Leading-order Equations Research Articles

Boundary Layer Thickness and Friction Velocity by Symmetry Arguments.

We derive analytically for the first time the downstream evolution of the boundary layer thickness and the friction velocity of the zero-pressure-gradient turbulent boundary layer (ZPGTBL). Lie groups were used to derive the downstream evolution and to obtain the full set of the similarity variables and the leading-order similarity equations. An approximate leading-order solution was obtained using matched asymptotic expansions. The similarities and differences between ZPGTBL and turbulent channel flows in terms of the similarity equations are discussed to support the notion of leading-order universality of the near-wall layer.

Matrix-valued Allen–Cahn equation and the Keller–Rubinstein–Sternberg problem

In this paper, we consider the sharp interface limit of a matrix-valued Allen–Cahn equation, which takes the form: $$\begin{aligned} \partial _t{\textbf{A}}=\Delta {\textbf{A}}-\varepsilon ^{-2}({\textbf{A}}{\textbf{A}}^{\textrm{T}}{\textbf{A}}-{\textbf{A}})\quad \text {with}\quad {\textbf{A}}:\Omega \subset {\mathbb {R}}^m\rightarrow {\mathbb {R}}^{n\times n}. \end{aligned}$$ We show that the sharp interface system is a two-phases flow system: the interface evolves according to the motion by mean curvature; in the two bulk phase regions, the solution obeys the heat flow of harmonic maps with values in $$O^+(n)$$ and $$O^-(n)$$ (represent the sets of $$n\times n$$ orthogonal matrices with determinant $$+1$$ and $$-1$$ respectively); on the interface, the phase matrices on two sides satisfy a novel mixed boundary condition. The above result provides a solution to the Keller–Rubinstein–Sternberg’s problem in the O(n) setting. Our proof relies on two key ingredients. First, in order to construct the approximate solutions by matched asymptotic expansions, as the standard approach does not seem to work, we introduce the notion of quasi-minimal connecting orbits. They satisfy the usual leading order equations up to some small higher order terms. In addition, the linearized systems around these quasi-minimal orbits needs to be solvable up to some good remainders. These flexibilities are needed for the possible “degenerations” and higher dimensional kernels for the linearized operators on matrix-valued functions due to intriguing boundary conditions at the sharp interface. The second key point is to establish a spectral uniform lower bound estimate for the linearized operator around approximate solutions. To this end, we introduce additional decompositions to reduce the problem into the coercive estimates of several linearized operators for scalar functions and some singular product estimates which are accomplished by exploring special cancellation structures between eigenfunctions of these linearized operators.

A Blasius boundary layer study for liquids and gasses with variable thermo-physical characteristics and radiative heat flux

A numerical study is presented to discuss the temperature dependent viscosity and the thermo-physical properties near a heated flat surface. Temperature dependent viscosity for liquids show decrease in mean velocity profile and enhancement in mean temperature profile, whereas for gasses, it shows reverse effect. Moreover, the thermo-physical properties are treated as variable as they affect the flow phenomena. The basic flow can be modeled in terms of the scaling transformation for the large Reynolds number. Using similarity solutions, the leading order equations are converted into a set of ordinary differential equations which have been solved by fifth order Runga-Kutta integration method and Newton Raphson method. All of the obtained results are investigated for the base flow, concentration and temperature profiles and physical interpretation has been provided in the form figures and tables. The impact of viscosity for gases show an increase behavior in velocity profile but for liquids opposite impact is noticed. The variable conductivity and mass diffusivity parameters rise the temperature and concentration profiles. The temperature profile increases for increasing values of radiation parameter. Comparison has been made with past literature and the results show good agreement.

A viscous active shell theory of the cell cortex

The cell cortex is a thin layer beneath the plasma membrane that gives animal cells mechanical resistance and drives most of their shape changes, from migration, division to multicellular morphogenesis. It is mainly composed of actin filaments, actin binding proteins, and myosin molecular motors. Constantly stirred by myosin motors and under fast renewal, this material may be well described by viscous and contractile active-gel theories. Here, we assume that the cortex is a thin viscous shell with non-negligible curvature and use asymptotic expansions to find the leading-order equations describing its shape dynamics, starting from constitutive equations for an incompressible viscous active gel. Reducing the three-dimensional equations leads to a Koiter-like shell theory, where both resistance to stretching and bending rates are present. Constitutive equations are completed by a kinematical equation describing the evolution of the cortex thickness with turnover. We show that tension and moment resultants depend not only on the shell deformation rate and motor activity but also on the active turnover of the material, which may also exert either contractile or extensile stress. Using the finite-element method, we implement our theory numerically to study two biological examples of drastic cell shape changes: osmotic shocks and cell division. Our work provides a numerical implementation of thin active viscous layers and a generic theoretical framework to develop shell theories for slender active biological structures.

On the propagation of nonlinear waves in the atmosphere

Starting from the general equations of fluid dynamics that describe the atmosphere, and using asymptotic methods, we present the derivation of the leading-order equations for nonlinear wave propagation in the troposphere. The only simplifying assumption is that the flow in the atmosphere exists in a thin shell over a sphere. The systematic approach adopted here enables us to find a consistent balance of terms describing the propagation, and to identify the temperature and pressure gradients that drive the motion, as well as the heat sources required. This produces a new nonlinear propagation equation that is then examined in some detail. With the morning glory in mind, we construct a few exact solutions, which, separately, describe breezes, bores and oscillatory motion.

Electrohydrodynamic instability of viscous liquid films: Electrostatically modified second-order Benney and Kuramoto–Sivashinsky models

Long wave evolution on viscous films flowing under gravity down an inclined plate in the presence of an electrostatic field acting normal to the plate is described using new nonlinear models. First, an asymptotic expansion is applied to derive a strongly nonlinear evolution equation for the interfacial position in the long-wave limit, which retains terms up to second order in a small film parameter (ε) and thus brings in dispersive effects, inclination-angle corrections for hydrostatic pressure, second-order contributions to inertia and electric stresses as well as the interaction of the last two. Two weakly nonlinear models (WNMs) for the disturbances with O(ε2) amplitude are then derived, which account for distinct effects of hydrostatic pressure, inertia and dispersion for shallow and steep angles. We compare the linear stability results based on the second-order electrified Benney equation (BE) and the WNMs with realistic parameter values. It is shown that the growth rates coincide, while the phase speed from the WNMs is a good approximation of that from the BE for small wavenumbers only. The comparison of exact dispersion relations from the Orr–Sommerfeld (OS) problem with the linear results of the second-order BE takes a step towards understanding the validity range of the BE with non-local terms that approximates the full electrohydrodynamic coupling system describing the inclined electrified film flow. The pertinent OS results demonstrate that a decrease in electrode distance can diminish the critical Reynolds number (Re). Furthermore, at subcritical Re with a slightly supercritical electric field, the finite wavelength of the least stable mode has been estimated using an energy balance, whose amplitude will be governed by two coupled Ginzburg–Landau (GL) equations, derived using a multiple-scale analysis by taking higher-order problems into account. The numerical solutions of the leading-order GL equation demonstrate that close to the bifurcation, it can fully govern the temporal behaviour of the system.

Singularities in nearly uniform one-dimensional condensates due to quantum diffusion

Dissipative systems can often exhibit wavelength-dependent loss rates. One prominent example is Rydberg polaritons formed by electromagnetically-induced transparency, which have long been a leading candidate for studying the physics of interacting photons and also hold promise as a platform for quantum information. In this system, dissipation is in the form of quantum diffusion, i.e., proportional to $k^2$ ($k$ being the wavevector) and vanishing at long wavelengths as $k\to 0$. Here, we show that one-dimensional condensates subject to this type of loss are unstable to long-wavelength density fluctuations in an unusual manner: after a prolonged period in which the condensate appears to relax to a uniform state, local depleted regions quickly form and spread ballistically throughout the system. We connect this behavior to the leading-order equation for the nearly-uniform condensate -- a dispersive analogue to the Kardar-Parisi-Zhang (KPZ) equation -- which develops singularities in finite time. Furthermore, we show that the wavefronts of the depleted regions are described by purely dissipative solitons within a pair of hydrodynamic equations, with no counterpart in lossless condensates. We close by discussing conditions under which such singularities and the resulting solitons can be physically realized.

On a consistent rod theory for a linearized anisotropic elastic material II. Verification and parametric study

We have derived a rod theory by an asymptotic reduction method for a straight and circular rod composed of linearized anisotropic material in part I of this series. In the current work, we first verify the derived rod theory through five benchmark Saint-Venant’s problems. Then, under a specific loading condition (line force at the lateral surface with two clamped ends), we apply the rod theory to conduct a parametric study of the effects of elastic moduli on the deformation of a rod composed of four types of anisotropic materials including cubic, transversely isotropic, orthotropic, and monoclinic materials. Analytical solutions for the displacement, axial twist angle, stress, and principal stress have been obtained and a systematic investigation of the effects of elastic moduli on these quantities is conducted, which is the main feature of this paper. It is found that these elastic moduli arise in a certain form and in a certain order in the solutions, which gives information about how to appropriately choose moduli to adjust the deformation. Among the four anisotropic materials, it turns out that the monoclinic material presents the most remarkable mechanical behavior owing to the existence of a coupling coefficient: it yields coupled leading-order rod equations, non-trivial axial twist angle, non-negligible transverse shear deformation, and a more adjustable principal stress along the axis.

On the propagation of waves in the atmosphere.

The leading-order equations governing the unsteady dynamics of large-scale atmospheric motions are derived, via a systematic asymptotic approach based on the thin-shell approximation applied to the ellipsoidal model of the Earth’s geoid. We present some solutions of this single set of equations that capture properties of specific atmospheric flows, using field data to choose models for the heat sources that drive the motion. In particular, we describe standing-waves solutions, waves propagating towards the Equator, equatorially trapped waves and we discuss the African Easterly Jet/Waves. This work aims to show the benefits of a systematic analysis based on the governing equations of fluid dynamics.

Prescaling and far-from-equilibrium hydrodynamics in the quark-gluon plasma

We use the leading-order QCD kinetic equations with quarks and gluons to study the quark-gluon plasma in the far-from-equilibrium state. At weak coupling, we demonstrate the emergence of early hydrodynamic behavior—prescaling—around a nonthermal fixed point, which is qualitatively different from the more conventional hydrodynamics around equilibrium. Prescaling is a far-from-equilibrium phenomenon which describes the rapid establishment of a universal scaling form of distributions much before the universal values of their scaling exponents are realized. The dynamics in the prescaling regime is encoded in a few time-dependent scaling exponents, whose slow evolution gives rise to far-from-equilibrium hydrodynamic description.

Turbulent flow through a random rigid submerged vegetation over a sinusoidal bed

The present study investigates turbulent flow over a periodic small amplitude bed form with rigid submerged vegetation. The equations governing the flow are formulated using Biot’s poro-elasticity theory. Further, the Reynolds stress is modeled employing a linear term of the velocity gradient. Governing equations for vegetated and non-vegetated layers are non-dimensionalized and a perturbation parameter is identified. The leading order equation is solved employing power series method and first order equation is solved semi-analytically. The velocity components in the flow and transverse directions and the vertical shear stress distribution are evaluated. The velocity measurements are also carried out within the array over the perturbed bed using acoustic Doppler velocimeter. The experimental results are utilized to develop a correlation for the unknown coefficient present in the shear stress model, which is adopted for modelling the Reynolds stress.

Nonlinear states and dynamics in a synthetic frequency dimension

Recent advances in the study of synthetic dimensions revealed a possibility to employ the frequency space as an additional degree of freedom which allows for investigating and exploiting higher-dimensional phenomena in a priori low-dimensional systems. However, the influence of nonlinear effects on the synthetic frequency dimensions was studied only under significant restrictions. In the present paper, we develop a generalized mean-field model for the optical field envelope inside a single driven-dissipative resonator with quadratic and cubic nonlinearities, whose frequencies are coupled via an electro-optical resonant temporal modulation. The leading-order equation takes the form of a driven Gross-Pitaevskii equation with a cosine potential. We numerically investigate the nonlinear dynamics in such a microring resonator with a synthetic frequency dimension in the regime where parametric frequency conversion occurs. We observe that the modulation brings additional control to the system, enabling one to readily create and manipulate bright and dark dissipative solitons inside the cavity. In the case of anomalous dispersion, we find that the presence of electro-optical mode coupling confines and stabilizes the chaotic modulation instability region. This leads to the appearance of an unconventional type of stable coherent structure which emerges in the synthetic space with restored translational symmetry, in a region of parameters where conventionally only chaotic modulation instability states exist. This structure appears in the center of the synthetic band and, therefore, is referred to as the band soliton. Finally, we extend our results to the case of multiple modulation frequencies with controllable relative phases creating synthetic lattices with nontrivial geometry. We show that an asymmetric synthetic band leads to the coexistence of chaotic and coherent states of the electromagnetic field inside the cavity, i.e., dynamics that can be interpreted as chimeralike states. Recently developed ${\ensuremath{\chi}}^{(2)}$ microresonators can open the way to experimentally exploring our findings.

High Weissenberg number boundary layer structures for UCM fluids

We describe three distinct stress boundary layer structures that can arise in the high Weissenberg number limit for the upper convected Maxwell (UCM) model. One is a single layer structure previously noted by M. Renardy, High Weissenberg number boundary layers for the upper convected Maxwell fluid J. Non-Newtonian Fluid Mech.68 (1997), 125–132. The other two are double layer structures. These latter two structures extend the core flows that can be accommodated by the UCM model in the high Weissenberg regime. The three structures taken together, represent the main dominant balances that occur for the UCM equations near solid boundaries. For each structure, the leading order equations are derived in each region together with particular exact solutions when available. Importantly, the matching conditions between respective regions for each structure are also derived and explained. These stress boundary layers can arise in order one Reynolds number flows and are independent of the velocity boundary layers that can arise in high Reynolds number flows.

Dynamic analysis of large strain deformation of flexible pipes conveying two-phase fluids Part I: linear vibration analysis

This work presents dynamic response of large strain deformed subsea flowlines and jumpers conveying two-phase fluid. Large strain deformation theory is used to analyze the effects of high pressure-high temperature two-phase flow on critical velocity in flexible pipes. Starting with natural frequency of simply supported flexible pipes, corresponding nonlinear transport equations are derived and subsequently expanded using the method of multiple scales perturbation. Frequency response obtained from the linearized leading order equations established that buckling and flutter-like instabilities attend critical velocities. Furthermore, the buckling velocity of transverse pipe decreases with increasing temperature, pressure or tension. Clearly, the transverse buckling velocity is higher in large strain deformation model compared with small strain model. In addition; whereas the critical velocity in the longitudinal direction is independent of temperature, pressure and tension for small strain models; these variables determine the path to failure in large strain deformed pipes.Keywords: Large strain deformation, small strain deformation, simply supported pipe

A singular vorticity wave packet within a rapidly rotating vortex: spiralling versus oscillating motions

This paper considers a free vorticity wave packet propagating within a rapidly rotating vortex in the quasi-steady regime, a long time after the wave packet strongly and unsteadily interacted with the vortex. We study a singular, nonlinear, helical and asymmetric shear mode inside a linearly stable, columnar and axisymmetric vortex on the $f$-plane. The amplitude-modulated mode enters resonance with the vortex at a certain radius $r_{c}$, where the phase angular speed is equal to the rotation frequency. The singularity in the modal equation at $r_{c}$ strongly modifies the flow in the three-dimensional helical critical layer, the region around $r_{c}$ where the wave/vortex interaction occurs. This interaction generates a vertically sheared three-dimensional mean flow of higher amplitude than the wave packet. The chosen envelope regime assumes the formation of a mean radial velocity of the same order as the wave packet amplitude, leading to the streamlines exhibiting a spiral motion in the neighbourhood of the critical layer. Radar images frequently show such spiral bands in tropical cyclones or tornadoes. Through matched asymptotic expansions, we find an analytical solution of the leading-order equations inside the critical layer. The generalized Batchelor integral condition applied to the quasi-steady, three-dimensional motion inside the separatrices yields a leading-order, non-uniform three-dimensional vorticity. The critical-layer pattern, strongly deformed by the mean radial velocity, loses its symmetries with respect to the azimuthal and radial directions, which makes the leading-order mean radial wave fluxes non-zero. Finally, a stronger wave/vortex interaction occurs with respect to previous studies where a steady neutral vortical mode or an envelope of larger extent was involved.

QCD evolution of superfast quarks

Recent high-precision measurements of nuclear deep inelastic scattering at high x and moderate 6 < Q$^2$ < 9GeV$^2$ give a rare opportunity to reach the quark distributions in the {\it superfast} region, in which the momentum fraction of the nucleon carried by its constituent quark is larger than the total fraction of the nucleon at rest, x>1. We derive the leading-order QCD evolution equation for such quarks with the goal of relating the moderate-Q$^2$ data to the two earlier measurements of superfast quark distributions at large 60 < Q$^2$ < 200~GeV$^2$. Since the high-Q$^2$ measurements gave strongly contradictory estimates of the nuclear effects that generate superfast quarks, relating them to the high-precision, moderate-Q$^2$ data through QCD evolution allows us to clarify this longstanding issue. Our calculations indicate that the moderate-Q$^2$ data at $x\lesssim 1.05$ are in better agreement with the high-Q$^2$ data measured in (anti)neutrino-nuclear reactions which require substantial high-momentum nuclear effects in the generation of superfast quarks. Our prediction for the high-Q$^2$ and x>1.1 region is somewhat in the middle of the neutrino-nuclear and muon-nuclear scattering data.

Gravitating scalar stars in the large D limit

We develop the large D limit of general relativity for spherically symmetric scalar fields in both asymptotically flat and asymptotically anti-de Sitter spaces. The leading order equations in the 1/D expansion can be solved analytically, providing a large D description of oscillating soliton stars. When the amplitude reaches a critical threshold, certain divergences occur which we interpret as signal of horizon formation. We estimate the size of the resulting black hole and obtain a scaling exponent. We speculate on some connections to Choptuik critical collapse.

On the phase structure of vector-matrix scalar model in four dimensions

The leading-order equations of the 1/N – expansion for a vector-matrix model with interaction gphi _a^*phi _bchi _{ab} in four dimensions are investigated. This investigation shows a change of the asymptotic behavior in the deep Euclidean region in a vicinity of a certain critical value of the coupling constant. For small values of the coupling the phion propagator behaves as free. In the strong-coupling region the asymptotic behavior drastically changes – the propagator in the deep Euclidean region tend to some constant limit. The phion propagator in the coordinate space has a characteristic shell structure. At the critical value of coupling that separates the weak and strong coupling regions, the asymptotic behavior of the phion propagator is a medium among the free behavior and the constant-type behavior in strong-coupling region. The equation for a vertex with zero transfer is also investigated. The asymptotic behavior of the solutions shows the finiteness of the charge renormalization constant. In the strong-coupling region, the solution for the vertex has the same shell structure in coordinate space as the phion propagator. An analogy between the phase transition in this model and the re-arrangement of the physical vacuum in the supercritical external field due to the “fall-on-the-center” phenomenon is discussed.

Energy density from second shape variations of the von Neumann entropy

We compute the local second variation of the von Neumann entropy of a region in theories with a gravity dual. For null variations our formula says that the diagonal part of the quantum null energy condition (QNEC) is saturated in every state, thus providing an equivalence between energy and entropy. We prove that the formula holds at leading order in $1/N$ and further argue that it will not be affected at higher orders. We conjecture that the QNEC is saturated in all interacting theories. We also discuss the special case of free theories, and the implications of our formula for the averaged null energy condition, quantum focusing conjecture (QFC), and gravitational equations of motion. We show that the leading-order gravitational equations of motion, Einstein's equations, are equivalent to the leading-order saturation of the QFC for Planck-width deformations.

Three dimensional fluid-kinetic model of a magnetically guided plasma jet

A fluid-kinetic model of the collisionless plasma flow in a convergent-divergent magnetic nozzle is presented. The model combines the leading-order Vlasov equation and the fluid continuity and perpendicular momentum equation for magnetized electrons, and the fluid equations for cold ions, which must be solved iteratively to determine the self-consistent plasma response in a three-dimensional magnetic field. The kinetic electron solution identifies three electron populations and provides the plasma density and pressure tensor. The far downstream asymptotic behavior shows the anisotropic cooling of the electron populations. The fluid equations determine the electric potential and the fluid velocities. In the small ion-sound gyroradius case, the solution is constructed one magnetic line at a time. In the large ion-sound gyroradius case, ion detachment from magnetic lines makes the problem fully three-dimensional.