Classical Fourier Heat Conduction Research Articles

Numerical study of electroconductive non-Newtonian hybrid nanofluid flow from a stretching rotating disk with a Cattaneo–Christov heat flux model

Motivated by emerging technologies in smart functional nanopolymeric coating systems in pharmaceutical and robotics applications, the convective heat transfer characteristics in swirl coating with a magnetic hybrid power-law rheological nanofluid polymer on a radially stretching rotating disk are examined theoretically. An axial magnetic field is imposed. Copper (Cu) and aluminium alloy (AA7075) nanoparticles with sodium alginate (C6H9NaO7) as a base fluid are considered, and a hybrid volume fraction model is deployed. A non-Fourier (Cattaneo–Christov) heat flux model is deployed to inspire thermal relaxation impacts absent in the classical Fourier heat conduction formulation. Through similarity proxies and the Von Karman transformations, the partial derivative systems of equations with associated boundary constraints are reduced to a system of derivative boundary value problems, which is solved numerically via the weighted residual method with an integral Galerkin scheme, executed in the MAPLE symbolic software. Validation with special cases from articles is captured. The computations revealed that radial, tangential and axial velocity are depleted, whereas temperature is enhanced with increasing magnetic parameter ( M). High thermal transfers are obtained for the hybrid nanofluid relative to the AA7075 alloy nanofluid. A strong increment in temperature is also produced with a greater thermal relaxation effect.

Grain refinement mechanism in adiabatic shear band of solution treated Ti–10V–2Fe–3Al alloy under high strain rate

The grain refinement mechanism in the adiabatic shear band (ASB) of solution treated Ti–10V–2Fe–3Al alloy was investigated under high strain rates using a split Hopkinson pressure bar and hat-shaped specimens. The TEM results show that elongated and parallel grains are the major characteristics of the transition region between the matrix and the centre of the ASB. The central region of the ASB consists of ultrafine and equiaxed recrystallized grains with a typical size of 100 nm. According to the classical one-dimensional Fourier heat conduction equation and recrystallization theory calculations, it is proved that ultrafine and equiaxed recrystallized grains did not undergo significant growth during the cooling stage after deformation. The thermodynamics and kinetics calculated results indicate that instant grain refinement within the ASB is due to the rotational dynamic recrystallization (RDR) that occurs during the deformation process.

Numerical investigation of heat and mass transfer of variable viscosity Casson nanofluid flow through a microchannel filled with a porous medium

Thermal behaviours and hydrodynamics of non-Newtonian nanofluids flow through permeable microchannels have large scale utilizations in industries, engineering and bio-medicals. Therefore, this paper presents the numerical investigation of heat and mass transfer of variable viscosity Casson nanofluid flow through a porous medium microchannel with the Cattaneo-Christov heat flux theory. The highly nonlinear PDEs corresponding to the continuity, momentum, energy and concentration equations are formulated and solved numerically via the second order implicit finite difference scheme known as the Keller-Box method. Accordingly, the numerical simulations reveal that variable viscosity parameter, thermal Grashof number, solutal Grashof number, thermophoresis parameter, Schmidt number and Casson fluid parameter show increasing effects on both velocity and temperature of the nanofluid. Furthermore, the temperature profile escalates with increasing values of the Eckert number and the thermal relaxation time parameter. Thus, the Cattaneo-Christov heat flux model is beneficial in warming the transport system of microfluidics when compared to that of the classical Fourier heat conduction law. The temperature profile however, indicates a retarding behavior with increasing values of the Brownian motion parameter, Prandtl number and porous medium parameters namely Forchheimer number and porous medium shape parameter and hence, the porous medium quite effectively controls the nanofluid temperature distribution which plays substantial roles in cooling the transport system of microfluidics. Moreover, the concentration profile shows an increasing pattern with escalating values of the Prandtl number, Schmidt number and thermophoresis parameter but it demonstrates a decreasing trend with the Casson fluid, variable viscosity, thermal relaxation time and solutal relaxation time parameters. It is also observed that coefficient of the skin friction increases with increasing values of the pressure gradient parameter, Eckert number, Forchheimer number and injection/suction Reynolds number. Besides, the heat transfer rate at both walls of the microchannel enhances with rising values of the Eckert number, variable viscosity, parameter and injection/suction Reynolds number. The Casson fluid and thermal relaxation time parameters reveal opposite scenarios on the heat transfer rate at the left and right walls of the microchannel. In addition, the mass transfer rate at both walls of the microchannel shows an increasing pattern as the Eckert number, variable viscosity parameter, Schmidt number and suction/injection Reynolds number increase.

Thermoelastic analysis of biological tissue during hyperthermia treatment for moving laser heating using fractional dual-phase-lag bioheat conduction

In this paper, a time-fractional dual-phase-lag (DPL) bio-thermoelastic model is presented to solve the thermoelastic response of a biological tissue during hyperthermia treatment by a moving laser heating. To improve the elimination rate of diseased tissue and reduce the side effects to surrounding normal tissue in hyperthermia, it is necessary to study the thermoelastic response of biological tissues under high-speed and high-intensity laser heating. The laser heating process causes a large instantaneous temperature difference and heat flow in the tissue, which leads to abnormal diffusion of heat conduction. The classical Fourier heat conduction model cannot describe this anomalous diffusion phenomenon due to infinite heat propagation velocity. In order to describe anomalous diffusion in tissue, the fractional derivative is introduced into the DPL heat conduction model. In order to solve the elastic deformation of tissue under heating, the biological fractional thermoelastic theory is established by combining the fractional DPL model with one-dimensional thermoelastic theory. The physical significance of the phase lags and fractional order is elucidated. The thermoelastic responses are compared with those based on existing models. Finally, the effects of blood perfusion rate and laser velocity on temperature distribution and thermal stress response during percutaneous laser thermal ablation are analyzed.

Numerical solution of the heat conduction problem with memory

It is necessary to use more general models than the classical Fourier heat conduction law to describe small-scale thermal conduction processes. The effects of heat flow memory and heat capacity memory (internal energy) in solids are considered in first-order integrodifferential evolutionary equations with difference-type kernels. The main difficulties in applying such nonlocal in-time mathematical models are associated with the need to work with a solution throughout the entire history of the process. The paper develops an approach to transforming a nonlocal problem into a computationally simpler local problem for a system of first-order evolution equations. Such a transition is applicable for heat conduction problems with memory if the relaxation functions of the heat flux and heat capacity are represented as a sum of exponentials. The correctness of the auxiliary linear problem is ensured by the obtained estimates of the stability of the solution concerning the initial data and the right-hand side in the corresponding Hilbert spaces. The study's main result is to prove the unconditional stability of the proposed two-level scheme with weights for the evolutionary system of equations for modeling heat conduction in solid media with memory. In this case, finding an approximate solution on a new level in time is not more complicated than the classical heat equation. The numerical solution of a model one-dimensional in space heat conduction problem with memory effects is presented.

A New Thermal Model for Predicted Discharge Craters in Micro/Nano-EDM Considering the Non-Fourier Effect

Micro/nano-electrical discharge machining is an alternative preparation method for surface micro/nano-structures, but it is difficult to precisely control the size of the micro/nano-structures due to its unclear material removal mechanism. Thus, it is useful to study its machining mechanism to achieve high-efficiency and controlled processing. At present, most of the established EDM thermal models for predicting the discharge crater size are based on the classical Fourier heat conduction law, assuming that the conduction velocity of heat energy is infinite. However, the single-pulse discharge time of micro/nano-EDM is transitory (<1 μs), and thus, the steady state heat balance condition cannot be achieved in a single-pulse discharge time. In order to predict the size of the micro/nano electrical discharge craters more accurately, the non-Fourier effect was considered to study the temperature field distribution of micro/nano-EDM of single-pulse discharge machining. Firstly, the classical Fourier heat conduction law was modified by introducing a relaxation time. Secondly, several key factors were considered to establish the thermal model of micro/nano-EDM in single-pulse discharge machining. Subsequently, numerical simulation software was used to solve the thermal model for obtaining the temperature field distribution of the workpiece material and predicting the size of the discharge craters. Finally, the predicting accuracy of the new thermal model was evaluated by comparing the relative error between the simulated values and experimental values. The comparison results show that considering the non-Fourier effect can reduce the average error of the thermal model from 33% to 10%. The non-Fourier effect is more obvious under the shorter discharge time of a single pulse.

Transient thermoelastic response of a size‐dependent nanobeam under the fractional order thermoelasticity

AbstractWith the miniaturization of structures, such as MEMS/NEMS, the size‐dependent effect has become an issue and attracted much attention. The well‐known theories describing the size‐dependent effect mainly include the nonlocal elasticity theory, the strain gradient theory and the modified coupled stress theory. Based on these theories, a number of works have been conducted to explore the size‐dependent behaviors of structures or devices in micro/nano‐scale, among them, majorities are on elastic performances, while, minorities are on thermoelastic performances. It is inevitable for structures suffering changeable temperature, as a consequence, thermal‐induced stress and deformation occur in structures and they are worth being fully concerned. For thermoelastic behaviors limited to small scale problems, the classical Fourier's heat conduction law may fail, meanwhile, new models, for example, fractional order heat conduction model, have been developed to modify Fourier's law. In present paper, the transient thermoelastic response of a nanobeam subjected to a ramp heating is investigated by combining the nonlocal elasticity theory and the fractional order heat conduction model. The governing equations are formulated and then solved by Laplace transform and its numerical inversion. The non‐dimensional temperature, displacement, stress, and deflection in the nanobeam are obtained and illustrated graphically. In calculation, the effects of the ramp‐heating time parameter, the nonlocal parameter and the fractional order parameter on the considered physical quantities are examined and discussed in detail.

Non-linear free vibrations of the column loaded with a mass element and a local heat source

In this paper, the problem of non-linear vibrations of an articulated column subjected to two loads was considered. The first load is axial compression associated with the action of the mass element. The second is the heat load in the form of a local heat source. In such a system, the problem analysis was divided into two parts. The first illustrates the influence of temperature on the physical and mechanical properties of the structure. The second - concerning non-linear vibrations of the system. The problem of heat flow in the column was formulated on the basis of the classical Fourier heat conduction equation and solved with the use of the Finite Element Method. Distribution of material properties (Young’s modulus) was obtained in subsequent moments of exposure of the column to the heat source. The obtained results were used in the problem of system vibrations, which was formulated on the basis of Hamilton’s principle. Due to the non-linearity, the perturbative small parameter method was used to finally derive the differential equations describing the behaviour of the system during vibrations. Numerical calculations were carried out on the basis of which the characteristic curves of the system (on the load - natural frequency plane) for various stages of heating the column were determined. Moreover, the influence of the heating time of the system on its non-linear natural frequency was presented. The significant influence of the longitudinal inertia of the mass element causing the load on the vibrations of the structure was shown. The mathematical model and the correctness of the performed numerical simulations were verified by experimental research.

Transient response of the hygro-thermo-elastic field in hollow cylinders under heat-moisture shock via hyperbolic non-Fourier heat conduction and non-Fick diffusion

The classical Fourier heat conduction theory as well as the widely used Fick’s diffusion law predicts an infinite wave velocity. This is physically unrealistic. By generalizing the classical Fourier’s and Fick’s laws, this paper presents a hyperbolic diffusion law to apply heat and moisture coupling. The transient response of the hygro-thermo-elastic field in infinitely long hollow cylinders subjected to sudden heat and moisture shock on the inner and outer surfaces is studied. With the aid of the Laplace transform and the decoupling technique, the closed-form solutions of temperature, moisture, elastic displacement and stresses are determined respectively. The analytical results show that the thermal and moisture relaxation time effect between temperature and moisture is significant for composites. Compared with the classic counterpart, the finite hygrothermal wave speed of the pipe is achieved and decreases with the relaxation time rising. The temperature, moisture, elastic displacement and stresses are calculated. Numerical results are displayed graphically to show the influence of the phase lag of heat/moisture flux on the response of the hygro-thermo-elastic fields. Non-Fourier and non-Fick effects are remarkable between the classic model and hyperbolic hygrothermal coupling model. Some drawbacks induced by the classical Fourier’s and Fick’s laws are averted.

Non-classical hygrothermal fracture behavior of a hollow cylinder with a circumferential crack

By extending the classical Fourier heat conduction law and Fick diffusion law, a hyperbolic hygrothermal coupled model is newly proposed to investigate the fracture problem of a hollow cylinder containing a circumferential crack. The cylindrical system is subjected to a hygrothermal shock at the inner surface, and its outer surface is insulated and damp-proof. The integral transform technique and decoupling technique are used to solve the hygrothermal and mechanical boundary problem. Finally, the effects of hygrothermal coupling, thermal and moisture relaxation time on the transient temperature, moisture, hygrothermal stress and stress intensity factors are calculated. It is found that the hyperbolic model gives more conservative results than the parabolic model for the safety design of structures against fracture under hygrothermal loads.

Size Dependent Thermo-Piezoelectricity for In-Plane Cracks

The finite element method (FEM) is developed to analyse the size effect (flexoeletricity) for 2-D crack problems in thermo-piezoelectricity. Flexoelectricity is observed in micro/nanoelectronic structures, where large strain gradients destroy the symmetric structure of atoms in crystals and thereby causing polarization, even in dielectric materials. In contrast to using classical Fourier heat conduction theory, a finite speed of the thermal wave is considered in the higher order transport equation. The variational principle is applied to derive the FEM equations and C1-continuous elements are employed in the implementation of the FEM. An example is presented to demonstrate the effect of the characteristic time parameter on the crack opening displacement and temperature distribution.

Transient hygrothermoelastic response in a porous cylinder subjected to ramp-type heat-moisture loading

The classical theory of heat conduction (Fourier theory) predicts an infinite speed for thermal disturbance propagation, which is physically unrealistic. By extending the classical Fourier heat conduction and Fick’s diffusion, this article develops hyperbolic diffusion/heat conduction laws with phase lags of heat/moisture flux to simulate coupled heat-moisture diffusion-propagation behavior with the Defour and Soret effects. A porous cylinder subjected to a ramp-type heating and humidifying at the surface is studied. The Laplace transform is used to obtain a closed-form solution of the temperature, moisture, displacements and stresses in the cylinder. Numerical results are calculated via the inversion of the Laplace transform. Obtained results show that the thermal/moisture relaxation time or phase lag plays a significant role in affecting transient hygrothermoelastic field. For a non-vanishing phase lag, non-Fourier and non-Fickian effects exist and hygrothermal waves have finite propagation speeds. The influences of the phase lag of heat/moisture flux and ramp-type time parameter on the transient response of hygrothermoelastic field are presented graphically. A comparison of the numerical results based on the classical model and the present one is made. The non-Fourier heat conduction and non-Fickian diffusion can effectively avoid the shortcomings induced by the classical Fourier and Fick laws.

Generalized Fractional Heat Conduction in a One-Dimensional Functionally Graded Material Layer

Functionally graded materials (FGMs) have been widely used. This work studies a heat conduction problem in an FGM layer with different exponential gradients. Based on a generalized fractional heat conduction theory with phase lag of heat flux, a mixed initial-boundary value problem is solved. Analytical expression for temperature change in the Laplace transform domain is derived. Numerical results of the transient temperature response in the time domain are evaluated by applying a numerical inversion of the Laplace transform. Two representative boundary conditions related to given temperature or heat convective transfer are discussed. For different Biot numbers, the effects of phase lag of heat flux, fractional order, and material properties on temperature field are illustrated graphically. A comparison of the temperature fields based on the non-Fourier model and classic Fourier model is made. The obtained results show that wave-like behaviors may occur for the generalized fractional heat transfer, which does not occur for the classical Fourier heat conduction. The material properties play an important role in the heat transfer process. The derived results are of benefit to the design of materials for thermoresistance and abstraction of heat.

Transient thermoelastic response in a cracked strip of functionally graded materials via generalized fractional heat conduction

This work is devoted to analyzing a thermal shock problem of an elastic strip made of functionally graded materials containing a crack parallel to the free surface based on a generalized fractional heat conduction theory. The embedded crack is assumed to be insulated. The Fourier transform and the Laplace transform are employed to solve a mixed initial-boundary value problem associated with a time-fractional partial differential equation. Temperature and thermal stresses in the Laplace transform domain are evaluated by solving a system of singular integral equations. Numerical results of the thermoelastic fields in the time domain are given by applying a numerical inversion of the Laplace transform. The temperature jump between the upper and lower crack faces and the thermal stress intensity factors at the crack tips are illustrated graphically, and phase lags of heat flux, fractional orders, and gradient index play different roles in controlling heat transfer process. A comparison of the temperature jump and thermal stress intensity factors between the non-Fourier model and the classical Fourier model is made. Numerical results show that wave-like behavior and memory effects are two significant features of the fractional Cattaneo heat conduction, which does not occur for the classical Fourier heat conduction.

Hygrothermoelastic response of a hollow cylinder based on a coupled time-fractional heat and moisture transfer model

In this paper, the time-fractional diffusion-wave equations are introduced to describe the coupled heat conduction and moisture diffusion in a hollow pipe. The outer surface of the hollow pipe is assumed to be insulated and damp-proof, while the inner surface is subjected to thermal and moisture shocks. Using the method of separation of variables and the Laplace transform, analytic expressions for transient hygrothermoelastic response of temperature change, moisture distribution, displacement and stress components in the hollow cylinder are obtained in terms of Mittag–Leffler functions, respectively. Numerical results are calculated, and the effects of fractional order on the dynamic response of temperature, moisture and hygrothermal stresses are illustrated graphically. Sub-diffusive and super-diffusive transport coupling behaviors as well as wave-like behaviors are shown. The results in the case of the fractional order $$\alpha =1$$ reduce to those for the classical Fourier heat conduction and the Fick’s moisture diffusion coupling model.

Numerical study of chemical reaction effects in magnetohydrodynamic Oldroyd-B: oblique stagnation flow with a non-Fourier heat flux model

Reactive magnetohydrodynamic flows arise in many areas of nuclear reactor transport. Working fluids in such systems may be either Newtonian or non-Newtonian. Motivated by these applications, in the current study, a mathematical model is developed for electrically conducting viscoelastic oblique flow impinging on stretching wall under transverse magnetic field. A non-Fourier Cattaneo–Christov model is employed to simulate thermal relaxation effects which cannot be simulated with the classical Fourier heat conduction approach. The Oldroyd-B non-Newtonian model is employed which allows relaxation and retardation effects to be included. A convective boundary condition is imposed at the wall invoking Biot number effects. The fluid is assumed to be chemically reactive and both homogeneous–heterogeneous reactions are studied. The conservation equations for mass, momentum, energy and species (concentration) are altered with applicable similarity variables and the emerging strongly coupled, nonlinear non-dimensional boundary value problem is solved with robust well-tested Runge–Kutta–Fehlberg numerical quadrature and a shooting technique with tolerance level of 10−4. Validation with the Adomian decomposition method is included. The influence of selected thermal (Biot number, Prandtl number), viscoelastic hydrodynamic (Deborah relaxation number), Schmidt number, magnetic parameter and chemical reaction parameters, on velocity, temperature and concentration distributions are plotted for fixed values of geometric (stretching rate, obliqueness) and thermal relaxation parameter. Wall heat transfer rate (local heat flux) and wall species transfer rate (local mass flux) are also computed and it is observed that local mass flux increases with strength of heterogeneous reactions whereas it decreases with strength of homogeneous reactions. The results provide interesting insights into certain nuclear reactor transport phenomena and furthermore a benchmark for more general CFD simulations.

Thermal shock fracture of an elastic half-space with a subsurface penny-shaped crack via fractional thermoelasticity

This paper analyzes a thermal shock problem of an elastic half-space with a penny-shaped crack near the surface based on a fractional thermoelasticity theory. The embedded crack is assumed to be insulated. The Hankel transform and Laplace transform are employed to solve an initial-boundary value problem associated with a fractional partial differential equation. Explicit expressions of temperature and thermal stresses induced by the penny-shaped crack are obtained by solving a system of singular integral equations. Numerical results of the thermoelastic fields in the time domain are given by applying the numerical inverse Laplace transform. The temperature jump between the upper and lower crack surfaces and the thermal stress intensity factors at the crack front are illustrated graphically for various relaxation times and fractional orders, as well as the distance between the crack plane and the half-space surface. A comparison of the temperature, thermal stresses, and their intensity factors is made when adopting the fractional heat conduction model and the classical Fourier heat conduction model. Numerical results show that the temperature overshooting phenomenon may occur for the fractional heat conduction model, whereas it does not occur for the classical Fourier heat conduction model.

Effect of Gravity Modulation on Single Component Convection in a Couple Stress Fluid with Maxwell-Cattaneo Law

The outset of convection in a thin layer of couple stress fluid is analyzed using the linear stability analysis when the fluid is heated from below. In order to assimilate the inertial effects Maxwell-Cattaneo law is used in lieu of the classical Fourier's heat conduction law. The normal mode analysis is used to arrive at the eigenvalues of the perturbed state and a regular perturbation method to find the analytical solutions. The effect of Cattaneo number, couple stress parameter and Prandtl number is discussed and it is concluded that gravity modulation can delay or advance the onset of convection.

Transient hygrothermoelastic response in a cylinder considering non-Fourier hyperbolic heat-moisture coupling

The classical theory of heat conduction (Fourier theory) predicts an infinite speed for thermal disturbance propagation, which is physically unrealistic. By extending the classical Fourier heat conduction and Fick’s diffusion, this paper presents a hyperbolic diffusion law with different phase lags of thermal and moisture fluxes to simulate coupled heat-moisture diffusion-propagation behavior. Transient hygrothermal and elastic response of an infinitely long cylinder subjected to sudden hygrothermal loadings at the surface is studied. By using Laplace transform and decoupling technique, a closed form solution of temperature, moisture, displacements and stresses is determined. The analytical results show that the thermal and moisture relaxation times or phase lags of heat and moisture fluxes play a significant role in the early stage of transient response after heat/moisture shock. The classical results corresponding to vanishing phase lags can be recovered from the present ones. For non-vanishing phase lags, hygrothermal waves have finite propagation speeds. Numerical results are calculated and displayed graphically to show the influence of the phase lags of heat and moisture fluxes on transient hygrothermoelastic fields. A comparison between classic model and hyperbolic hygrothermal coupling model is given. Based on the non-Fourier heat conduction and non-Fick diffusion, some shortcomings induced by the classical Fourier’s and Fick’s laws can be effectively avoided.

The Micromorphic Approach to Generalized Heat Equations

AbstractIn this paper, the micromorphic approach, previously developed in the mechanical context is applied to heat transfer and shown to deliver new generalized heat equations as well as the nonlocal effects. The latter are compared to existing formulations: the classical Fourier heat conduction, the hyperbolic type with relaxation time, the gradient of temperature or entropy theories, the double temperature model, the micro-temperature model or micro-entropy models. A new pair of thermodynamically-consistent micromorphic heat equations are derived from appropriate Helmholtz-free energy potentials depending on an additional micromorphic temperature and its first gradient. The additional micromorphic temperature associated with the classical local temperature is introduced as an independent degree of freedom, based on the generalized principle of virtual power. This leads to a new thermal balance equation taking into account the nonlocal thermal effects and involving an internal length scale which represents the characteristic size of the system. Several existing extended generalized heat equations could be retrieved from constrained micromorphic heat equations with suitable selections of the Helmholtz-free energy and heat flux expressions. As an example the propagation of plane thermal waves is investigated according to the various generalized heat equations. Possible applications to fast surface processes, nanostructured media and nanosystems are also discussed.