Flux Law Research Articles

Rayleigh Waves in a Thermoelastic Half-Space Coated by a Maxwell–Cattaneo Thermoelastic Layer

This paper investigates the propagation of in-plane surface waves in a coated thermoelastic half-space. First, it investigates a special case where the surface layer is described by the Maxwell–Cattaneo thermoelastic approach, while the half-space is filled by a thermoelastic material described by the classical Fourier law for the heat flux. The contact between the layer and the half-space is assumed to be welded, i.e., the displacements and the temperature, as well as the stresses and the heat flux are continuous through the interface of the layer and the half-space. The boundary and continuity conditions of the problem are formulated and then the exact dispersion relation of the surface waves is established. An illustrative numerical simulation is presented for the case of an aluminum thermoelastic layer coating a thermoelastic copper half-space, highlighting important aspects regarding the propagation of Rayleigh waves in such structures. The exact effective boundary conditions at the interface are also established replacing the entire effect of the layer on the half-space. The general case of the problem is also investigated when both the surface layer and the half-space are described by the Maxwell–Cattaneo thermoelasticity theory. This study helps to further understand the propagation characteristics of elastic waves in layered structures with thermal effects described by the Maxwell–Cattaneo approach.

Necessary and sufficient conditions for Kolmogorov’s flux laws on T2 and T3

Necessary and sufficient conditions for the third order Kolmogorov universal scaling flux laws are derived for the stochastically forced incompressible Navier Stokes equations on the torus in 2D and 3D. This paper rigorously generalises the result of (Bedrossian 2019 Commun. Math. Phys. 367 1045–75) to functions which are heavy-tailed in Fourier space or have local finite time singularities in the inviscid limit. In other words, we have rigorously derived the existence of the well known physical relationships, the direct and inverse cascades. Furthermore we show that the rate of the direct cascade is proportional to the amount of energy ‘escaping to infinity’ in spectral space as well as a measure of the total singularities within the solution. Similarly, an inverse cascade is proportional to the amount of energy that moves towards the k = 0 Fourier mode in the invisicid limit.

Hydrodynamic performance of multi-chamber oscillating water columns in a caisson array

A theoretical model based on the linear potential flow theory and eigenfunction expansion-matching method is developed to analyze the interaction between water waves and multi-chamber oscillating water columns (OWCs) embedded in a caisson array. The semi-analytical solution consists of the diffraction and radiation components of periodic OWCs. The velocity singularity at the tip of the OWC chamber walls is resolved by implementing a Chebyshev polynomial expansion. The unknown expansion coefficients are determined by the continuity equations of velocity and pressure at the interface of subdomains. The semi-analytical solution is verified using the Haskind relation and the conservation law of wave energy flux. It is found that multi-chamber OWCs have higher hydrodynamic efficiency over a broader frequency bandwidth than traditional single- and dual-chamber OWCs. The maximum hydrodynamic efficiency η increases with increasing of chamber number (J), from 0.5 (J = 1) to nearly 1.0 (J = 8 and 12). It is also found that a larger incident wave angle leads to a narrower frequency bandwidth for energy capturing. Increasing the incident wave angle from π/3 to 5π/12 results in a decrease of the first cutoff frequency kc from 2.41 to 2.02, and therefore results in the energy capture bandwidth narrowing accordingly. Both constructive and destructive hydrodynamic interactions between caissons array and OWCs are observed over the tested frequency range. Interestingly, when configuration of the OWC chambers is reversed, the transmission coefficient remains unchanged.

Optimal permeance ratio, flux direction and layer distribution in composite asymmetric membranes composed of sequences of layers obeying real-power flux laws

In this study, the optimal permeance ratio and layers distribution in multilayer membranes and thin films in which each layer obeys the following real-power law: Flux=πi(P1,ini−P2,ini) are determined for any positive real values of ni. Such an exponent has a physical meaning in several cases, such as: diffusion-controlled permeation of hydrogen in metal membranes i) under infinite-dilution conditions (n = 0.5), and ii) under higher-concentration ones (0.5 < n < 1.5), iii) desorption-limited permeation (n = 1) in the same type of membranes, Knudsen-type diffusional mechanism (n = 1) and viscous flow (n = 2) in microporous membranes. Furthermore, n is equal to 4 in radiation-driven thermal flux. The work considers two generic layers of different thicknesses and exponents, for which it is proven the existence of a general optimal permeance ratio π1/π2 maximising the flux ratio, which is found to be equal to the inverse of the ratio between the respective maximum theoretical driving forces that can be established in each layer. Such an optimal permeance ratio is also proven to be a global optimum. Afterwards, we provide a theoretical generalisation of this optimality to whatever types of driving forces. Then, we provide a demonstration that the highest species permeating flux is always established when feeding it from the layer side with the highest exponent independently of the permeance ratio. As an important consequence, to maximise flux, a multilayer membrane/thin film should be fabricated by depositing the layers in cascade according to the ascending or descending exponents, and should be used in a configuration in which the flux flows from the layer with the highest pressure exponent to the layer with the lowest one. It is also shown that there exists an optimal value of the lower exponent n1 that maximizes the flux ratio keeping constant the other parameters. Finally, we showed that a virial-type linear combination of polynomial driving forces can be approximated by the empirical flux law investigated. Overall, our results are valid not only for whatever type of composite membranes and thin films (metallic, ceramic, polymeric and hybrid ones), but also for any sequential system involving a flux of whatever physical entity obeying the aforementioned law, such as heat transfer in the presence of radiation, non-linear electrical resistances and possible others.

Optimizing the noise control in a two-layer conduit

In the modern world, noise pollution is a major concern due to its prevalence. This work focuses on optimizing noise control in a two-layer conduit. A conduit comprises an inlet and double-outlet zones (DOZ). The upper walls of the DOZ are lined with perforated absorbent material while the lower walls are layered with fibrous. Mathematically, the physical problem is formulated with a field wave equation together with rigid and impedance boundary conditions in the respective zones. Such governing boundary value problem (BVP) leads to the Sturm-Liouville (SL) category in which standard orthogonality relations (OR) are indispensable. The system of the linear equations of the BVP is acquired with semi-analytical Mode-Matching (MM) approach by implementing the continuity conditions of sound pressures and velocities at the matching junction with the aid of OR. These systems are truncated and solved numerically with computation learning to obtain the reflected and transmitted modal amplitudes in respective zones. Due to the lining's perforated upper walls, fibrous lower walls, and reversal in the DOZ, the analysis of the reflected and transmitted powers versus frequencies is significantly observed and is shown in graphical findings. The algebraic derivation is validated by satisfying the conservation law of power flux and matching continuity conditions for impedance, perforated, and fibrous lined boundaries.

Mathematical Model and Stability Analysis for Dusty-hybrid Nanofluid over a Curved Surface with Cattaneo–Christov Heat Flux and Fourier law with Slip Effect

Mathematical Model and Stability Analysis for Dusty-hybrid Nanofluid over a Curved Surface with Cattaneo–Christov Heat Flux and Fourier law with Slip Effect

A note on cascade flux laws for the stochastically-driven nonlinear Schrödinger equation

In this note we point out some simple sufficient (plausible) conditions for ‘turbulence’ cascades in suitable limits of damped, stochastically-driven nonlinear Schrödinger equation in a d-dimensional periodic box. Simple characterizations of dissipation anomalies for the wave action and kinetic energy in rough analogy with those that arise for fully developed turbulence in the 2D Navier–Stokes equations are given and sufficient conditions are given which differentiate between a ‘weak’ turbulence regime and a ‘strong’ turbulence regime. The proofs are relatively straightforward once the statements are identified, but we hope that it might be useful for thinking about mathematically precise formulations of the statistically-stationary wave turbulence problem.

The Kuznetsov and Blackstock Equations of Nonlinear Acoustics with Nonlocal-in-Time Dissipation

In ultrasonics, nonlocal quasilinear wave equations arise when taking into account a class of heat flux laws of Gurtin–Pipkin type within the system of governing equations of sound motion. The present study extends previous work by the authors to incorporate nonlocal acoustic wave equations with quadratic gradient nonlinearities which require a new approach in the energy analysis. More precisely, we investigate the Kuznetsov and Blackstock equations with dissipation of fractional type and identify a minimal set of assumptions on the memory kernel needed for each equation. In particular, we discuss the physically relevant examples of Abel and Mittag–Leffler kernels. We perform the well-posedness analysis uniformly with respect to a small parameter on which the kernels depend and which can be interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the limiting behavior of solutions with respect to this parameter, and how it is influenced by the specific class of memory kernels at hand. Through such a limiting study, we relate the considered nonlocal quasilinear equations to their limiting counterparts and establish the convergence rates of the respective solutions in the energy norm.

Thermal convection with a Cattaneo heat flux model

The problem of thermal convection in a layer of viscous incompressible fluid is analysed. The heat flux law is taken to be one of Cattaneo type. The time derivative of the heat flux is allowed to be a material derivative, or a general objective derivative. It is shown that only one objective derivative leads to results consistent with what one expects in real life. This objective derivative leads to a Cattaneo–Christov theory, and the results for linear instability theory are in agreement with those for a material derivative. It is further shown that none of the theories allow a standard nonlinear, energy stability analysis. A further heat flux due to P.M. Mariano is added and then an analysis is performed for stationary convection, oscillatory convection, and fully nonlinear theory. For the material derivative case, the analysis proceeds and global nonlinear stability is achieved. For Cattaneo–Christov theory, it appears necessary to add a regularization term in the equation for the heat flux, and even then the analysis only works in two space dimensions, and is conditional upon the size of the initial data. For the three-dimensional situation, it is shown how a nonlinear stability analysis may be achieved with a Navier–Stokes–Voigt fluid rather than a Navier–Stokes one.

First-Principle Validation of Fourier's Law: One-Dimensional Classical Inertial Heisenberg Model.

The thermal conductance of a one-dimensional classical inertial Heisenberg model of linear size L is computed, considering the first and last particles in thermal contact with heat baths at higher and lower temperatures, Th and Tl (Th>Tl), respectively. These particles at the extremities of the chain are subjected to standard Langevin dynamics, whereas all remaining rotators (i=2,⋯,L-1) interact by means of nearest-neighbor ferromagnetic couplings and evolve in time following their own equations of motion, being investigated numerically through molecular-dynamics numerical simulations. Fourier's law for the heat flux is verified numerically, with the thermal conductivity becoming independent of the lattice size in the limit L→∞, scaling with the temperature, as κ(T)∼T-2.25, where T=(Th+Tl)/2. Moreover, the thermal conductance, σ(L,T)≡κ(T)/L, is well-fitted by a function, which is typical of nonextensive statistical mechanics, according to σ(L,T)=Aexpq(-Bxη), where A and B are constants, x=L0.475T, q=2.28±0.04, and η=2.88±0.04.

Buoyancy driven convection with a Cattaneo flux model

Abstract We review models for convective motion which have a flux law of Cattaneo type. This includes thermal convection where the heat flux law is a Cattaneo one. We additionally analyse models where the convective motion is due to a density gradient caused by a concentration of solute. The usual Fick’s law in this case is replaced by a Cattaneo one involving the flux of solute and the concentration gradient. Other effects such as rotation, the presence of a magnetic field, Guyer–Krumhansl terms, or Kelvin–Voigt theories are briefly introduced.

On a Cahn–Hilliard system with source term and thermal memory

A nonisothermal phase field system of Cahn–Hilliard type is introduced and analyzed mathematically. The system constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn–Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter is lost. We provide several mathematical results under general assumptions on the source term and the double-well nonlinearity governing the evolution: existence and continuous dependence results are shown for weak and strong solutions to the corresponding initial-boundary value problem.

Convective heat and mass transfer analysis on Casson nanofluid flow over an inclined permeable expanding surface with modified heat flux and activation energy

Due to the enormous advantages of nanofluid technology, the study of various nanofluid flows under different assumptions became an important research topic. Furthermore, Fourier proposed the basic heat flux law, and it has some initial disturbances. Later, Cattaneo and Christov modified the Fourier model by including the relaxation term and rectified the conflicts in the Fourier model. Current research explores the influence of activation energy on a chemically reactive Casson nanofluid over a stretching surface. Cattaneo–Christov heat and mass flux are considered for the investigation. The governing PDEs (partial differential equations) along with the boundary constraints are transformed into ODEs (ordinary differential equations) with the aid of suitable similarity variables and then solved numerically by the bvp5c MATLAB package. We have drawn the velocity, temperature, and concentration profiles. Also, plotted the isotherms and streamlines, and analyzed the effects of all the influential parameters on the flow regime. The increasing Casson fluid parameter leads to a decrease in the fluid velocity. Larger value of the Casson fluid parameter () recovers the results for the Newtonian fluid. The results obtained from the study show that the temperature shoots up for intensifying values of the magnetic parameter, radiation parameter, Eckert number, thermal Biot number, Brownian motion parameter, and the thermophoresis parameter; meanwhile the increase in the thermal relaxation parameter drops the temperature. The positive values of the heat source parameter decreases the rate of heat transfer by 10%. Also, we noticed that the effect of the growing values of the Schmidt number and the chemical reaction parameter is to decrease the concentration field. Moreover, a rise in the Reynolds and Brinkmann numbers pushes up the entropy generation.

A generalized thermal memory effect of Caputo–Fabrizio fractional integral on natural convection flow of hybrid nanofluid

In this paper, the natural convective flow of hybrid nanofluid over the vertical plate has been examined. Cattaneo law of thermal flux is used to characterize thermal transport. The novel model for fractional constitutive equation is expressed by the time fractional Caputo–Fabrizio integral. The analytical solution of the generalized convective flow of viscous fluid over the vertical plate along generalized conditions is obtained by using the Laplace transform. Mathcad is used to determine numerically the influence of the physical and fractional parameters on the temperature and velocity field to depict the results graphically.

The vanishing relaxation time behavior of multi-term nonlocal Jordan–Moore–Gibson–Thompson equations

The family of Jordan–Moore–Gibson–Thompson (JMGT) equations arises in nonlinear acoustics when a relaxed version of the heat flux law is employed within the system of governing equations of sound motion. Motivated by the propagation of sound waves in complex media with anomalous diffusion, we consider here a generalized class of such equations involving two (weakly) singular memory kernels in the principal and non-leading terms. To relate them to the second-order wave equations, we investigate their vanishing relaxation time behavior. The key component of this singular limit analysis are the uniform bounds for the solutions of these nonlinear equations of fractional type with respect to the relaxation time. Their availability turns out to depend not only on the regularity and coercivity properties of the two kernels, but also on their behavior relative to each other and the type of nonlinearity present in the equations.

Optimal Temperature Distribution for a Nonisothermal Cahn–Hilliard System with Source Term

In this note, we study the optimal control of a nonisothermal phase field system of Cahn–Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. The system couples a Cahn–Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter, typical of the classic Cahn–Hilliard equation, is no longer satisfied. In this paper, we analyze the case that the double-well potential driving the evolution of the phase transition is differentiable, either (in the regular case) on the whole set of reals or (in the singular logarithmic case) on a finite open interval; nondifferentiable cases like the double obstacle potential are excluded from the analysis. We prove the Fréchet differentiability of the control-to-state operator between suitable Banach spaces for both the regular and the logarithmic cases and establish the solvability of the corresponding adjoint systems in order to derive the associated first-order necessary optimality conditions for the optimal control problem. Crucial for the whole analysis to work is the boundedness property stating that the order parameter attains its values in a compact subset of the interior of the effective domain of the nonlinearity. While this property turns out to be generally valid for regular potentials in three dimensions of space, it can be shown for the logarithmic case only in two dimensions.

The generalized Fourier’s and Fick laws effects on MHD free convection flows of Maxwell fluids by employing Caputo–Fabrizio time-fractional integral

The relevance of time-dependent magneto-free convection and its consequences for mass and energy transport are being increasingly understood in science. Unfortunately, very little is known about how the fractional generalized technique would affect a complete analysis of Maxwell fluid dynamics over a porous plate. Using the Caputo–Fabrizio time-fractional integral, the Fourier thermal flux law and the fractionally generalized Fick’s equation of mass flow are both generalized. Using the appropriate similarity transformations allows us to characterize the structured governing equations, which are nondimensionalized. The dimensionless energy, concentration, and velocity distribution problem is solved using the Laplace transform method. The graph demonstrates how physical and fractional parameters are affected. Fractional derivatives may be employed to accurately represent the rheology of such fluids. The Maxwell generalized fluid across an oscillating sheet was studied by Zheng et al. 3

Magnetic Discontinuities in the Inner Heliosphere: Do Intermediate Shocks Exist?

Magnetic discontinuities are fundamental structures in space and laboratory plasmas where the changes in magnetic and velocity fields are constrained by Rankine–Hugoniot relations. Due to the absence of precise measurements for particles, some issues therein are hardly investigated. The nature of discontinuities driven by the magnetohydrodynamics (MHD) turbulence, and the intermediate shock are two puzzles to be solved. The MHD turbulence generates numerous discontinuities with both small normal magnetic fields and nearly constant magnetic field magnitudes in statistics. By utilizing the data from the Parker Solar Probe, we identify among the turbulence-driven discontinuities two components that exhibit diverse statistical characteristics of the plasma density, and reveal that these discontinuities comprise 80.2% rotational and 19.8% tangential discontinuities. Then, we note a special class of discontinuities within 0.35 au that have jump conditions similar to that of the rotational discontinuity and the shock simultaneously, including (1) positively correlated jumps in the plasma density and temperature, (2) a small change in the magnetic field magnitude, and (3) opposite tangential magnetic fields on two sides. These features conform to the theoretical intermediate shock, which previous studies have found to not practically exist due to the breakdown of the evolutionary condition. By the conservation law of the mass flux across a boundary, we calculate their propagation speeds and find three intermediate shock candidates with super-Alfvénic upstream and sub-Alfvénic downstream flows. This work can improve our understanding of plasma intermittencies and suggests reassessing conclusions based on ideal MHD Rankine–Hugoniot relations.

Flexural gravity wave interaction with an articulated heterogeneous plate within the paradigm of blocking dynamics

Surface gravity waves interact with the flexural waves to generate the flexural gravity waves whose characteristics are triggered for higher values of lateral compressive stress to generate multiple propagating wave modes. This investigation examines the scattering of obliquely incident flexural gravity waves due to articulation in two semi-infinite heterogeneous floating elastic plates in finite water depth within a blocking dynamics regime. The dispersion curve demonstrates the existence of three propagating wave modes within the blocking limits. The canonical eigenfunction expansion method used for a single propagating mode is generalized to account for multiple propagating wave modes within the limits of blocking periods. The energy relation is established using the conservation of wave energy flux and Snell's law of refraction, which depends upon the angles and amplitude of the scattered waves along with the wave energy transfer rates. The amplitude of scattering coefficients (energy transfer rate) goes beyond the unit, where the corresponding energy transfer rate (scattering coefficients) diminishes for specific wave periods. Subsequently, complete wave reflection occurs for oblique waves beyond a critical angle of incidence for a fixed period and prior to a critical angle of incidence at a higher angle of incidence. Removable discontinuities occur at the blocking and saddle points, while a jump discontinuity appears due to a change in the incident wave mode in the scattering coefficients. Surface plots reveal the irregular pattern of plate deflection for the period within the blocking limits. Linear time-dependent plate displacement is simulated in two and three dimensions.

Analytical solutions of fractal and anomalous diffusion models for pressure and rate transient analysis

In this study, we examine the modeling of the naturally fractured reservoirs based on the fractal, Metzler, and Raghavan anomalous diffusion models which are different from Darcy flux law. The governing equations of these models are presented for a vertical well producing at either a constant rate (CR) or constant bottomhole pressure (CBHP) production in an infinite-acting reservoir. New analytical and early- and late-time asymptotic approximate solutions were derived for the finite-wellbore problem for the Raghavan anomalous model. The approximate solutions identify the fractal and anomalous diffusion parameters that influence behaviors of the early- and late-time pressure and rate transient data. The method of Laplace transformation is used to find the solutions in dimensionless forms which are then inverted to the real-time domain using a numerical inversion algorithm. In addition, we compare the solutions for the fractal, Metzler anomalous, and Raghavan anomalous models and delineate the similarities and differences among these solutions. We show the effect of fractal parameters on diffusion models, the relationship between the fractal model, and Raghavan and Metzler anomalous models, in terms of dimensionless pressure and dimensionless pressure derivative responses in the reservoir. We show that the “dimensionless” pressure defined by Raghavan is not a dimensionless quantity unless the Euclidean dimension d=2 (or for a fractal structure with the mass fractal dimension df=2). To the best of our knowledge, in the literature, there exists no study providing a detailed review and comparing the applicability of each diffusion model on real field well-test data as well as providing new analytical solutions and early- and late-time asymptotic approximate solutions for the anomalous diffusion for both the CR and CBHP cases for the finite-wellbore problem. Hence, this study fills this gap. In addition, we apply the three diffusion models to an interference well test data from the Kizildere geothermal reservoir in Turkey by using the ensemble smoother with multiple data assimilation (ES-MDA) method. The results show that the fractal model in terms of the goodness of fit (root-mean-square error – RMSE) best matches the field test data.