Unconditional nonlinear stability for double‐diffusive convection in a porous medium with temperature‐dependent viscosity and density

In this study, fluid flow in a porous medium is analyzed using a Forchheimer model. The problem of double‐diffusive convection is addressed in such a porous medium. We utilize a higher‐order approximation for viscosity‐temperature and density‐temperature, such that the perturbation equations contain more nonlinear terms. For unconditional stability, nonlinear stability has been achieved for all initial data by utilizing the or norms. It also shows that the theory of is not sufficient for such unconditional stability. Both linear instability and nonlinear energy stability thresholds are tested using three‐dimensional (3D) simlations. If the layer is salted above and salted below then stationary convection is dominant. Thus the critical value of the linear instability thresholds occurs at a real eigenvalue , and our results show that the linear theory produces the actual threshold. Moreover, it is known that with the increase of the salt Rayleigh number, R_c, the onset of convection is more likely to be via oscillatory convection as opposed to steady convection. The 3D simulation results show that as the value of R_c increases, the actual threshold moves towards the nonlinear stability threshold, and the behavior of the perturbation of the solutions becomes more oscillatory.     

Three-dimensional numerical simulation of free convection and entropy generation in a cubic cavity containing a heat source

The present article provides a three-dimensional numerical investigation of thermal convection and entropy generation. The lattice Boltzmann method, coupled with the finite difference approach, is applied to perform numerical simulations. The validation of these numerical approaches for thermal convection simulation and entropy calculation is performed by comparing our numerical results with those in the published literature for the case of benchmark problems. The physical geometry studied in this paper concerns a hot obstacle having the shape of a plus sign (+) placed in the center of a cubic enclosure. This cube is filled with air of a Prandtl number of 0.71 and characterized by two cold vertical walls. The heat exchange between the fluid and the hot body is studied as a function of the Rayleigh number (${10}^3\le \mathit{Ra}\le {10}^7$). The performed simulations show that the heat transfer rate can be increased by about 429% by switching from $\mathit{Ra}={10}^3$ to ${10}^7$. The entropy generation due to fluid friction, heat transfer, and total entropy are also calculated and discussed. For an irreversibility coefficient $\mathit{\phi }={10}^{-4}$, the analysis of the results showed that for low values of the Rayleigh number ($\mathit{Ra}={10}^3$), the entropy production due to temperature gradients predominates over that produced by viscous effects. In the cases of $\mathit{Ra}={10}^4$ and ${10}^5$, entropy generation is due to both fluid friction and heat transfer. However, when the Rayleigh number becomes large ($\mathit{Ra}{\ge 10}^6$), entropy generation due to viscosity predominates over entropy production related to heat exchange. These results have important implications for the optimization and design of heat transfer systems in various industrial applications.     

The onset of Darcy‐Brinkman convection in a porous medium layer with vertical throughflow and variable gravity field effects

In this study, the influence of constant throughflow and varying downward gravity field on the onset of Darcy‐Brinkman convection in a porous medium layer is examined numerically. We considered two cases of gravity field variations: (a) linear and (b) parabolic. A higher‐order Galerkin‐weighted residual technique and QZ procedure are used to get the numerical solution for the entire Darcy numbers amid $\text{Da}=0$ (Horton‐Rogers‐Lapwood convection) and $\text{Da}\to \infty$ (Rayleigh Bénard Convection). It is obtained that the throughflow, the downward gravity field and the Darcy number are to holdup the beginning of convection. It is also obtained that the system is more stable for linear variation of gravity field in comparison to the parabolic variation.     

Lattice Boltzmann simulation of natural convection and heat transfer from multiple heated blocks

This study is aimed to investigate the natural convection heat transfer from discrete heat sources (similar to heated microchips) using Bhatnagar‐Gross‐Krook lattice Boltzmann method via graphics process unit computing. The simulation is carried out separately for three and six heated blocks model for different Rayleigh numbers and fixed Prandtl number, $Pr=0.71$ (air). The uniformly heated blocks are placed at the bottom wall inside a rectangular enclosure. The enclosure is maintained by the cold temperature at its left and right walls. The top and bottom surface is maintained by adiabatic conditions apart from the regions where blocks are attached to the bottom wall. The numerical code is validated with the benchmark heat transfer problem of side‐heated square cavity as well as with an experimental study for one discrete heat source. The rate of heat transfer is presented in terms of the local Nusselt and average Nusselt number for each block. It is found that the heat transfer rate becomes maximized in the leftmost and rightmost blocks due to the adjacent cold walls. It is found that the number of blocks and their positions play a substantial role in determining their collective performance on the heat transfer rate.     

Effect of viscous dissipation on finding dual solutions for mixed convection boundary layer flow for nanofluid

The effect of viscous dissipation on mixed convection boundary layer flow for Ag‐water nanofluid under steady‐state condition has been studied numerically for both the buoyancy assisting and opposing flows over a vertical semi‐infinite flat plate. A new co‐ordinate system has been introduced to transform the governing partial differential equations (PDEs) to facilitate the numerical calculations. Then, the local similarity method has been used for approximating the transformed PDEs to ordinary differential equations. Further, the quasi‐linearization method has been introduced to linearize the nonlinear equations and then numerical integration has been carried out using implicit trapezoidal rule along with the principle of superposition. For higher Pr, the coupled differential equations behave like stiff differential equations. To overcome the situation, orthonormalization process has been introduced. The effects of solid volume fraction of nanoparticles , the mixed convection parameter , Prandtl number , and Eckart number have been analyzed on the heat transfer and flow characteristics. It has been observed that the dual solutions are obtained for buoyancy opposing flow only and the range of dual solutions have become wider with the increases in . Further, nanofluids enhance the heat transfer process as compared to conventional heat transfer fluids . Moreover, the addition of viscous dissipation causes less heat transfer in the boundary.     

Study of turbulent natural convection coupled with thermal radiation in a vertical cavity having a large form factor

In this study, a numerical simulation study of turbulent natural convection coupled with thermal radiation in a vertical cavity differentially heated and filled with air assumed as a transparent fluid was carried out. The cavity has a variable form factor which can reach large values. The vertical walls are subjected to constant temperatures (T_c and T_f), whereas the horizontal walls are assumed adiabatic. The flow inside the cavity is turbulent and turbulence was modeled by using the $K?\epsilon$ model, and to take into account of the radiative transfer, the discrete ordinate model (DO) was introduced. To solve the different equations, Ansys‐Fluent software based on the finite volume method was used. Some numerical results obtained for the Rayleigh number value of 10¹¹ have been validated by some existing results in the theory. It is found that the thermal radiation has a significant influence on the flow structure and temperature variation where the flow becomes reinforced. It accelerates the airflow inside the cavity and gives the formation of significant velocity and temperature gradients along the walls of the cavity. Taking into account of the surface, thermal radiation is essential in the correct evaluation of temperature in the cavity.     

Nanoscale heat transfer investigation of an array of impinging jet systems with different working fluids under crossflow with and without pin fins

In the current numerical study, the thermal and flow field performance of an array of confined multiple jets with air, water, and water‐Al₂O₃ nanofluid in the maximum crossflow configuration over the target plate with and without pin fins is investigated. The numerical results are validated with the experimental data; it is found that a reasonable prediction related to heat transfer can be made. For this study, steady‐state Reynolds‐averaged Navier‐Stokes simulations with the shear‐stress transport $k‐\omega$ turbulence model in ANSYS Fluent were performed. The simulations are performed with volumetric concentration $?=0.2\%$ to 3% and the jet's Reynolds number Re = 15 000 to 35 000. In all cases, the jet outlet‐to‐target plate distance $Z/D$ is 3. It is found that the increase in values of the volumetric concentration of nanoparticles results in a decrease of the Nusselt number and an increase of the convective heat transfer coefficient. This is because there is an increase in thermal conductivity of the working fluid with the increase in the volumetric concentration of nanoparticles for the same Reynolds number. About 81.5% and 89.1% enhancement in the average heat transfer flux values is observed for flat and pin fin‐roughened target plates, respectively, for $?=3\%$.     

Mixed convection flow of a Maxwell nanofluid with Hall and ion‐slip impacts employing the spectral relaxation method

This article investigates the Hall and ion‐slip impacts on the mixed convection flow of a Maxwell nanofluid over an expanding surface in a permeable medium. The impacts of Brownian movement and thermophoresis parameters, Soret, Dufour, viscous dissipation, chemical reaction, and suction parameters, are, moreover, considered. Using the similitude changes, the partial differential equations with regard to the momentum, energy, and concentration equations are transformed to an arrangement of nonlinear ordinary differential equations, which are handled numerically utilizing a spectral relaxation method (SRM). The impacts of noteworthy physical parameters on the velocities, thermal, and concentration distributions are investigated graphically. Moreover, the numerical values of skin‐friction coefficients, local Nusselt number, and Sherwood number for different values of the mixed convection parameter $\left(\gamma \right),$ Deborah number $\left(\lambda \right),$ Hall parameter $({\beta }_{\mathrm{H}}),$ ion‐slip parameter $({\beta }_{\mathrm{i}}),$ Dufour number (Du), and Soret number $\left(\mathit{Sr}\right)$ are computed and tabulated. It is discovered that ascent in Deborah number reduces both the stream and transverse velocity profiles, while the inverse pattern is seen with augmentation in the mixed convection parameter. In addition, inverse patterns of the stream and transverse velocity profiles are seen with expansion in magnetic, Hall, and ion‐slip parameters. Besides this, the temperature and concentration disseminations decline with augmentation in Dufour number and chemical reaction parameters, respectively. It is likewise seen that both the skin‐friction coefficients lessen with expansion in Deborah number, and they ascend with upgrade in blended convection and ion‐slip parameters, while the opposite condition is noticed with augmentation in Hall parameter. Furthermore, the reverse trends of local Nusselt and Sherwood numbers are discovered with expansion in the Dufour and Soret numbers.     

Laminar natural convection of power‐law fluids over a horizontal heated flat plate

Natural convection of power‐law fluids over a horizontal flat plate with constant heat flux is studied. The stretching transformations relating the similarity forms of the boundary layer velocity, pressure, and temperature profiles are applied to the governing boundary layer equations. The resultant set of coupled ordinary differential equations are solved analytically and numerically using the integral method and the finite difference method, respectively. The results are presented for the details of the velocity and temperature fields for various values of the non‐Newtonian power‐law viscosity index (n) and the generalized Prandtl number (Pr*). At a fixed value of the viscosity index, increasing the Prandtl number increases the skin friction and wall temperature. For Pr* > 1, a lower viscosity index results in larger wall skin friction, temperature scale, and thermal boundary layer thickness, and thus lower Nusselt number. The reverse trend is observed for Pr* < 1. By using an integral solution and the numerical results, a semi‐analytical correlation for the Nusselt number is obtained, valid for and .     

Nonlinear mixed convection flow of a tangent hyperbolic fluid with activation energy

In this communication, the dynamics of a non‐Newtonian tangent hyperbolic fluid with nanoparticles past a nonuniformly thickened stretching surface is discussed. We examine the impact of nonlinear mixed convection flow of a hyperbolic tangent fluid with the Cattaneo‐Christov heat and mass diffusion model past a bidirectional stretching surface. The effects of activation energy and magnetic field are incorporated in the analysis. The variables of transformations are used to change the nonlinear partial differential equations into ordinary differential equations (ODEs). Then, these ODEs are numerically solved using the Matlab routine of the bvp4c algorithm. The derailed analysis of the influences of the governing parameters on velocities along the x‐ and y‐axes, temperature and concentration profiles are presented using tables and figures. The outcomes of these parameters reveal that the velocities along the x‐ and y‐axes are decreased for the values of We increasing but the opposite behavior is observed as the value of $A$ increases. The results also show that the values of $e$ and $Nb$ rise as the temperature profiles increase. Similar influences are observed on the profile of concentration as the values of $F$ and $f$ rise. As the values of $N1$ go from 0.27 to 0.25, the skin‐friction coefficient increases, and similarly, as $Nb$ goes from 0.3 to 0.1, $?\theta \prime (0)$ is enhanced.     

Natural convection coupled to surface radiation in an air-filled square cavity containing two heat-generating bodies

The present numerical study focuses on the cooling by natural convection and surface radiation of two electronic components generating two different and uniform volumetric powers. These components are modeled by two square bodies placed inside a closed square cavity with a cold straight wall. Two configurations are analyzed based on the position of the two heat-generating bodies. In the first one (horizontal position configuration), the two bodies are located at the same height of the cavity, while they are placed at different heights in the second case (vertical position configuration). The effects of two Rayleigh numbers ($0\le ({\mathit{Ra}}_1,{\mathit{Ra}}_2)\le {10}^6$), the conductivity ratio ($0.01\le K\le 100$), and the emissivity ($0\le \epsilon \le 1$) on the heat transfer characteristics and the flow structure are analyzed. The data is displayed as streamlines, isotherms, velocity, and maximum temperature profiles, and local heat transfer on the active wall. The obtained results indicate that the choice of the appropriate configuration depends mainly on the deviation between the two Rayleigh numbers. Furthermore, the maximum temperature of a specific block decreases as the quantity of heat generated by the other block rises. We can also see that the maximum temperature of the two blocks decreases by about $50\%$ with the increase in the emissivity (from $0$ to $1$) or the conductivity ratio (from $0.1$ to $1$).     

Computation of entropy generation in dissipative transient natural convective viscoelastic flow

Entropy generation is an important aspect of modern thermal polymer processing optimization. Many polymers exhibit strongly non‐Newtonian effects and dissipation effects in thermal processing. Motivated by these aspects in this study, a numerical analysis of the entropy generation with viscous dissipation effect in an unsteady flow of viscoelastic fluid from a vertical cylinder is presented. The Reiner‐Rivlin physical model of grade 2 (second‐grade fluid) is used, which can envisage normal stress variations in polymeric flow‐fields. Viscosity variation is included. The obtained governing equations are resolved using implicit finite difference method of Crank‐Nicolson type with well imposed initial and boundary conditions. Key control parameters are the second‐grade viscoelastic fluid parameter (), viscosity variation parameter (), and viscous dissipation parameter (). Also, group parameter (), Grashof number (Gr), and Prandtl number (Pr) are examined. Numerical solutions are presented for steady‐state flow variables, temperature, time histories of friction, wall heat transfer rate, entropy, and Bejan curves for distinct values of control parameters. The results specify that entropy generation decreases with augmenting values of , , and Gr. The converse trend is noticed with increasing Pr and . Furthermore, the computations reveal that entropy and Bejan lines only occur close to the hot cylinder wall.     

Tangent hyperbolic nanofluid with mixed convection flow: An application of improved Fourier and Fick's diffusion model

This article presents a tangent hyperbolic fluid with the effect of the combination of forced and natural convection flow of nanoparticle past a bidirectional extending surface. Modified Fick's and Fourier's diffusion theories are incorporated into concentration and energy equations, respectively. Convective boundary conditions and second‐order slip flow are taken in the boundary condition. Nonlinear partial differential equations result after boundary layer approximations of the mathematical formulation of the flow problem. Nonlinear high order ordinary differential equations (ODEs) are formed by applying similarity transformation on the nonlinear partial differential equations. The transformed equations are solved with the bvp4c algorithm from Matlab. The numerical solution of ODEs was obtained and the effect of interesting parameters, dimensionless velocity component along x‐ and y‐axis, temperature, and concentration particle, Re_x, Re_y, , and , were presented through tables and graphs and discussed thoroughly. The results indicated that a decrease in velocity along with the y‐axis results from the increasing behavior of S, M, and n. Decrease in both temperature and concentration results in an increase of but their elongation is a result of increase in Bi. An increase in concentration results in decrease of N and S but a decrease in concentration results in the widening of Sc, Nb, and . Furthermore, enlargement of and results in increase of and modules and elongation of both and results in increase of and (Sc and Nb), respectively. A comparison with previously published literature was performed and a good agreement was found.     

Study of magnetohydrodynamic convection between two rotating cylinders filled with casson fluid and the viscous dissipation effect

We have studied the forced convection of a viscous incompressible and electrically conducting non‐Newtonian Casson fluid between two rotating cylinders with viscous dissipation effect. An angular velocity and gives to inner and outer cylinders, respectively, and constant heat flux presented on the inner cylinder surface. Also, the outer cylinder is taken as insulated. Here, we have discussed three cases: (i) The inner cylinder is rotating with a constant angular velocity while the outer cylinder is at rest; (ii) Both the cylinders rotate in the identical direction with equal angular velocity; and (iii) Both the cylinders are rotating with equal angular speed but the outer cylinder rotates in the opposite direction of the inner cylinder. The governing equations are solved by numerical techniques using MATLAB Software and the results are obtained graphically.     

Natural convection boundary layer flow of nanofluids around different stations of the sphere and into the plume above the sphere

The main intent of the present study is to investigate the natural convection boundary layer flow of nanofluids around different stations of the sphere and eruption of the fluid from the boundary layer in to the plume above the sphere. It is pertinent to point out that in this study heated sphere is treated as point source. The system of transport boundary layer equations is based on the effects of Brownian motion and thermophoresis. The system of dimensioned boundary layer equations is transformed into nondimensional form. Later, the nondimensional form of the mathematical model is solved numerically by using implicit finite difference method. The solution of the problem depends on a controlling parameters Prandtl number P_r, Lewis number , thermophoresis parameter , and Brownian motion parameter . Particularly, it is observed that for Lewis number , Prandtl number P_r, Brownian motion parameter , and thermophoresis parameter the velocity profile is maximum at station and minimum at station . On the other hand temperature distribution is uniform at each station around the sphere and slightly reduced for . It is also observed that nanoparticles concentration is maximum at station and minimum at station We also established the result that with the increase of skin friction is reduced while the heat and mass flux are increased in the plume region‐III.     

Numerical simulation of nanoparticle shape and thermal ray on a CuO/C2H6O2–H2O hybrid base nanofluid inside a porous enclosure using Darcy's law

In this study, the physical aspects of magnetohydrodynamic flow and heat transfer of a hybrid base nanofluid in a porous medium under the effect of the shape, thermal radiation, and Lorentz force have been examined using the finite element method. Copper oxide (CuO) of various shapes was dispersed into ethylene glycol 50%‐water 50% (likewise for Fe₃O₄). The Darcy model is chosen because of the porous medium. The effect of changeable, diverse parameters, for example, Hartmann number (Ha), volume fraction (), radiation parameter (), and buoyancy force (Ra), on the streamlines, temperature gradient, and Nusselt number are shown through contours. Outputs show that the Fe₃O₄ nanoparticles have a smaller temperature gradient than that of CuO nanoparticles. The Nusselt number decreases for a larger (Ha) number, but increases for a larger Ra, Rd. The blade shaped nanoparticle has a larger impact on increasing compared with that of other shapes.     

Conjugate local thermal nonequilibrium and non-Darcy flow inside porous enclosure: Analysis of localized heating and cooling arrangements

This study covers a simulation on conjugate free convective in a porous enclosure containing a side wall thickness and partially heated and cooled from sides under the considerations of local thermal nonequilibrium (LTNE) and non-Darcy flow. Interest has been focused on how the side wall thickness and the locations of cooled and heated parts affect the effectiveness of the Nusselt number (Nu). Three different cases of localized heating and cooling locations have been implemented for the following ranges: scaled heat transfer coefficient ($0.1\le H\le 100$), wall to fluid thermal conductivity ratio ($0.1{\le R}_k\le 100$), modified Rayleigh number ($200\le \mathit{Ra}*\le 1000$), wall width ($0.1\le \hat{Z}\le 0.5$), inertial parameter (${10}^{-4}{\le F}_s/P_r^{*}\le {10}^{-2}$), and thermal conductivity ratio ($0.1\le K_r\le 100$). Outcomes show that $\hat{Z}$ and the locations of cooled and heated parts have remarkable impacts on all the Nusselt numbers. The intensity of LTNE region considerably relies on $\mathit{Ra}*$, $K_r$ and $H$. The total average Nu_T is highly dependent on $R_k$, $\mathit{Ra}*$, $\hat{Z}$, $F_s/P_r^{*}$, and $K_r$ as compared to H. The increase in $\hat{Z}$ leads to change of the convective mechanism to conductive mode. The rise in $R_k$ guides to increase Nu, where $R_k$ can control the flow strength. The actions of $F_s/P_r^{*}$ on Nu_f is more evident than Nu_s. For low H and K_r, the size of LTNE zone is considerably affected by H as compared to K_r although K_r has a high influence on Nu. For high K_r and H, the LTNE zone has closely vanished. Findings display that the Case 2 provided the highest Nu for all tested parameters except the case of $K_r=0.1$. Finally, it is evident that for the problems that employed solid conduction wall with localized heating and cooling sections, Case 2 is recommended for future use in the applications that implement a porous medium and depend on free convection.     

A semianalytical technique for MHD peristalsis of pseudoplastic nanofluid with temperature‐dependent viscosity: Application in drug delivery system

An analysis has been implemented to study the influences of nonconstant viscosity and magnetohydrodynamics on pseudoplastic nanofluid through a porous medium. Ohmic dissipation, chemical reaction, and heat generation are taken into consideration. The current problem is debated under the molds of tiny or zero and approximation. Two models of nonconstant viscosity are deliberated. Model (I)—all parameters are nondimensional and have been measured as constants inside the flow. Model (II)—all these stated nondimensional parameters have been considered to differ with the temperature. Comparison among the solutions achieved by utilizing numerical results and multi‐step differential transform method (Ms‐DTM) is displayed in excellent agreement. Attention is dedicated to , ‐, and parameters. The governing equations for each case have been explained by an easy and highly perfect series established seminumerical Ms‐DTM utilizing Mathematica 11, which uses mathematical software package. Nanofluids are active for drug carrying and drug delivery systems because of the control with the velocity of fluid.     

Modeling and validation of heat transfer in packed bed with internal heat generation

Several researchers have modeled the heat transfer in a packed bed, heated externally, and determined its effective thermal conductivity ( $k_{\mathit{eff}}$). But till date, very few researchers have studied the heat transfer of the pebble bed, where the heat is generated inside the bed; and the effective thermal conductivity of the packed bed with internal heat generation has not yet been reported. In the present work, heat generation inside the bed has been imitated by inductively heating randomly placed steel balls with lithium titanate ( ${\text{Li}}_2{\text{TiO}}_3$) pebbles. The system has been modeled and validated with experimental results. The $k_{\mathit{eff}}$ of the ${\text{Li}}_2{\text{TiO}}_3$ pebble bed is determined for various process conditions. A correlation has been developed to calculate the $k_{\mathit{eff}}$ based on various process parameters such as pebble diameter, air flow rate, and induction temperature. The result presented in this study will be used for the design and scale‐up studies of future fusion reactors.