Fourier Equation Research Articles

Pressure-Driven Nitrogen Flow in Divergent Microchannels with Isothermal Walls

Gas flow and heat transfer in confined geometries at micro-and nanoscales differ considerably from those at macro-scales, mainly due to nonequilibrium effects such as velocity slip and temperature jump. Nonequilibrium effects increase with a decrease in the characteristic length-scale of the fluid flow or the gas density, leading to the failure of the standard Navier–Stokes–Fourier (NSF) equations in predicting thermal and fluid flow fields. The direct simulation Monte Carlo (DSMC) method is employed in the present work to investigate pressure-driven nitrogen flow in divergent microchannels with various divergence angles and isothermal walls. The thermal fields obtained from numerical simulations are analysed for different inlet-to-outlet pressure ratios (1.5≤Π≤2.5), tangential momentum accommodation coefficients, and Knudsen numbers (0.05≤Kn≤12.5), covering slip to free-molecular rarefaction regimes. The thermal field in the microchannel is predicted, heat-lines are visualised, and the physics of heat transfer in the microchannel is discussed. Due to the rarefaction effects, the direction of heat flow is largely opposite to that of the mass flow. However, the interplay between thermal and pressure gradients, which are affected by geometrical configurations of the microchannel and the applied boundary conditions, determines the net heat flow direction. Additionally, the occurrence of thermal separation and cold-to-hot heat transfer (also known as anti-Fourier heat transfer) in divergent microchannels is explained.

EXPERIMENTAL INVESTIGATION OF CYLINDRICAL FOOD PRODUCT DURING FORCED CONVECTION COOLING

The objective of this work is to study the transient heat transfer behavior during precooling of cylindrical food product at different air flow rates. Temperature measurement is carried out at the center of the specimen at different air flow rates (3.8, 4.0, 4.2 & 4.5 m/s). These temperature distributions are plotted with time. The terms of Fourier equation were evaluated and thermal diffusivity was plotted against the temperature variation during cooling of the cylindrical food product.Surface film conductance values is also calculated in present work, at different air flow rates. These values are effective values, which included the additional cooling effect due to moisture evaporation from the exposed product surface. Variation of Reynolds number and Nusselt number is also plotted with different air velocities and it is found that more favorable results of thermal conductivity and thermal diffusivity is obtain at 3.8 m/s. Which ultimately saves energy and provide optimum cooling for the specimen at the given air velocity.

An analytical approach to the cooling of a flat plate

The present work deals with the cooling of a flat plate having an initial non-homogeneous temperature distribution, which undergoes a sudden temperature drop at its surfaces. Hence, starting from an arbitrary steady-state temperature distribution, the two surfaces of the slab are suddenly cooled down to the value θ0, resulting in a strong perturbation of the initial temperature distribution. A general analytical solution is provided by adopting the variable separation technique to the Fourier equation in transient conditions, thus obtaining a general solution in terms of a very fast converging series, which includes a definite integral of the initial steady-state distribution. Common engineering applications are investigated and the related results, obtained thanks to an extremely low computational effort, are presented and discussed.

Extended lattice Boltzmann model for gas dynamics

We propose a two-population lattice Boltzmann model on standard lattices for the simulation of compressible flows. The model is fully on-lattice and uses the single relaxation time Bhatnagar–Gross–Krook kinetic equations along with appropriate correction terms to recover the Navier–Stokes–Fourier equations. The accuracy and performance of the model are analyzed through simulations of compressible benchmark cases including Sod shock tube, sound generation in shock–vortex interaction, and compressible decaying turbulence in a box with eddy shocklets. It is demonstrated that the present model provides an accurate representation of compressible flows, even in the presence of turbulence and shock waves.

Revisiting the Lattice Boltzmann Method Through a Nonequilibrium Thermodynamics Perspective

Abstract In this paper, we incorporate a nonequilibrium thermodynamics perspective that is consistent with the Onsager reciprocity principle into the lattice Boltzmann framework to propose a novel regularized lattice Boltzmann formulation for modeling the Navier–Stokes–Fourier equations. The new method is applied to one-dimensional (1D) isothermal situations wherein the advantages of incorporating such a nonequilibrium perspective can be explicitly appreciated. In such situations, the nonequilibrium contribution of the lattice populations obtained by the new method completely vanishes, and the lattice update is entirely reduced to evaluating the equilibrium distribution function. Such a counterintuitive 1D mesoscopic description is not obtained in any other existing lattice Boltzmann scheme. We therefore numerically test the proposed formulation on two complex problems, namely, shockwave and nonlinear wave propagation, and compare results with analytical results along with six existing lattice Boltzmann schemes; it is found that the new method indeed yields results that are more stable and accurate. These results highlight the potency of the nonequilibrium thermodynamics-based approach for obtaining accurate and stable lattice Boltzmann computations, and provide new insights into established lattice Boltzmann simulation methods.

Improved compressible hybrid lattice Boltzmann method on standard lattice for subsonic and supersonic flows

A D2Q9 Hybrid Lattice Boltzmann Method (HLBM) is proposed for the simulation of both compressible subsonic and supersonic flows. This HLBM is an extension of the model of Feng et al. [1], which has been found, via different test cases, to be unstable for supersonic regimes. To circumvent this limitation, we propose:: (1) a new discretization of the lattice closure correction term that makes possible the simulation of supersonic flows, (2) a corrected viscous stress tensor that takes into account polyatomic gases, and (3) a novel discretization of the viscous heat production term fitting with the regularized formalism. The result is a hybrid method that resolves the mass and momentum equations with an LBM algorithm, and resolves the entropy-based energy equation with a finite volume method. This approach fully recovers the physics of the Navier–Stokes–Fourier equations with the ideal gas equation of state, and is valid from subsonic to supersonic regimes. It is then successfully assessed with both smooth flows and flows involving shocks. The proposed model is shown to be an efficient, accurate, and robust alternative to classic Navier–Stokes methods for the simulation of compressible flows.

Controlling thermal waves of conduction and convection

Electromagnetic wave refractions have been widely explored due to their profound physics and practical applications, especially the negative refraction. However, the hyperbolic feature of the Maxwell equations makes it difficult to extend the related physics to the Fourier conduction because the Fourier equation is parabolic. To solve the problem, we study thermal waves of conduction and convection because convection can additionally introduce the hyperbolic feature to heat transfer. We further design convection-assisted metamaterials to realize the positive, vertical, and negative refractions of thermal waves, which have a phenomenal analog of electromagnetic wave refractions despite different mechanisms. The negative refraction of thermal waves means that the incident and refractive thermal waves are at the same side of the normal, but the wave vector and energy flow are still along the same direction. As a model application, we apply the refractive behaviors to design a thermal wave concentrator which can increase wave numbers and energy flows. This work provides insights into thermal wave manipulation, which may have potential applications in thermal wave imaging.

Mathematical Modeling of Technological Modes of Operation of Liquid Stamping of Aluminum Alloy AK7

In this paper we examine a mathematical model of solidification of the melt in a two-phase zone during the formation of forgings through liquid stamping. From the simultaneous solution of the Fourier equations of thermal conductivity for the melt, the two-phase zone, and the solid crust, the solidification kinetics has been established and the effect of external pressure on the temperature fields and components of the solidification time of hollow thin-walled forgings has been thoroughly assessed.

Zero viscosity and thermal diffusivity limit of the linearized compressible Navier–Stokes–Fourier equations in the half plane

We study the zero viscosity and thermal diffusivity limit of an initial boundary problem for the linearized Navier–Stokes–Fourier equations of a compressible viscous and heat conducting fluid in the half plane. We consider the case that the viscosity and thermal diffusivity converge to zero at the same order. The approximate solution of the linearized Navier–Stokes–Fourier equations with inner and boundary expansion terms is analyzed formally first by multiscale analysis. Then the pointwise estimates of the error terms of the approximate solution are obtained by energy methods. Thus establish the uniform stability for the linearized Navier–Stokes–Fourier equations in the zero viscosity and heat conductivity limit. This work is based on (Comm. Pure Appl. Math. 52 (1999), 479–541) and generalize their results from isentropic case to the general compressible fluid with thermal diffusive effect. Besides the viscous layer as in (Comm. Pure Appl. Math. 52 (1999), 479–541), the thermal layer appears and couples with the viscous layer linearly.

Numerical Modeling of Heat and Mass Transfer during Cryopreservation Using Interval Analysis

In the paper, the numerical analysis of heat and mass transfer proceeding in an axially symmetrical articular cartilage sample subjected to the cryopreservation process is presented. In particular, a two-dimensional (axially symmetrical) model with imprecisely defined parameters is considered. The base of the heat transfer model is given by the interval Fourier equation and supplemented by initial boundary conditions. The phenomenon of cryoprotectant transport (Me2SO) through the extracellular matrix is described by the interval mass transfer equation. The liquidus-tracking (LT) method is used to control the temperature, which avoids the formation of ice regardless of the cooling and warming rates. In the LT process, the temperature decreases/increases gradually during addition/removal of the cryoprotectant, and the articular cartilage remains on or above the liquidus line so that no ice forms, independent of the cooling/warming rate. The discussed problem is solved using the interval finite difference method with the rules of directed interval arithmetic. Examples of numerical computations are presented in the final part of the paper. The obtained results of the numerical simulation are compared with the experimental results, realized for deterministically defined parameters.

Modelling and validation of a new hybrid scheme for predicting the performance of U-pipe borehole heat exchangers during distributed thermal response test experiments

The reliable design of Ground Coupled Heat Pumps (GCHP) requires accurate models for both predicting the long term behavior of the borehole (BHE) field and their sub-hourly response during the thermal response test experiments. Several literature models have been proposed in the past to cope with this goal: resistance/capacitance (RC) models, finite difference (FD) thermal descriptions of both inner BHE and surrounding ground, FEM models based on commercial software packages. The present model, as Fortran code, is a hybrid approach that employs an RC scheme (to single or double U pipes) BHEs and solves the full Fourier equation in the (even layered) ground volume. The fluid vertical energy transfers are treated with an upwind scheme. New thermal parameters have been introduced and used as indicators of the physical and geometrical conditions of the U-BHE. Crucial of the RC part is the proper estimation of local resistances and the correct position of the inner BHE thermal capacitance, to be located somewhere in between the pipe surface and the BHE periphery. The extensive validation with experimental TRT data allowed the model to be refined in order to provide very close agreement among predictions and measurements (root mean square error within 0.17 °C).

Nonlinear axisymmetric Taylor–Couette flow in a dilute gas: multiroll transition and the role of compressibility

Abstract

Local well-posedness of the two-dimensional Dirac–Klein–Gordon equations in Fourier–Lebesgue spaces

The local well-posedness problem is considered for the Dirac–Klein–Gordon system in two space dimensions for data in Fourier–Lebesgue spaces [Formula: see text], where [Formula: see text] and [Formula: see text] and [Formula: see text] denote dual exponents. We lower the regularity assumptions on the data with respect to scaling improving the results of d’Ancona et al. in the classical case [Formula: see text]. Crucial is the fact that the nonlinearities fulfill a null condition as detected by these authors.

Role of Atwood number on flow morphology of a planar shock-accelerated square bubble: A numerical study

The Atwood number plays a critical role in describing the physics of fluids behind the hydrodynamic instabilities in gas dynamics. In order to investigate the impacts of the Atwood number (At), the evolution of a shock-accelerated square bubble containing either SF6, Kr, Ar, Ne, or He and surrounded by N2 is investigated numerically. For this purpose, the unsteady compressible Navier–Stokes–Fourier equations are solved using an explicit modal discontinuous Galerkin method. For validation, the numerical results are compared with available experimental results and are found to be in good agreement. The results demonstrate that the Atwood number has a significant influence on flow morphology with wave patterns, vortex creation, vorticity generation, and bubble deformation. For At > 0, the speed of the shock wave traveling along with the bubble inner surface is often less than that of the incident shock wave and greater than that of the transmitted shock wave. Moreover, vortex pairs from the upstream and downstream corners are generated, and the former vortex pair ultimately dominates the flow morphology. For At ≈ 0, the incident and transmitted shock waves move at the same speeds, whereas for At < 0, the transmitted shock wave travels faster than the incident shock wave. Moreover, only one vortex pair at the upstream corners is generated, which dominates the flow morphology. Furthermore, a detailed study of Atwood number impacts is investigated through the vorticity generation at interfaces. A quantitative analysis based on the shock trajectories, the interface features, and the integral diagnostics is also studied in detail to investigate the impacts of the Atwood number on the flow structure. Finally, a comparative study of the flow physics between the shock-accelerated square and cylindrical bubbles is conducted to examine their natural differences.

Analysis of Maxwell–Stefan systems for heat conducting fluid mixtures

The global-in-time existence of bounded weak solutions to the Maxwell–Stefan–Fourier equations in Fick–Onsager form is proved. The model consists of the mass balance equations for the partial mass densities and the energy balance equation for the total energy. The diffusion and heat fluxes depend linearly on the gradients of the thermo-chemical potentials and the gradient of the temperature and include the Soret and Dufour effects. The cross-diffusion system exhibits an entropy structure, which originates from the thermodynamic modeling. The lack of positive definiteness of the diffusion matrix is compensated by the fact that the total mass density is constant in time. The entropy estimate yields the a.e. positivity of the partial mass densities and temperature. Also diffusion matrices are considered that degenerate for vanishing partial mass densities.

Norm Inflation for Nonlinear Schrödinger Equations in Fourier–Lebesgue and Modulation Spaces of Negative Regularity

We consider nonlinear Schr{\"o}dinger equations in Fourier-Lebesgue and modulation spaces involving negative regularity. The equations are posed on the whole space, and involve a smooth power nonlinearity. We prove two types of norm inflation results. We first establish norm inflation results below the expected critical regularities. We then prove norm inflation with infinite loss of regularity under less general assumptions. To do so, we recast the theory of multiphase weakly nonlinear geometric optics for nonlinear Schr{\"o}dinger equations in a general abstract functional setting.

Competition of natural convection and thermal creep in a square enclosure

Although natural convection and thermal creep have been well recognized in the continuum and rarefied regimes, respectively, the study of the competition of them in a wide flow regime is very scarce. From a theoretical point of view, natural convection can be described by Navier–Stokes–Fourier (NSF) equations at the macroscopic level, while thermal creep needs descriptions at the molecular level. Therefore, it is quite challenging to capture these two effects simultaneously. In this work, we employ the unified stochastic particle Bhatnagar–Gross–Krook (USP-BGK) method to investigate thermally driven gas flow in a square enclosure. The simulation results obtained by the USP-BGK method are validated by comparing to those from NSF solutions and direct simulation Monte Carlo method for the continuum and transitional regimes, respectively. We find that the flow patterns in the whole flow regime cannot be determined by just one nondimensional parameter, i.e., the Rayleigh number (Ra), but needs two nondimensional parameters, i.e., the Knudsen number (Kn) and the Froude number (Fr), or Kn and Ra. Specifically, small Knudsen and Froude numbers tend to generate natural convection, while large Knudsen and Froude numbers tend to cause thermal creep. Moreover, our simulation results and analyses demonstrate that when Kn < 0.12, thermal creep is dominant if Ra < 1.0, while natural convection is dominant if Ra/Fr > 0.28, or equivalently, L/L* > 1.0, where L is the characteristic length of the system and L* is the equivalent characteristic length of molecules. These findings provide useful guidance for better understanding of the complex gas flows resulting from the competition of natural convection and thermal creep under microscale or low-density conditions such as on Mars.

Uniform analytic solutions for fractional Navier–Stokes equations

In this paper, we present the existence of the uniform analytic solution of the Cauchy problem for fractional incompressible Navier–Stokes Equations in critical Fourier–Herz spaces. The main strategies are to prove that the existence of the uniform analytic solution is equivalent to the boundedness of convolution inequality on Herz space.

Preparation of a solid product with high water content and water retention properties

Abstract This work presents the development of a solid product with a high water content (99.08%) and water retention properties. Water was chosen as a potential carrier of a volatile active substance and water retaining properties of material were studied at a temperatures and relative air humidity values with the support of the theory of drying. The study first confirmed the role of Gibbs’ phase rule in the research of solid-gas phase equilibrium, and second presented drying kinetics developed from Fick’s second law and expressed with the first term of the Fourier equation. Solutions of equations for phase equilibrium and mass transfer enabled the calculation of Luikov’s parameters, which are important for equilibrium relations and for the diffusivity of water in a solid for mass transfer prediction. The obtained thermodynamic and kinetic parameters enabled product characterisation that may be important for the prediction of retention times.

Development of precipitation model with the use of the Lattice Boltzmann Method and its application for the rolling process

A new precipitation model based on the Lattice Boltzmann Method with elements of Cellular Automata is presented in the paper. The model consists of three submodels: nucleation, precipitation growth, and diffusion of niobium and carbon.The nucleation model used classical nucleation theory and considers homo- and heterogeneous nucleation. The nucleation rate depends on the number of possible nucleation sites, which are defined by dislocation density, i.e. deformation conditions. The nucleation rate depends also on the Zeldowicz parameter, the factor of condensation rate, precipitation work, temperature, and other variables. The nucleation rate is realized in the model by calculating the probability of nucleon appearance in every cell, free of precipitation, and depends on the nucleation rate, the cell volume, and the time step.The growth or dissolution of the precipitation depends on the model on particle size and prevailing local conditions near the particle. The temperature and concentration define the equilibrium size of the particle.LBM model is applied to solve the Fourier equation for niobium and carbon diffusion. Near the precipitations equilibrium concentration of niobium and carbon is established according to precipitations size and temperature, while free diffusion is considered in the other space.Strain-induced precipitation at different temperatures and with the different constant cooling rate was simulated. Results are coincident with theory and experimental researches reported in the scientific literature.Modeling of precipitation during the rolling process with consideration of recrystallization shows an ability of the model to simulate and study the more complex processes that strain-induced precipitation in single deformation. It can be applied for multi-stages deformation processes to design or optimize them in view of precipitation and microstructure evolution.