Aim. The investigation into the applicability of classical macroscopic approximations to obtain the nonequilibrium local distribution function inside the structure of a strong shock wave. Methodology. This paper examines the capability of various macroscopic models (the Navier – Stokes – Fourier equations, the Burnett equations, and the original and regularized 13-moment Grad equations) to approximate a nonequilibrium molecular velocity distribution function. Results. The locally reconstructed distribution functions obtained from the flow macro-parameters for the considered models are compared with each other and with a benchmark solution at different locations within the structure of a planar shock wave. The benchmark solution is provided by the Direct Simulation Monte Carlo (DSMC) method, which supplies the flow macro-parameters required for the reconstruction of the distribution function. Research implications. All considered models predict the distribution function rather poorly in the supersonic part of the shock-wave structure, where strong oscillations and nonphysical negative values are observed.

Near-space hypersonic vehicles encounter significant rarefaction effects during the flight through the atmosphere, causing the classical Navier–Stokes–Fourier (NSF) equations to break down and posing challenges for the evaluation of surface drag and heat flux. In this paper, the nonlinear momentum and heat transfer in a hypersonic transitional boundary layer are analysed based on the generalized hydrodynamic equations (GHE), and the generality of the derived formulae is also discussed. The leading transport relations are obtained by estimating the relative orders of the various terms in GHE according to the hypersonic flow and boundary-layer requirements. Local non-equilibrium parameters characterising the shear non-equilibrium effect ( $K_\sigma$ ) and thermal-gradient non-equilibrium effect ( $K_q$ ) are introduced, and a set of correlation formulae for local surface pressure, shear stress and heat flux are proposed as corrections to continuum-based solutions. The correction function depends only on the non-equilibrium parameters $K_\sigma$ and $K_q$ , and the continuous solutions can be either analytical formulae or NSF simulation results. This enables us to predict the surface aerothermodynamics with enhanced accuracy while still using the solutions of the NSF equations. The proposed formulae are carefully verified by comparing with direct simulation Monte Carlo (DSMC) results of different hypersonic rarefied flows, including flat-plate, sharp-wedge, cylinder and blunt-cone flows, and partial experimental data are also given. The results demonstrate that the proposed formulae can significantly enhance the accuracy of the continuum-based solutions, and show good agreement with DSMC simulations and experimental measurements in the near-continuum regime.

External body forces influence the evolution of the distribution function in mesoscopic methods through the particle–velocity–space gradient term in the Boltzmann equation. Simplified forcing models for continuum flows can be categorized into moment-expansion-based and Chapman–Enskog-expansion-based models, while their performance in compressible turbulence remains insufficiently explored. Using the partial internal energy double-distribution-function (DDF) framework, this study addresses that gap. We systematically derive the moment constraints for any forcing model to recover the Navier–Stokes–Fourier equations with arbitrary bulk viscosity for the first time. Within the partial internal energy DDF framework, we novelly propose a moment-expansion-based forcing model for compressible flows, which accounts for the higher-order structure of the distribution function without requiring increased Gauss–Hermite quadrature accuracy. For comparison, we also consider two classical Chapman–Enskog-based models, the He model [He et al., J. Comput. Phys. 146, 282 (1998)], which requires higher-order quadrature, and the Kupershtokh model [Kupershtokh et al., Comput. Math. Appl. 58, 965 (2009)], which neglects higher-order structural effects, even though both models satisfy the same velocity moment constraints. All three models yield stable simulations of forced compressible turbulence using partial internal energy DDF-based discrete unified gas kinetic scheme. However, nonlinear interactions lead to observable, though small, differences in turbulence statistics under identical initial conditions. Overall, the results indicate that each model is capable of providing reasonable turbulence statistics.

The study examines the concentration of cryoprotectant (CPA) in an articular cartilage sample during cryopreservation by computing the effective diffusion coefficient using different material models-homogeneous and porous. The mass transfer phenomenon is coupled to the effective diffusion coefficient, which is determined by three different approaches. The first and second models, based on the Einstein-Stokes equation and the Arrhenius expression, respectively, treat the sample as a homogeneous material, whilst the third considers it as a porous medium. The effective diffusion coefficient is additionally weakly coupled to the heat transfer phenomenon described by the Fourier equation, and the third variant is also strongly coupled to the concentration of CPA. The final section of the article presents example calculations for the selected cryopreservation method, and the results are compared with the experimental results. Depending on the method applied to estimate the effective diffusion coefficient, the maximal relative errors in relation to experimental results are equal to 15.82%, 5.20%, and 24.96%, respectively. A decrease in temperature and an increase in the concentration of dimethyl sulfoxide (DMSO) cause a reduction of the effective diffusion coefficient. Moreover, in the model considering the porosity of the sample, the lowest values of the effective diffusion coefficient were obtained. This study's novelty lies in its comparative analysis of homogeneous and porous models, as well as its explicit coupling of temperature, concentration, and diffusion processes during cryopreservation.

The Nepetoideae and the Lamioideae are the largest subfamilies of the family Lamiaceae comprising more than half of the family’s species including mostly shrubs or herbs, many of which are applied as culinary and medicinal herbs. The results of previous analysis of twenty-one species in the subfamily Nepetoideae indicated a relationship of nutlet size and shape with the taxonomic position, with larger nutlets in the tribe Elsholtzieae, intermediate in the tribe Ocimeae, and smaller in the tribe Mentheae, except for Salvia species. This work focuses on 49 species, 34 in the Nepetoideae and 15 in the Lamioideae, and shows differences in nutlet size and shape between both subfamilies. The nutlets of the Lamioideae are larger and have higher aspect ratio and lower circularity, roundness, and solidity values. Elliptic Fourier Transform equations were obtained allowing the representation of average outlines for each of these species. Curvature analysis indicates higher absolute values of maximum and average curvature in the subfamily Lamioideae. Morphometric analysis including geometric measurements, curvature, and the coefficients of Fourier equations reveals differences between genera and species in both subfamilies of the Lamiaceae that can be of applied interest in the taxonomy of this family.

In this paper, we derive a fourth-order entropy stable extension of the Navier–Stokes–Fourier equations into the transition regime of rarefied gases. We do this through a novel reformulation of the closure of conservation equations derived from the Boltzmann equation that subsumes existing methods such as the Chapman–Enskog expansion. We apply the linearized version of this extension to the stationary heat problem and the Poiseuille channel and compare our analytical solutions to asymptotic and numerical solutions of the linearized Boltzmann equation. In both model problems, our solutions compare remarkably well in the transition regime. For some macroscopic variables, this agreement even extends far beyond the transition regime.

Purpose In this paper, the mathematical and numerical models of heat transfer processes based on the dual-phase lag equation (DPLE; the energy equation with two delay times) are considered. This type of equation is, as a rule, applied for mathematical description of thermal processes taking place in the microdomains (microscale heat transfer) and also for the modeling of problems associated with the heat transfer in biological tissue domains, which follows from the specific tissue structure. The purpose of this paper is to show the correct form of boundary conditions supplementing the DPL equation (equations). The solutions of different DPLE variants discussed in the literature are obtained using the classical form of boundary conditions (as in the Fourier equation), which is not completely correct. In this paper, the proper mathematical form of the Neumann, Robin and continuity conditions is presented. Design/methodology/approach The second part of the paper is devoted to the numerical aspects of solving the problems basing on the mathematical description formulated in this way. At the stage of computations, the finite difference method in an implicit scheme is applied (1D and axially-symmetrical problems are considered). One of the examples was also solved using the generalized boundary element method. For numerical computations, an authorial computer program was developed, which performs simulations related to the modeling of thermal processes based on DPL as well as on the Cattaneo–Vernotte and Fourier equations. Findings The results of numerous simulations concerning both microscale heat transfer and bioheat transfer are shown, including a comparison of solutions using the classical approach to boundary-initial conditions and the approach presented in this paper. Research limitations/implications Delay times values are not known for all materials, whereas the values presented in the literature sometimes differ from each other (especially in the case of biological tissue). In some works, it is emphasized that there are some limitations concerning the delay times of material considered, which assure the physical correctness of DPLE. Originality/value The correct formulations of boundary and initial conditions supplementing the dual-phase lag model are presented.

The propagation of linear waves in non-ideal compressible fluids plays a crucial role in numerous physical and engineering applications, particularly in the study of instabilities, aeroacoustics and turbulence modelling. This work investigates linear waves in viscous and heat-conducting non-ideal compressible fluids, modelled by the Navier–Stokes–Fourier equations and a fully arbitrary equation of state (EOS). The linearised governing equations are derived to analyse the dispersion relations when the EOS differs from that of an ideal gas. Special attention is given to the influence of non-ideal effects and various dimensionless numbers on wave propagation speed and attenuation. By extending classical results from Kovásznay (1953 J. Aeronaut. Sci. vol. 20, no. 10, pp. 657–674) and Chu (1965 Acta Mech. vol. 1, no. 3, pp. 215–234) obtained under the ideal gas assumption, this study highlights the modifications introduced by arbitrary EOSs to the linear wave dynamics in non-ideal compressible flows. This work paves the path for an improved understanding and modelling of wave propagation, turbulence and linear stability in arbitrary viscous and heat-conducting fluids.

The main objective of this study was to investigate boiling heat transfer during refrigerant flow in a mini-channel heat sink. The test section consisted of multiple parallel mini-channels, each with a depth of 1 mm. The working fluid was heated by a thin layer of Haynes-230 alloy with a thickness of 0.1 mm. The outer surface temperature of the heater was measured using infrared thermography, while other thermal and flow-based parameters were recorded via a dedicated data acquisition system. The mini-channel heat sink was tested in seven different spatial orientations, with inclination angles relative to the horizontal plane of 45°, 60°, 75°, 90°, 105°, 120°, and 135°. The analysis focused on the early stage of the experiment, corresponding to the forced convection, boiling incipience, and subcooled boiling region. A time-dependent, two-dimensional model of heat transfer during flow boiling of a refrigerant in asymmetrically heated mini-channels was developed. The temperatures of both the heating foil and the working fluid (Fluorinert FC-770) were described using appropriate forms of the Fourier–Kirchhoff equation, subject to relevant boundary conditions. Two sets of time-dependent Trefftz functions were employed to solve the governing equations: one set corresponding to the two-dimensional Fourier equation and another, newly derived, for the energy equation in the fluid. The results include thermographic images of the heated surface, temperature distributions, fluid temperatures, local heat-transfer coefficients, and boiling curves. A comparison of the heat-transfer coefficients obtained using the Trefftz-based approach and those calculated using Fourier’s law demonstrated satisfactory agreement.

Abstract There is a lack in the current state‐of‐the‐art for the evaluation of the temperature profile of adhesive anchors and post‐installed rebars (PIR) exposed to fire. Although the problem is generally governed by Fourier's partial differential equation of heat transfer, at the current state, the only way to determine the temperature along the anchorage length is through finite‐element analysis (FEM), which is very time‐consuming and often impractical for many design scenarios. To address this, a first Level‐of‐Approximation (LoA‐I) is introduced, following the Levels‐of‐Approximation approach widely used by the fib community. Essentially, LoA‐I employs a closed‐form solution of Fourier's equation for uni‐axial heat flow in concrete elements, adapted for both a single anchor embedded in concrete and PIR. As a key element of the method, an equivalent constant thermal diffusivity is introduced on the basis of sound engineering assumptions. Validation of LoA‐I is achieved by comparing the analytically derived temperatures to both numerically derived (LoA‐II) and experimental temperatures for five benchmark cases. Additionally, pull‐out axial load‐bearing capacities are analytically derived using the Resistance Integration Method. Although LoA‐I results are generally more conservative than those from LoA‐II, the reduction in computational costs is considerable. Therefore, LoA‐I is recommended for quick design assessment as well as for all design cases in which pullout failure is expected to be not decisive.

Abstract In this paper, we study the Cauchy problem for the Hall‐MHD equations in critical Fourier–Herz space . Making use of the Fourier localization method and Banach fixed point theorem, we established the global well‐posedness of a mild solution with small initial data and local well‐posedness for any initial data. Furthermore, we conclude the weak–strong uniqueness.

The problem of obtaining a mathematical model of the process of thermal molding of structures from polymer composite materials in an autoclave, taking into account its technical capabilities and types of molding equipment, is considered in the article. The main task of the work was to obtain the dependence, showing the change of temperature of the internal structure of the polymerized package depending on the working environment of the autoclave and the initial conditions of the process, depending on the type of heating device and type of molding equipment. The initial realization of the problem was based on the use of Fourier equations for a triple package (auxiliary equipment, product, main molding equipment). The proposed approach used values of the initial temperature jump depending on the mass of auxiliary equipment (tooling) and the type of composite material being formed, which were obtained experimentally. The study involved real processes of manufacturing PCM products in a «Scholz» autoclave. Since the practical part of the study was carried out for a device equipped with a powerful fan unit, an assumption was made about instantaneous convection of heated air throughout the working volume of the autoclave. The second stage of the work involved solving the problem of the unsteady heat conduction problem by the method of elementary balances approximated to the classical finite element method. For this task, the method was transformed, which allowed for taking into account the heat exchange of the environment with the body. In this way, it is possible to obtain dependencies for corner points, points lying on the edges and faces of the formed package. This allows us to determine not only the amount of heat entering the package from the most developed surfaces, as is done in the analytical solution, but also from the sides of the package. In this model, the thermal conductivity coefficient and specific heat capacity are taken as linear functions of temperature. When developing the calculation model, the change in the physical parameters of the package over time during heating was taken into account, which also increased the probability of finding a more accurate solution. The main difference from other software products used to solve similar problems is the ability to solve the inverse problem, which consists in determining the required temperature of the environment for a given mode.The obtained dependencies for the total heat flows allow us to find the total heat value used for polymerization processes, the temperature of the environment in the autoclave and the corresponding value of the temperature of the middle layer of the package.

The ability to focus at near is achieved by dynamic changes in the shape of the lens of the eye. The Helmholtz hypothesis of accommodation proposes that, at distance gaze, all of the lenticular supporting zonules are at maximal tension. To bring a near object into focus, this tension is reduced by action of the ciliary muscle. The resultant release of tension allows the elastic lens capsule to mold the lens into a more rounded shape, increasing both its central thickness and central optical power (COP). Based upon Helmholtz's hypothesis, complete removal of these zonules should result in a rounded shaped lens of maximal COP. Schachar has offered an alternative mechanism of accommodation based upon the distinct actions of the three different groups of lenticular zonules. Schachar believes that for distant objects, all the zonules are under the minimum tension required to maintain lens stability; however, during lenticular accommodation, equatorial zonular tension increases while, simultaneously, the anterior and posterior zonular tension decreases. The selective increase in equatorial zonular tension results from the unique orientation of the different ciliary muscle fiber groups. With this increase in equatorial zonular tension, the peripheral lens surfaces flatten, central surfaces steepen and central lens thickness and COP increase. Schachar's hypothesis would anticipate that with zonular removal, the COP of the isolated lens would be minimal and diametrically opposite to the high lens COP expected with the Helmholtz hypothesis. In order to determine the COP of the isolated human lens, we obtained, through the kindness of the authors of an independent research study, the x-y coordinates of the central sagittal lens profiles of 10 freshly isolated human lenses (donors aged 20-30 years). These coordinate data were then mathematically utilized by fitting them into Chien, Forbes, Fourier, and elliptical equations. Additionally, the coordinate data was smoothed and fit to third-degree polynomials (S4W 3rd Poly). Independent of which of these equations was employed, within central optical zone diameters of [Formula: see text] 3 mm, the COP was found to be minimal. Since the S4W 3rd Poly provided the best fit, it was used to represent lens surfaces in optically modeled eyes. In all modeled eyes, Zernike spherical aberration (SA) coefficients were positive. These findings are consistent with in vivo measurements of SA obtained from human eyes while viewing distant visual objects. Having thus demonstrated that freshly removed human lenses, free of zonular tension, have their least COP, it is likely that this condition mimics the physiologic status of the human lens in the eye while attending to the most distant visual objects. In an independent, companion paper, we observed, using interferometric measurements of surface radius of curvatures of 12 fresh, isolated human lenses, obtained from donors aged 20-30 years, that the minimal COP was also associated with the unaccommodated state in vivo.

Time-Asymptotic Stability of Generic Riemann Solutions for Compressible Navier–Stokes–Fourier Equations

The GENERIC-13 moment equations (general equation for the non-equilibrium reversible-irreversible coupling) [Struchtrup & Öttinger, Phys. Fluids 34, 017105 (2022)] were developed to have complete thermodynamic structure, in contrast to Grad's 13-moment equations which are not accompanied by a suitable formulation of the second law of thermodynamics and loose hyperbolicity for larger deviations from equilibrium. With GENERIC-13 constructed to agree with Grad-13 to second order in the Knudsen number, both sets are considered and compared for hyperbolicity and plane heat transfer, and Couette and Poiseuille flows. It is shown that the GENERIC-13 equations are unconditionally hyperbolic. Jump and slip boundary conditions for GENERIC-13 are developed from the second law with coefficients adapted from kinetic theory. Additional asymptotically vanishing boundary conditions are constructed such that solutions of the GENERIC-13 equations reduce to those of Grad-13 to second and of Navier–Stokes–Fourier equations to first order in the Knudsen number.

This work examines the evaporation and condensation phenomena at small scales, focusing on how surface deformations affect mass and heat transfer under temperature-driven and pressure-driven conditions. The rarefaction effects arising at these scales cannot be accurately captured by the classical continuum theories such as Navier–Stokes–Fourier equations. To address this limitation, the coupled constitutive relations (CCR) are employed to describe the process. The thermodynamically admissible boundary conditions for both partial and complete evaporation and condensation are presented by determining the reciprocity coefficients for the CCR model. The problem is solved using the method of fundamental solutions (MFS), which is a meshless numerical scheme. Numerical results from the MFS are validated with an analytic solution for a circular cross section of an evaporating jet. Furthermore, the effect of cross-sectional deformations on evaporation and condensation is demonstrated by evaluating mass and heat transfer characteristics wherein spherical harmonics are used to generate deformed shapes. An error analysis is performed to showcase the accuracy and convergence of the MFS. The results provide an understanding of the modeling of phase-change phenomena in micro- and nanoscale systems.

Real-time heat distribution and phase transformation based on operating conditions and material properties can be estimated using heat equations. The corresponding characteristic functions are used to analyze heat conduction processes in various fields, including laser and electron beam processing. A powerful universal analytical and numerical method that transforms partial differential equations into a coupled system of ordinary differential equations is the wavelet transform method. Fourier and non-Fourier heat equations can be implemented for both equilibrium and non-equilibrium thermodynamic processes, including a wide range of processes such as the two-temperature model, ultrafast laser irradiation, and biological processes. The ultrafast laser heating process of nanofilms is characterized by ultrashort duration and ultrasmall spatial size, in which the classical Fourier law based on the local equilibrium hypothesis is no longer applicable. Based on the Cattaneo-Vernotte model and the double phase delay model, two-dimensional analytical solutions of thermal conductivity in two-dimensional structures under the action of ultrafast laser are obtained using the integral transform method. The results show that there is a thermal wave phenomenon inside the film, which becomes increasingly obvious as the temperature gradient delay time elapses. In this paper, non-Fourier heat conduction problems with temperature and heat flux relaxations are studied based on the wavelet finite element method and solved by the central difference scheme for one-dimensional and two-dimensional media. The heat wave model and the double phase delay model are used to formulate the finite elements, and a new formulation of the wavelet finite element solution is proposed to solve the computational optimization problem. Compared with the current methodologies for the heat wave model and the dual phase delay model, the present model is a direct model that describes the thermal behavior with a single equation with respect to temperature. The developed method can be used for arbitrary shapes. A new iteration update methodology is also proposed for the dual phase delay model to solve the computationally efficient problems. The time iteration algorithms do not use the global stiffness matrix. This allows for optimized calculations. Numerical calculations were performed in comparison with the classical finite element method and the spectral finite element method. The comparisons in accuracy, efficiency, flexibility and applicability confirm that the developed method is an effective and alternative tool for thermal analysis of local volumes of two-dimensional materials.

Introduction. For many modern manufacturing processes, induction heating provides an attractive combination of speed, consistency and control. Multi-inductor (zone) systems with continuous billets feed are the most promising, which keep the billet cross sectional average temperature equal. It allows to avoid overheating at low throughputs and reduces the number of rejected billets. Problem. With zone induction heating systems for metal billets developing it is necessary, at the design stage, to perform a quantitative analysis of the main characteristics of the electrothermal process and provide recommendations for optimal parameters and heating modes selections. Accurate calculations for induction heating systems involve considering the distribution of the magnetic field, current density, and changes of material properties throughout volume of the heated billet. The goal of the work is to develop the numerical model and analyze the coupled electromagnetic and thermal processes in zone induction heating system for metal billets to determine the optimal power ratio of the inductors and choose rational heating modes for the billets. Methodology. The spatiotemporal distribution of the electromagnetic field and temperature throughout the volume of the billet during the induction heating process is described by the system of Maxwell and Fourier equations. For numerical calculations by the finite element method, the COMSOL Multiphysics 6.1 software package was used. All three methods of heat transfer are taken into account – conduction, convection, and radiation. Multiphysics couplings use electromagnetic power dissipation as a heat sources, and the billet material properties are specified by temperature functions. The operation of the inductors’ coils is modeled using the «Multi-Turn Coil» function, which uses a homogenized model. The translational motion of the billet is modeled by using the «Translational Motion» function. Results. The numerical 3D-model of coupled electromagnetic and thermal processes in the zone induction heating system for metal billets has been developed. Modeling was carried out for the design of a four-inductor system with the nominal capacity of 5000 kg/h. Data on the spatial distribution of the electromagnetic and temperature fields in the moving heated steel billet were obtained. Originality. Three-dimensional graphs of electrical conductivity and relative magnetic permeability change inside the moving heated steel billet are presented. Results of the temperature distribution calculations along the length of the steel billet for different inductors power ratios are provided. It is shown how the change in the power distribution of the inductors affects the billet heating parameters. Practical value. Analysis of the obtained data allows to determinate the necessary inductors powers to ensure the required heating mode. The results make it possible to reduce the time and resources required for the development, optimization of the design and improvement of the technological process of zone induction heating for metal billets. References 20, table 1, figures 13.

A mathematical model is developed that describes the shock wave structure in a viscous flow of a mixture containing carbon dioxide and noble gases, particularly argon, neon, and helium. The proposed three-temperature model takes into account several mechanisms of vibrational relaxation in polyatomic gases, diffusion, heat conductivity associated with different vibrational modes, shear, and bulk viscosity. A continuum approach based on the generalized Chapman–Enskog method is applied to derive a self-consistently closed set of extended Navier–Stokes–Fourier equations. The peculiarity of the model is that we use neither phenomenological approaches when deriving constitutive relations for the transport fluxes nor widely known approximations for thermodynamic and transport properties; the energy and specific heats for various vibrational modes are calculated explicitly; the transport coefficients are found as solutions of corresponding transport linear systems; and the expression for the diffusion velocity is free of common limitations of the Fick law. The model is implemented to the in-house finite-volume flow solver. The effects of free-stream thermal nonequilibrium, mixture composition, diffusion, and bulk viscosity on the shock structure are discussed. While in the CO2–Ar mixture diffusion is negligible, it is dominating in the CO2–He mixture. The contribution of bulk viscosity is generally weak compared to other effects. In CO2–Ar mixture, there is a compensation effect between the heat fluxes due to diffusion and vibrational relaxation; these contributions are, however, small compared to the flux of translational–rotational energy. In CO2–He, the heat flux due to diffusion is significant, making more than a half of the total heat flux.

This work results from a numerical investigation of the thermochemical non-equilibrium effects on the surface properties of a hypersonic body. Non-equilibrium within an air mixture composed of 11 chemical species was considered when solving the Navier–Stokes–Fourier equations using a density-based algorithm in OpenFOAM. The influence of thermal and chemical non-equilibrium on the surface properties of a hypersonic double-cone test body was studied by considering two types of surfaces. It was found that the heat flux and pressure distribution along the surface are higher under non-equilibrium free-stream conditions. Unlike what was observed at the impingement point, where the vibrational non-equilibrium effects on the surface properties are almost independent of the surface type, at the stagnation point, these effects are highly dependent on the catalytic activity of the surface. At the stagnation point, the vibrational non-equilibrium effects are more pronounced on a fully catalytic surface than on a non-catalytic surface. Under the studied conditions, the vibrational non-equilibrium reduces the heat flux by 18% for a non-catalytic surface, while for a fully catalytic surface, it reduces the heat flux by 38%. Additionally, the presence of vibrational non-equilibrium in the free-stream reduces the pressure by 24% for a non-catalytic surface, while for a fully catalytic surface, it is reduced by 42%.