Search Results (37)

The Second Law of Thermodynamics states that entropy S increases in a spontaneous process in an ideal isothermal and isolated system. Real systems are influenced by external forces and fields, including the temperature field. In this case, only entropy is not enough, and we suggest using a new function, $L_s$, which is analogous to the Lagrangian in classical mechanics. It includes total potential energy but instead of mechanical kinetic energy, $L_s$ includes the product ST, and the system always evolves towards increasing this modified Lagrangian. It reaches an equilibrium when total potential force is balanced by both entropic and thermal forces. All forces have the same units, Newton/mol, and may be added or subtracted. For condensed systems with friction forces, it is a molecular transport velocity, and not acceleration, which is proportional to the acting force. Our approach has several advantages compared to Onsager’s non-equilibrium thermodynamics with its thermodynamic forces, which may have different units, including 1/T for energy transport. For isolated systems, the description is reduced to Second Law and Clausius inequality. It easily explains diffusion, Dufour effect, and Soret thermodiffusion. The combination of electric, thermal, and entropic forces explains thermoelectric phenomena, including Peltier–Seebeck and Thomson (Lord Kelvin) effects. Gravitational and entropic forces together inside a black hole may lead to a steady state or the black hole evaporation. They are also involved in and influenced by solar atmospheric processes. Full article

This article reports an investigation of the Soret and Dufour effects on the double-diffusive natural convection of $A{l}_2{O}_3$-$H_{2}O$ nanofluids in a U-shaped porous enclosure. Numerical problems were resolved using the multiple-relaxation-time (MRT) lattice Boltzmann method (LBM). The indented part of the U-shape was cold, and the right and left walls were heated, while the bottom and upper walls were adiabatic. The experimental data-based temperature and nanoparticle size-dependent correlations for the $A{l}_2{O}_3$-water nanofluids are used here. The benchmark results thoroughly validate the graphics process unit (GPU) based in-house compute unified device architecture (CUDA) C/C++ code. Numeral simulations were performed for a variety of dimensionless variables, including the Rayleigh number, ($Ra$ = ${10}^4,{10}^5,{10}^6$), the Darcy number, ($Da$ = ${10}^{-2},{10}^{-3},{10}^{-4}$), the Soret number, ($Sr$ = $0.0,0.1,0.2$), the Dufour number, ($D_f$ = $0.0,0.1,0.2$), the buoyancy ratio, ($-2\le Br\le 2$), the Lewis number, ($Le$ = $1,3,5$), the volume fraction, ($0\le \varphi \le 0.04$), and the porosity, $ϵ$ = ($0.2-0.8$), and the Prandtl number, $Pr$ = $6.2$ (water) is fixed to represent the base fluid. The numerical results are presented in terms of streamlines, isotherms, isoconcentrations, temperature, velocity, mean Nusselt number, mean Sherwood number, entropy generation, and statistical analysis using a response surface methodology (RSM). The investigation found that fluid mobility was enhanced as the $Ra$ number and buoyancy force increased. The isoconcentrations and isotherm density close to the heated wall increased when the buoyancy force shifted from a negative magnitude to a positive one. The local $Nu$ increased as the Rayleigh number increased but reduced as the volume fraction augmented. Furthermore, the mean $Nu$ ($\overline{Nu}$) decreased by $3.12\%$ and $6.81\%$ and the $\overline{Sh}$ increased by $83.17\%$ and $117.91\%$ with rising Lewis number for ($Ra={10}^5$ and $Da={10}^{-3}$) and ($Ra={10}^6$ and $Da={10}^{-4}$), respectively. Finally, the $Br$ and $Sr$ demonstrated positive sensitivity, and the $Ra$ and $\varphi$ showed negative sensitivity only for higher values of $\varphi$ based on the RSM. Full article

The present review analyzes classes of exact solutions for the convection and thermal diffusion equations in the Boussinesq approximation. The exact integration of the Oberbeck–Boussinesq equations for convection and thermal diffusion is more difficult than for the Navier–Stokes equations. It has been shown that the exact integration of the thermal diffusion equations is carried out in the Lin–Sidorov–Aristov class. This class of exact solutions is a generalization of the Ostroumov–Birikh family of exact solutions. The use of the class of exact solutions by Lin–Sidorov–Aristov makes it possible to take into account not only the inhomogeneity of the pressure field, the temperature field and the concentration field, but also the inhomogeneous velocity field. The present review shows that there is a class of exact solutions for describing the flows of incompressible fluids, taking into account the Soret and Dufour cross effects. Accurate solutions are important for modeling and simulating natural, technical and technological processes. They make it possible to find new physical mechanisms of momentum transfer for the design of new types of equipment. Full article

We present a new exact solution of the thermal diffusion equations for steady-state shear flows of a binary fluid. Shear fluid flows are used in modeling and simulating large-scale currents of the world ocean, motions in thin layers of fluid, fluid flows in processes, and apparatuses of chemical technology. To describe the steady shear flows of an incompressible fluid, the system of Navier–Stokes equations in the Boussinesq approximation is redefined, so the construction of exact and numerical solutions to the equations of hydrodynamics is a very difficult and urgent task. A non-trivial exact solution is constructed in the Lin-Sidorov-Aristov class. For this class of exact solutions, the hydrodynamic fields (velocity field, pressure field, temperature field, and solute concentration field) were considered as linear forms in the x and y coordinates. The coefficients of linear forms depend on the third coordinate z. Thus, when considering a shear flow, the two-dimensional velocity field depends on three coordinates. It is worth noting that the solvability condition given in the article imposes a condition (relation) only between the velocity gradients. A theorem on the uniqueness of the exact solution in the Lin–Sidorov–Aristov class is formulated. The remaining coefficients of linear forms for hydrodynamic fields have functional arbitrariness. To illustrate the exact solution of the overdetermined system of Oberbeck–Boussinesq equations, a boundary value problem was solved to describe the complex convection of a vertical swirling fluid without its preliminary rotation. It was shown that the velocity field is highly stratified. Complex countercurrents are recorded in the fluid. Full article

Activation energy can be elaborated as the minimal energy required to start a certain chemical reaction. The concept of this energy was first presented by Arrhenius in the year 1889 and was later used in the oil reservoir industry, emulsion of water, geothermal as well as chemical engineering and food processing. This study relates to the impacts of mass transfer caused by temperature differences (Soret) and heat transport due to concentration gradient (Dufour) in a Carreau model with nanofluids (NFs), mixed convection and a magnetic field past a stretched sheet. Moreover, thermal radiation and activation energy with new mass flux constraints are presumed. All chemical science specifications of nanofluid are measured as constant. As a result of the motion of nanofluid particles, the fluid temperature and concentration are inspected, with some physical description. A system of coupled partial differential frameworks is used mathematically to formulate the physical model. A numerical scheme named the Runge–Kutta (R-K) approach along with the shooting technique are used to solve the obtained equations to a high degree of accuracy. The MATLAB R2022b software is used for the graphical presentation of the solution. The temperature of the nanofluid encompasses a quicker rate within the efficiency of a Dufour number. An intensifying thermal trend is observed for thermophoresis and the Brownian motion parameter. The Soret effect causes a decline in the fluid concentration, and the opposite trend is observed for rising activation energy. In addition, the local Nusselt number increases with the Prandtl number. Further, the comparative outcomes for drag force are established, with satisfying agreement with the existing literature. The results acquired here are anticipated to be applied to improving heat exchanger thermal efficiency to maintain thermal balancing control in compact heat density equipment and devices. Full article

The current study used a novel Casson model to investigate hybrid Al₂O₃-Cu/Ethylene glycol nanofluid flow over a moving thin needle under MHD, Dufour–Soret effects, and thermal radiation. By utilizing the appropriate transformations, the governing partial differential equations are transformed into ordinary differential equations. The transformed ordinary differential equations are solved analytically using HAM. Furthermore, we discuss velocity profiles, temperature profiles, and concentration profiles for various values of governing parameters. Skin friction coefficient increases by upto 45% as the Casson parameter raised upto 20%, and the heat transfer rate also increases with the inclusion of nanoparticles. Additionally, local skin friction, a local Nusselt number, and a local Sherwood number for many parameters are entangled in this article. Full article

A numerical investigation of unsteady boundary layer flow with heat and mass transfer of non-Newtonian fluid model, namely, Jeffrey fluid subject, to the significance of Soret and Dufour effects is carried out by using the local nonsimilarity method and homotopy analysis method. An excellent agreement in the numerical results obtained by both methods is observed and we establish a new mathematical approach to obtain the solutions of unsteady-state flow with heat and mass transfer phenomenons. Similarity transformation is applied to governing boundary layer partial differential equations to obtain the set of self-similar, nondimensional partial differential equations. Graphical results for different emerging parameters are discussed. The dimensionless quantities of interest skin friction coefficient, Sherwood number, and Nusselt number are discussed through tabulated results. The main novelty of the current work is that the average residual error of the $mth$-order approximation of the OHAM scheme for steady-state solution is decreased for higher-order approximation. Further, a rapid development of the boundary layer thickness with the increasing values of dimensionless time $\tau$ is observed. It is noted that for large values of $\tau$, the steady state in the flow pattern is gained. It is worth mentioning that the magnitude of Sherwood number is increased with the increasing values of Schmidt number $Sc$ and Dufour number $D_f$. The magnitude of local Nisselt number is increased for the increasing values of Soret number, $Sr$. Full article

The study is devoted to investigating the effect of an unsteady non-Newtonian Casson fluid over a vertical plate. A mathematical analysis is presented for a Casson fluid by taking into consideration Soret and Dufour effects, heat generation, heat radiation, and chemical reaction. The novelty of the problem is the physical interpretation of Casson fluid before and after adding copper water-based nanoparticles to the governing flow. It is found that velocity was decreased and the temperature profile was enhanced. A similarity transformation is used to convert the linked partial differential equations that control flow into non-linear coupled ordinary differential equations. The momentum, energy, and concentration formulations are cracked by means of the finite element method. The thermal and solute layer thickness growth is due to the nanoparticles’ thermo-diffusion. The effects of relevant parameters such as the Casson fluid parameter, radiation, Soret and Dufour effects, chemical reaction, and Prandtl number are discussed. A correlation of the average Nusselt number and Sherwood number corresponding to active parameters is presented. It can be noticed that increasing the Dufour number leads to an uplift in heat transfer. Fluid velocity increases with Grashof number and decreases with magnetic effect. The impact of heat sources and radiation is to increase the thermal conductivity. Concentration decreases with the Schmidt number. Full article

The focal interest in this article is to investigate the Stefan blowing and Dufour and Soret effects on hybrid nanofluid (HNF) flow towards a stretching cylinder with thermal radiation. The governing equations are converted into ODE by using suitable transformations. The boundary value problem solver (bvp4c), which is a package in the MATLAB, is used to solve the resulting ODE equations. Results show that rise in the Stefan blowing enhances velocity, temperature, and concentration profiles. Heat transfer rate increases by up to 10% in the presence of 4% nanoparticle/HNF but mass transfer rate diminishes. Additionally, skin friction coefficient, Nusselt number and Sherwood number are examined for many parameters entangled in this article. Additionally, results are deliberatively discussed in detail. Full article

In this study, an analysis of the rotating flow of viscoelastic Oldroyd-B fluid along with porous medium featuring the Soret–Dufour effects is explored. The heat transport mechanism is discussed with the involvement of thermal radiation and heat source/sink. Additionally, the thermophoresis of particle deposition and chemical reaction are taken into the concentration equation in order to investigate the mass transportation in the liquid. To formulate the non-linear ordinary differential equations, the von Karman similarity approach is used in the system of partial differential equations and then integrated numerically by the bvp midrich scheme in Maple programming. Results are provided by graphical framework and tabular form. A quick parametric survey is carried out concerning flow field, thermal, and solutal distributions through graph representation. The curves show that increasing the values of the retardation time parameter decreases the radial velocity while increasing the angular velocity. Additionally, when the relaxation time parameter becomes powerful, the magnitude of the velocity curves decreases considerably in the radial and axial directions. The presence of a radiation parameter indicates that the fluid will absorb a greater amount of heat, which is equivalent to a higher temperature. Further, an increase in the stretching parameter leads to a reduction in the temperature components. Full article

Heat-transfer enhancement in microchannel heat sinks (MCHS) has been a hot topic in the last decade. However, most published works did not focus on the heat sources that are discrete, as in most microelectronic devices, and the enhancement of heat and mass transfer (HMT) due to the Soret and Dufour effects being ignored. Based on a heterogeneous two-phase model that takes into consideration the Soret and Dufour effects, numerical simulations have been performed for various geometries and heat sources. The numerical results demonstrate that the vortices induced by a heat source(s) can enhance the heat transfer efficiency up to 2665 W/m²·K from 2618 W/m²·K for a discrete heat source with a heat flux q = 10⁶ W/m². The Soret effect can affect the heat transfer much more than the Duffour effect. The integrated results for heat transfer due to the Soret and Dufour effects are not sampled superpositions. Discrete heat sources (DHS) arranged in microchannels can enhance heat transfer, especially when the inlet velocity of the forced flow is less than 0.01 m/s. This can provide a beneficial reference for the design of MCHS with DHS. Full article

In this paper, we study the magnetohydrodynamics of Darcy flow in a non-Newtonian liquid. The influence of thermophoresis on particle deposition is examined in the Darcy flow of a Maxwell nanofluid. In our model, the temperature distribution is generated by the Fourier law of heat conduction with nonlinear thermal radiation and heat sink/source. We also examine the Soret–Dufour effects in the mass concentration equations. The Brownian and thermophoretic diffusions are assumed to be generated by nanoparticle dispersion in the fluid. The similarity method is used to transform the partial differential equations into nonlinear ordinary differential equations. The transformed flow equations were solved numerically using the BVP Midrich scheme. The results of the computation are displayed graphically and in tabular form. The results obtained show that increasing the Deborah number leads to a decline in radial and angular motion and a decrease in the magnitude of axial flow. As expected, the strength of the heat source and the values of the thermal radiation parameters determine the temperature of the liquid. We also found that as the Soret number rises (or the Dufour number falls), so does the mass transfer rate. Full article

Here, our main aim is to examine the impacts of Dufour and Soret in a radiative Darcy–Forchheimer flow. Ohmic heating and the dissipative features are outlined. The characteristics of the thermo-diffusion and diffusion-thermo effects are addressed. A binary chemical reaction is deliberated. To examine the thermodynamical system performance, we discuss entropy generation. A non-linear differential system is computed by the finite difference technique. Variations in the velocity, concentration, thermal field and entropy rate for the emerging parameters are scrutinized. A decay in velocity is observed for the Forchheimer number. Higher estimation of the magnetic number has the opposite influence for the velocity and temperature. The velocity, concentration and thermal field have a similar effect on the suction variable. The temperature against the Dufour number is augmented. A decay in the concentration is found against the Soret number. A similar trend holds for the entropy rate through the radiation and diffusion variables. An augmentation in the entropy rate is observed for the diffusion variable. Full article

The widespread application of hybrid nanofluid in real applications has been accompanied by a large increase in computational and experimental research. Due to the unique characteristics of hybrid nanofluid, this study aspires to examine the steady two-dimensional mixed convection stagnation point flow of a hybrid nanofluid past a vertical plate with radiation, Dufour, and Soret effects, numerically. The formulations of the specific flow model are presented in this study. The model of fluid flow that is expressed in the form of partial differential equations is simplified into ordinary differential equations via the transformation of similarity, and then solved numerically by using the boundary value problem solver known as bvp4c in MATLAB, which implements the finite difference scheme with the Lobatto IIIa formula. Two possible numerical solutions can be executed, but only the first solution is stable and meaningful from a physical perspective when being evaluated via a stability analysis. According to the findings, it is sufficient to prevent the boundary layer separation by using 2% copper nanoparticles and considering the lesser amount of Dufour and Soret effects. The heat transfer rate was effectively upgraded by minimizing the volume fraction of copper and diminishing the Dufour effect. Stronger mixed convection would lead to maximum skin friction, mass transfer, and heat transfer rates. This important preliminary research will give engineers and scientists the insight to properly control the flow of fluids in optimizing the related complicated systems. Full article

The significance of radiation, Soret and Dufour’s effects on MHD flow in a porous media near a stagnation point past a vertical plate with slip, temperature, and concentration boundary conditions were investigated. Local similarity variables are used in the solution, which reduces the PDEs into analogous boundary value problem for ODEs. Symmetry analysis can be used to detect these variations in local similarity. To numerically explain the problem, a shooting approach and the MATLAB bvp4c solver are utilized. As the magnetic field and porous medium parameters are raised, the skin friction increases, and the temperature increases as the radiation pointer is increased. As the Soret number grows, the concentration profile rises. Full article