Lyapunov exponent (FTLE) and study its fluctuations; we find that chaotic measures could be quantified accurately even at small simulation box sizes where for comparative sizes spectral measures would be inconclusive. We further highlight applications of chaotic measures in analyzing phase transition behavior in turbulent flow and two-dimensional thin-layer turbulent systems. This work shows that chaotic measures are an excellent tool that can be used alongside spectral measures in studying turbulent flow.
The work explores the dynamics of a spherically symmetric perturbation of viscous modified Chaplygin gas (VMCG) in different gravity theories within the spherical top hat collapse framework (SC-TH). The study investigates the behaviour of perturbed quantities such as the δ, θ, w, w c , c s ², c e ², and h using numerical and graphical analysis. Our findings reveal that VMCG generates quintessential dark energy without crossing over to the phantom barrier in most of the gravity models considered here. Further, in all the gravity models considered here, VMCG remained classically stable. This research offers new insights into the evolution of VMCG in different gravitational contexts. In this paper, we have examined the collapse of viscous modified Chaplygin gas in the context of (i) Einstein’s gravity, (ii) Loop quantum cosmology, (iii) generalised Rastall gravity, and (iv) the fractal universe. We have also addressed their comparative analysis.
Material thermal conductivity is a key factor in various applications, from thermal management to energy harvesting. With microstructure engineering being a widely used method for customizing material properties, including thermal properties, understanding and controlling the role of extended phonon-scattering defects, like grain boundaries, is crucial for efficient material design. However, systematic studies are still lacking primarily due to limited tools. In this study, we demonstrate an approach for measuring grain boundary thermal resistance by probing the propagation of thermal waves across grain boundaries with a temperature-sensitive scanning probe. The method, implemented with a spatial resolution of about 100 nm on finely grained Nb-substituted SrTiO 3 ceramics, achieves a detectability of about 2 × 10 −8 K m 2 W −1 , suitable for chalcogenide-based thermoelectrics. The measurements indicated that the thermal resistance of the majority of grain boundaries in the STiO 3 ceramics is below this value. While there are challenges in improving sensitivity, considering spatial resolution and the amount of material involved in the detection, the sensitivity of the scanning probe method is comparable to that of optical thermoreflectance techniques, and the method opens up an avenue to characterize thermal resistance at the level of single grain boundaries and domain walls in a spectrum of microstructured materials.
The paramount importance of mechanical forces in morphogenesis and embryogenesis is widely recognized, but understanding the mechanism at the cellular and molecular level remains challenging. Because of its simple internal organization, Caenorhabditis elegans is a rewarding system of study. As demonstrated experimentally, after an initial period of steady elongation driven by the actomyosin network, muscle contractions operate a quasi-periodic sequence of bending, rotation, and torsion, that leads to the final fourfold size of the embryos before hatching. How actomyosin and muscles contribute to embryonic elongation is investigated here theoretically. A filamentary elastic model that converts stimuli generated by biochemical signals in the tissue into driving forces, explains embryonic deformation under actin bundles and muscle activity, and dictates mechanisms of late elongation based on the effects of energy conversion and dissipation. We quantify this dynamic transformation by stretches applied to a cylindrical structure that mimics the body shape in finite elasticity, obtaining good agreement and understanding of both wild-type and mutant embryos at all stages.
The formation of the electrolyte–electrode interface is essential for the performance of Li-ion batteries. This study aims to explore the wetting characteristics of an electrolyte within a porous electrode positioned between a current collector and a separator. By utilizing the Shan-Chen-based lattice Boltzmann method, an in-house code has been developed and thoroughly validated. This code integrates actual contact angles at the interfaces between the electrolyte and battery components. Furthermore, code acceleration through GPU-based parallel programming facilitates adequate meshing, underscoring the novelty and originality of our approach. The results of this study provide insights into the overall saturation curves and imbibition rates and clarify the primary mechanisms of electrolyte wetting within the porous matrix via local wetting rates. The electrode-current collector interface emerged as a critical factor influencing the imbibition rate and gas entrapment tendencies. Pore types at the interface have been categorized, focusing on how the material contact angle variations between the current collector and electrolyte influence wetting dynamics. Notably, it is observed that higher contact angles (90°) between the electrolyte and current collector increase the risk of trapping gas. Conversely, lower angles (15° and 35°) improve overall saturation; however, the enhancement of the wetting rate is particularly noticeable when interconnected pores are present at the interfaces of the electrode and battery components. This study underscores the combined influence of the separator and current collector in comprehending electrolyte wetting behavior, thus contributing to the advancement of battery technology.
Within micrometeorology the term modelling is not uniquely defined. It refers to various methods covering a range of complexity extending from simple regressions up to complicated numerical models. In applied meteorology (agrometeorology and hydrometeorology) simple analytical models are very common. Modelling of evaporation is particularly important but sophisticated numerical methods are not yet widely used in this research area.—The following section describes different types of models and their limitations beginning with simple analytical methods up to numerical models of near-surface energy and matter transport. Special attention is given to large-eddy simulation, which is increasingly used in urban boundary layer modelling, flow around wind turbines, and also in basic micrometeorological research. The application of models in heterogeneous terrain receives special attention and related flux averaging approaches are addressed in a separate section.
The present study aims to develop a parameter-free staggered-grid Lagrangian scheme that avoids the empirical parameters needed for artificial viscosity and anti-hourglass force. The artificial viscosity is equal to the pressure jump, which is constructed by the relationship between pressure and velocity. The anti-hourglass force is established using pressure compensation, which is the difference in pressure between sub-cells belonging to the same primary cell. The scheme maintains the conservation of total mass, momentum, and energy. Numerical experiments show that the scheme is highly robust for extreme flow problems with different fine meshes such as Sedov, Noh, Saltzmann triple point and Rayleigh–Taylor instability. The robust parameter-free scheme is well suited for multi-physics problems and engineering applications.
Code: https://github.com/xihua9s/CSRC
The power exchange between fluid and structure plays a significant role in the force production and flight efficiency of flapping wings in insects and artificial flyers. This work numerically investigates the performance of flapping wings by using a high-fidelity fluid-structure interaction solver. Simulations are conducted by varying the hinge flexibility (measured by the Cauchy number, Ch, i.e. the ratio between aerodynamic and torsional elastic forces) and the wing shape (quantified by the radius of the first moment of area, r_1. Results show that the lift production is optimal at 0.05 < Ch < 0.2 and larger r_1 where the minimum angle of attack is around 45 degree at midstroke. The power economy is maximised for wings with lower r_1 near Ch=0.2. Power analysis indicates that the optimal performance measured by the power economy is obtained for those cases where two important power synchronisations occur: anti-synchronisation of the pitching elastic power and the pitching aerodynamic and inertial powers and nearly in-phase synchronisation of the flapping aerodynamic power and the total input power of the system. While analysis of the kinematics for the different wing shapes and hinge stiffnesses reveals that the optimal performance occurs when the timing of pitch and stroke reversals are matched, thus supporting the effective transfer of input power from flapping to passive pitching and into the fluid. These results suggest that careful optimisation between wing shapes and hinge properties can allow insects and robots to exploit the passive dynamics to improve flight performance.
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