Heat Flux Law Research Articles

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

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.

Sensitivity analysis on optimizing heat transfer rate in hybrid nanofluid flow over a permeable surface for the power law heat flux model: Response surface methodology with ANOVA test

<abstract> <p>Joule dissipation has an important role in the conversion of mechanical energy to heat within a fluid due to the internal friction and viscosity. Moreover, Darcy friction is a measure of the resistance to flow in a porous medium. In response to the efficient heat transfer performance, a robust statistical approach was established to optimize the heat transfer rate in a two-dimensional flow of a nanofluid over a permeable surface embedded with a porous matrix. The electrically conducive fluid affected the flow phenomena to include a carbon nanotube nanoparticle in the conventional liquid water for the enhanced heat transfer properties; additionally, the power-law heat flux model was considered. Appropriate transformation rules were adopted to obtain a non-dimensional system that brought a developed model equipped with several factors. The traditional numerical technique (i.e., shooting based Runge-Kutta) was proposed to handle the coupled nonlinear system. Furthermore, the statistical response surface methodology (RSM) was adopted to obtain an efficient optimized model for the heat transportation rate of the considered factors. An analysis of variance (ANOVA) was utilized to validate the result of the regression analysis. However, it was evident that the nanoparticle concentrations were useful to augment the fluid velocity and the temperature distributions; the statistical approach adopted for the heat transfer rate displayed an optimized effect as compared to a conventional effect.</p> </abstract>

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.

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.

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.

Bioconvective nonlinear radiated thermal transport of magnetized third-grade nanoparticles in accelerated frame with implementation of modified fourier heat flux laws

The contribution of nanoparticles is important to enhance energy resources and improve the efficiency of solar management systems. Owing to impressive thermal consequences, various applications of nanoparticles are observed in engineering processes, chemical and mechanical industries, thermal systems and many more. It is further emphasized that the decomposition of nanoparticles with non-Newtonian materials is more dynamic from an application point of view. The current research communicates the improvement in thermal transportation phenomenon due to interaction of nanoparticles along with decomposing of third-grade base fluid. The nanofluid is immersed with microorganism to justify the applications of bioconvection. The thermal observations are carried out with the existence of some extra thermal consequences like activation energy and radiative phenomenon with nonlinear approach. The modified relations of Fourier and Fick theories have been used to entertain the thermal results. The confined flow is due to the accelerated regime with stretching characteristics. The dynamical system is formulated with the attention of partial differential equations. The physical impact of parameters is addressed for specific range of parameters like [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text]. The analytical outcomes are listed via homotopy analysis scheme. The role of parameters fluctuating in the various profiles is graphically observed. The results conveyed via the current model are effective in enhancing the thermal processes, solar systems, heat transfer, control of cooling and heating phenomenon, biofuels, fertilizers, etc.

Numerically analysis of Marangoni convective flow of hybrid nanofluid over an infinite disk with thermophoresis particle deposition

This study discusses the flow of hybrid nanofluid over an infinite disk in a Darcy–Forchheimer permeable medium with variable thermal conductivity and viscosity. The objective of the current theoretical investigation is to identify the thermal energy characteristics of the nanomaterial flow resulting from thermo-solutal Marangoni convection on a disc surface. By including the impacts of activation energy, heat source, thermophoretic particle deposition and microorganisms the proposed mathematical model becomes more novel. The Cattaneo-Christov mass and heat flux law is taken into account when examining the features of mass and heat transmission rather than the traditional Fourier and Fick heat and mass flux law. MoS2 and Ag nanoparticles are dispersed in the base fluid water to synthesize the hybrid nanofluid. PDEs are transformed to ODEs by using similarity transformations. The RKF-45th order shooting method is used to solve the equations. With the use of appropriate graphs, the effects of a number of non-dimensional parameters on velocity, concentration, microorganism, and temperature fields are addressed. The local Nusselt number, density of motile microorganisms and Sherwood number are calculated numerically and graphically to derive correlations in terms of the relevant key parameters. The findings show that as we increase the Marangoni convection parameter, skin friction, local density of motile microorganisms, Sherwood number, velocity, temperature and microorganisms profiles increase, whereas Nusselt number and concentration profile exhibit an opposite behavior. The fluid velocity is reduced as a result of enhancing the Forchheimer parameter and Darcy parameter.

A novel numerical method for predicting the hydration heat of concrete based on thermodynamic model and finite element analysis

In this study, a rational theoretical expression and effective numerical realization of the hydration temperature evolution of concrete are presented. The proposed novel numerical method is based on the Fourier's law for heat flux, the heat source function which is obtained from the dissolution of cement clinker by the modified Parrot and Killoh model, and also takes into account the Arrhenius equation. Experimental temperature measurements performed on three different specimens under semi-adiabatic and adiabatic conditions are used to validate the proposed model. The results show that the measured semi-adiabatic and adiabatic temperatures of the SCM-blended concrete mixtures are in general agreement with the simulation results. The method proposed in this study can also predict the temperature field of mass concrete, providing guidance for practical engineering.

Chaotic Cattaneo-LTNE porous convection

The chaotic convection in a porous medium has been studied with a local thermal nonequilibrium (LTNE) model and hyperbolic-type heat transport equation in the solid arising due to Cattaneo heat flux law. The Brinkman-extended Darcy model is considered to describe the flow in a porous medium. A system of autonomous nonlinear ordinary differential equations is derived by using a truncated double Fourier series expansion such that the linear stability theory results are the same as those for the full two-dimensional problem. The stability of equilibrium points of the system is discussed in detail for the governing parameters involved therein. The transition to chaos in different phase planes and the transient behavior of heat transfer are analyzed by solving the model equations numerically using the Runge-Kutta-Fehlberg technique. The effect of increasing scaled solid thermal relaxation time parameter is to speed up the transition to chaos and to decrease the heat transfer.

Analysis of Fractional Bioconvection with Hybrid Nanoparticles in Channel Flow

In this paper, MHD Brinkman-type fluid flow containing titanium dioxide and silver nanoparticle hybrid nanoparticles with generalized Mittag–Leffler kernel-based fractional derivative is investigated in the presence of bioconvection. The governing equations with dimensional analysis and fractional approach are obtained by using the fractional Fourier’s law for heat flux and Fick’s law for diffusion. As a result, the bioconvection Rayleigh number, which is responsible for the declining in the fluid velocity and fractional parameters used to control the thermal and momentum boundary layers thickness of fluid properties. The obtained solutions can be beneficial for proper analysis of real data and provide a tool for testing possible approximate solutions where needed.

Modified Homogeneous and Heterogeneous Chemical Reaction and Flow Performance of Maxwell Nanofluid with Cattaneo–Christov Heat Flux Law

Modified Homogeneous and Heterogeneous Chemical Reaction and Flow Performance of Maxwell Nanofluid with Cattaneo–Christov Heat Flux Law

Heat Transfer of Nanomaterial over an Infinite Disk with Marangoni Convection: A Modified Fourier’s Heat Flux Model for Solar Thermal System Applications

The demand for energy due to the population boom, together with the harmful consequences of fossil fuels, makes it essential to explore renewable thermal energy. Solar Thermal Systems (STS’s) are important alternatives to conventional fossil fuels, owing to their ability to convert solar thermal energy into heat and electricity. However, improving the efficiency of solar thermal systems is the biggest challenge for researchers. Nanomaterial is an effective technique for improving the efficiency of STS’s by using nanomaterials as working fluids. Therefore, the present theoretical study aims to explore the thermal energy characteristics of the flow of nanomaterials generated by the surface gradient (Marangoni convection) on a disk surface subjected to two different thermal energy modulations. Instead of the conventional Fourier heat flux law to examine heat transfer characteristics, the Cattaneo–Christov heat flux (Fourier’s heat flux model) law is accounted for. The inhomogeneous nanomaterial model is used in mathematical modeling. The exponential form of thermal energy modulations is incorporated. The finite-difference technique along with Richardson extrapolation is used to treat the governing problem. The effects of the key parameters on flow distributions were analyzed in detail. Numerical calculations were performed to obtain correlations giving the reduced Nusselt number and the reduced Sherwood number in terms of relevant key parameters. The heat transfer rate of solar collectors increases due to the Marangoni convection. The thermophoresis phenomenon and chaotic movement of nanoparticles in a working fluid of solar collectors enhance the temperature distribution of the system. Furthermore, the thermal field is enhanced due to the thermal energy modulations. The results find applications in solar thermal exchanger manufacturing processes.

Onset about non-isothermal flow of Williamson liquid over exponential surface by computing numerical simulation in perspective of Cattaneo Christov heat flux theory

In view of increaing significance of non-isothermal flow of non-Newtonian fluids over exponential surfaces in numerous industrial and technological procedures such as film condensation, extrusion of plastic sheets, crystal growth, cooling process of metallic sheets, design of chemical processing equipment and various heat exchangers, and glass and polymer industries current disquisition is addressed. For comprehensive examination Williamson model expressing the attributes of shear thickening and thinning liquids is taken under consideration. The physical aspects of magnetic field applied in transverse direction to flow is also accounted. Heat transfer aspects are incorporated and analyzed by employing Cattaneo-Christov heat flux model. Mathematical formulation of problem is conceded in the form of PDE’s by implementing boudary layer approach and later on converted into ODE’s with the assistance of transformation procedure. The resulting equations are solved numerically using shooting and Runge–Kutta methods. Impact of involved parameters on flow distributions is displayed through graphs. From the analysis it is inferred that Cattaneo Christov heat flux law exhibits hyperbolic equation which follows the causality principle and make the problem more compatible to real world applications. It is also deduced that magnetic field suppresses the velocity field and associated boundary layer region. Decrease in temperature profile and heat transfer coefficient is found against inciting magnitude of thermal relaxation parameter. Substantial decrease in velocity is found against increasing magnitude of Williamson fluid parameter and magnetic field parameter whereas skin friction coefficient increments. Confirmation about present findings is executed by making comparison with existing literature.

Navier slip effect on the thermal-flow of Walters’ liquid B flow due to porous stretching/shrinking with heat and mass transfer

The current work investigates the steady, 2D laminar flow for effect of mass transpiration and Navier slip on the flow of Walters’ liquid B flow due to porous stretching/shrinking with heat and mass transfer. The governing equations which are a set of highly nonlinear partial differential equations are converted into a set of highly nonlinear ordinary differential equations by adopting similarity transformations and solved analytically to get velocity solution exponential form. Furthermore, the heat transfer in the flow is balanced with heat source/sink, and solved analytically to obtain solutions for temperature and mass transfer in terms of hypergeometric function upon considering the two types of boundary conditions, prescribed power law surface temperature (PST) and prescribed power law heat flux (PHF), prescribed power law surface concentration (PSC) and prescribed mass flux (PMF). The effect of mass transfer, porous media, Reynolds number, viscoelastic parameter and Navier slip on momentum and effect of Prandtl number, radiation parameter, and Schmidt number on heat and mass transfer is studied. The results are compared with previous works and were well matched with those works. Present analysis of heat and mass transfer with porous media is motivated by a wide range of technological/engineering applications which includes ground pollution by chemicals which are non-Newtonian like lubricants and polymers, in the treatment of sewage sludge in drying beds and in liquid -based systems involving stretchable/shrinkable materials.

Computational modelling of nanofluid flow over a curved stretching sheet using Koo–Kleinstreuer and Li (KKL) correlation and modified Fourier heat flux model

• Flow and heat transfer characteristics of nanoliquid by using Koo–Kleinstreuer and Li (KKL) correlation is studied. • Modified Fourier heat flux law is utilised to model the temperature equation. • Viscous dissipation, activation energy and chemical reaction effects are used to study heat and mass transfer characteristics. • Accurate numerical solutions are obtained using Runge-Kutta-Fehlberg fourth fifth order along with shooting technique. The objective of the current paper is to study the two-dimensional, incompressible nanofluid flow over a curved stretching sheet coiled in a circle. Further, the impact of dispersion of nanoparticle CuO in base liquid water on the performance of flow, thermal conductivity and mass transfer using KKL model in the presence of Cattaneo-Christov heat flux and activation energy is deliberated. A curvilinear coordinate system is used to develop the mathematical model describing the flow phenomena in the form of partial differential equations . Further, by means of apt similarity transformations the governing boundary value problems are reduced to ordinary differential equations. Mathematical computations are simplified using Runge-Kutta-Fehlberg-45(RKF-45) process by adopting shooting method. Graphical illustrations of velocity, temperature, concentration gradients for various pertinent parameters are presented. The result reveals that, the heightening of porosity parameter heightens the thermal gradient but converse trend is depicted in velocity gradient. The enhancing values of Schmidt number and chemical reaction rate parameter declines concentration gradient whereas converse trend is depicted for upsurge in activation energy parameter.

Two-phase flow of dusty fluid with suspended hybrid nanoparticles over a stretching cylinder with modified Fourier heat flux

The current paper explores the influence of hybrid nanoparticles on the dusty liquid flow through a stretching cylinder by employing the modified Fourier heat flux law. Two phase model is implemented in the present research to characterise the fluid flow. Molybdenum disulphide and silver are used as nanoparticles suspended in base fluid water. The equations which represent the described flow are changed into a set of ordinary differential equations by opting appropriate similarity variables. The reduced dimensionless nonlinear ODEs are numerically solved out by using Runge–Kutta–Fehlberg fourth fifth order inclusive of shooting approach. The impact of several dimensionless parameters over velocity and thermal gradients are deliberated by using graphs. Graphical illustrations for skin friction are also executed. Result outcome reveals that, rise in values of mass concentration of particle declines the velocity and thermal gradient of both dust and fluid phases and cumulative in curvature parameter upsurges the velocity and thermal gradient within the boundary. Further, the heightening of thermal relaxation parameter enhances the thermal profile of both fluid and dust phases.

Micropolar Couple Stress Nanofluid Flow by Non-Fourier’s-Law Heat Flux Model past a Stretching Sheet

In this investigation, thermal radiation effect on MHD nonlinear convective micropolar couple stress nanofluid flow by non-Fourier’s-law heat flux model past a stretching sheet with the effects of diffusion-thermo, thermal-diffusion, and first-order chemical reaction rate is reported. The robust numerical method called the Galerkin finite element method is applied to solve the proposed fluid model. We applied grid-invariance test to approve the convergence of the series solution. The effect of the various pertinent variables on velocity, angular velocity, temperature, concentration, local skin friction, local wall couple stress, local Nusselt number, and local Sherwood number is analyzed in both graphical and tabular forms. The range of the major relevant parameters used for analysis of the present study was adopted from different existing literatures by taking into consideration the history of the parameters and is given by 0.07 ≤ Pr ≤ 7.0 , 0.0 ≤ λ , ε ≤ 1.0 , 0.0 ≤ R d , D f , S r , K , ≤ 1.5 , 0.0 ≤ γ E ≤ 0.9 , 0.9 ≤ S c ≤ 1.5 , 0.5 ≤ M ≤ 1.5 , 0.0 ≤ β ≤ 1.0 , 0 . 2 ≤ N b ≤ 0 . 4 , 0 . 1 ≤ N t ≤ 0 . 3 . The result obtained in this study is compared with that in the available literatures to confirm the validity of the present numerical method. Our result shows that the heat and mass transfer in the flow region of micropolar couple stress fluid can be enhanced by boosting the radiation parameters.

Double diffusive LTNE porous convection with Cattaneo effects in the solid

AbstractThe impact of Cattaneo heat flux law in the solid on the onset of double‐diffusive Darcy porous convection with local thermal nonequilibrium temperatures is investigated. The Fourier law of heat transfer is invoked for the fluid, whereas the Cattaneo heat flux law used to transfer heat in solid skeleton alters the temperature equation from parabolic to hyperbolic. The results are obtained for porous skeletons of aluminum and copper oxides. Both Cattaneo and solute concentration effects reinforce in controlling the onset of oscillatory convection and some novel consequences are observed. Compared with the results perceived in the absence of solute concentration, a manifestation of oscillatory convection with scaled‐interphase heat transfer coefficient as well as solid thermal relaxation time parameter initiates earlier in its presence. The effect of increasing interphase heat transfer coefficient and the Lewis number is to delay and hasten the onset of stationary and oscillatory convection. Besides, the increase in the value of solid thermal relaxation time parameter advances the oscillatory onset. Although the increase in the solute Darcy–Rayleigh number is to delay the stationary onset, it shows a twofold behavior on the onset of oscillatory convection. Before the onset of oscillatory convection, the size of the convection cell gets narrower and after which it becomes much wider. The existing results are retrieved as limiting cases from the current study.