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The purpose of the current research is to inspect the free convection of the nanofluid (Cu–water) within a hexagonal cavity containing a square obstacle with isothermal vertical walls at Th{T}_{{\rm{h}}} and Tc{T}_{{\rm{c}}}, and insulated horizontal walls. The aim of this study is to analyze the interaction between the Rayleigh number (103

The primary purpose of this research is to examine how the presence of thermal features variation affects the velocity and heat transfer rate of nanofluids composed of sodium alginate and molybdenum disulfide [Na-Alg/MoS2]m and sodium alginate and molybdenum disulfide and graphene oxide [Na-Alg/MoS2 + GO]h, respectively, flowing between two rotating, permeable plates. Both centripetal and Coriolis forces, which act on a spinning fluid, are taken into account. The impacts of magnetized force, thermal radiative flux, heat source (sinking), and varied pressure in the Darcy–Forccheimer material are considered. Using the physical vapor deposition method, single and hybridity nanofluid thin films of thickness 150 ± 5 nm may be created. The controlling mathematical equations of the suggested model are solved using the Keller-box technique in MATLAB software. The surface friction coefficient of a hybrid nanofluid is less, and the heat transfer rate is greater than that of a regular nanofluid. The rate of heat transmission is slowed by the rotational parameter. The thermal efficiency of mono nanofluids is as low as 6.16% and as high as 21.88% when compared to those of hybrid nanofluids. In particular, the findings of density functional theory (DFT) calculations reveal that the energy bandgap ΔEgOpt\Delta {E}_{{\rm{g}}}^{{\rm{Opt}}} drops from 1.641 eV for conventional nanofluid to 0.185 eV for hybridity nanofluid. Based on the findings, the addition of graphene oxide nanoparticles to the base nanofluid converts it from a semi-conductor to a hybridity nanofluid as a superconductor.

Thermal energy from the Sun comes mostly from sunlight. These energies might be used in photovoltaic cells, sustainable power systems, solar light poles, and water-collecting solar pumps. This age studies solar energy and how direct sunshine might improve solar panel efficiency. Solar energy, especially solar tiles, is widely used in manufacturing today. The literature includes a modified Buongiorno hybrid nanofluid prototype. There are no studies that have examined the impact of tri-hybrid and unique tetra hybridity nanomolecules integrated with the Buongiorno nanofluid prototype on liquid moving on a flexible surface. This study examines the effects of an improved Buongiorno tetra hybrid nanoliquid prototypical with Buongiorno and Tiwari–Das nanofluid on magnetized double-diffusive binary nanofluid with cross fluid and Maxwell liquid flowing with variant thermal conductance over a porous medium. Different profiles include diffusion thermo and thermo diffusion. The LobattoIIIA scheme’s convergence and stability are examined in terms of residual error, mesh points for ordinary differential equations (ODEs), and boundary conditions. Leading equations about liquid flow continuity, impetus, temperature, and concentricity are obtained using continuity, conservation of momentum, the second law of thermodynamics, Fick’s second law of diffusion, and boundary layer expectations. The system of partial differential equations obtained from the given assumption becomes a system of ODEs and well-established LobattoIII. Their numerical solution is obtained using a numerical technique. Statistical charts and tables provide numerical solutions. The heat transport rate of tetra-hybrid nanomolecules increases dramatically, unlike tri- and di-hybrid nanomolecules. The improved Buongiorno tetra hybrid nanofluid (BTHNF) model produces more heat when radiation Rd{\rm{Rd}}, Brownian diffusion Nb{\rm{Nb}}, and thermal conductivity are increased. The data show that the diffusion factor LL, Brinkman number Br{\rm{Br}}, and Reynolds number Re increase entropy production, but Bejan number reduces it owing to an increase in Be{\rm{Be}} and Re\mathrm{Re}. A statistical regression study shows that retaining the Maxwell fluid parameter constant and increasing the Weissenberg number We{\rm{We}} decrease the drag coefficient error. A BTHNF model containing tetra hybrid nanoparticles has not been utilized to examine heat and mass transferences in non-Newtonian fluids, considering diffusion, thermo, and thermo diffusion. Entropy generation in a binary fluid with tetra hybrid nanoparticles and BTHNF has not been studied. Tetra hybrid nanofluid is not mentioned in the literature. This effort aims to create a new tetra-hybrid nanofluid model. This article is novel because it investigates the effects of thermal radiation, thermal conductivity, porosity, Darcy–Forchheimer, and Buongiorno models on a tetra-hybrid nanofluid flow under an extensible sheet.

This article presents a comprehensive literature survey on fractional-order optimal control problems. Fractional-order differential equation is extensively used nowadays to model real-world systems accurately, which exhibit fractal dimensions, memory effects, as well as chaotic behaviour. These versatile features attract engineers to concentrate more on this, and it is widely used in the broad domain of science and technology. The mentioned numerical tools take the necessary optimal conditions into account, which makes it a two-point boundary value problem of non-integer order. In this review article, some numerical approaches for the approximation have been stated for obtaining the solution to fractional optimal control problems (FOCPs). Here, few numerical approaches including Grunwald-Letnikov approximation, Adams type predictor-corrector method, generalized Euler’s method, Caputo-Fabrizio method Bernoulli and Legendre polynomials method, Legendre operational method, and Ritz’s and Jacobi’s method are treated as an advanced method to obtain the solution of FOCP. Fractional delayed optimal control is selected for our investigation. It refers to a type of control problem where the control action is delayed by a fractional amount of time. In other words, the control input at a given time depends not only on the current state of the system but also on its past state at fractional times. The fractional delayed optimal control problem is formulated as an optimization problem that seeks to minimize a cost function subject to a set of constraints that represent the dynamics of the system and the fractional delay in the control input. The solution to this problem typically involves the use of fractional polynomials types, i.e. Chebyshev and Bassel polynomials.

This study introduces a three-dimensional numerical analysis of the mixing yield stress fluid inside stirred vessels. The Bingham–Papanastasiou model predicts the yield stress behavior of the working fluid. The implications of a new anchor impeller design are investigated; it involves certain modifications to a typical anchor impeller’s blade. Different curved shapes replace the straight blade of a classical anchor. The flow pattern and energy consumed inside the stirred system for various geometrical configurations, Reynolds numbers (0.1, 1, 10, and 100), and Bingham numbers (1, 10, 100, and 500) have been investigated. According to the findings, introducing this new geometrical configuration gives a significant acceleration of flow pattern and extension of the well-mixed zone, as well as decreased power consumption (Np{\rm{Np}}). Three configurations were introduced to the mixing system: Case 1 is a standard anchor impeller, and Cases 2 and 3 are curved anchor impellers with two shapes different. Case 3 (helical design of blade) has been found to be the less power consumption case by five times, i.e., for Case 1 and Case 2, Np=5{\rm{Np}}=5, whereas for Case 3, Np=1{\rm{Np}}=1. Based on the results, it is evident that the configuration (Case 3) demonstrates a superior geometric design in enhancing mixing characteristics compared to the other configurations.

In this study, the steady 2D flow of micropolar fluid via a vertical surface is taken into account. The magnetohydrodynamics applied normally to the flow direction at a vertical surface in the presence of temperature-dependent attributes. The effect of the chemical reaction under the generalized Fourier–Fick law is considered to investigate the heat transference rate at the vertical sheet. Under the flow assumptions, the boundary layer approximations were applied to the nonlinear differential equations and partial differential equations were obtained. The use of similarity modifications allows for a reduction in the number of partial differential equations. The resulting ordinary differential equations are then resolved numerically using a technique known as the homotopy analysis method. The results reveal that microparticle suspensions have a significant impact on the flowing domain when varied fluid characteristics are utilized. The effect of potential factors on flow, micro-rotation velocities, temperature, drag force factor, and heat transport rate is investigated. The obtained results show that the velocity profile and micropolar function increase for larger values of micropolar parameters. Drag force effects are also seen, and required outcomes are observed to be in outstanding accord with the available literature. Significant results of this work were toward the velocity function, which gets reduced with increasing magnetic field parameter values, but the velocity function enhances for higher values of β\beta and λ\lambda . On temperature distribution, it decreased for higher values of ϵ1{{\epsilon }}_{1} and temperature profile declines due to higher values of Pr\text{Pr}, γ2{\gamma }_{2} and γ1{\gamma }_{1} or both cases of δ>0\delta \gt 0 and δ

This demonstration of study focalizes the melting transport and inclined magnetizing effect of cross fluid with infinite shear rate viscosity along the Skan-Falkner wedge. Transport of energy analysis is brought through the melting process and velocity distribution is numerically achieved under the influence of the inclined magnetic dipole effect. Moreover, this study brings out the numerical effect of the process of thermophoresis diffusion and Brownian motion. The infinite shear rate of viscosity model of cross fluid reveals the set of partial differential equations (PDEs). Similarity transformation of variables converts the PDEs system into nonlinear ordinary differential equations (ODEs). Furthermore, a numerical bvp4c process is imposed on these resultant ODEs for the pursuit of a numerical solution. From the debate, it is concluded that melting process cases boost the velocity of fluid and velocity ratio parameter. The augmentation of the minimum value of energy needed to activate or energize the molecules or atoms to activate the chemical reaction boosts the concentricity.

The goal of this research is to consider the thermal impact on varied convection flow in hybrid nanofluids with heat generation over a two-dimensional heated flat around a stretchable sheet. The flow is considered steady and incompressible while the stretchable sheet is assumed an impermeable. Two distinctive nano-level particles are considered, namely copper (Cu) and silver (Ag) with ethylene glycol base fluid. The boundary layer was generated on a stretchable sheet surface by mixed convection flow in hybrid nanofluids. Ideally, the sink and source are thermal reservoirs of internal thermal capacities. This means you can extract or reject heat from them without changing their temperature. To make a study of thermodynamic systems like heat engines and refrigerator systems, the governing equations were solved numerically with Keller-box methodology depending on the implicit finite-difference technique. Research findings were worked with the parameters of mixed convection, Prandtl number, nanoparticle volume fraction, through various non-dimensional parameters, and heat generation. Especially for thermal generation enhancement, the fluidity and thermal dispersal get elevated. Even though the flowing behavior and the thermal dispersal of hybridity fluids with the combinations of Cu and Ag nanoparticles were similar, their values are distinct, which reflect in graphical displays. The hybrid nanofluidity gets improved with the volume variation of nanoparticles if the ϕ\phi value is 0.01≤ϕ≤0.050.01\le \phi \le 0.05 and if the flow profile value decreases ϕh{\phi }_{{\rm{h}}}, where 0.01≤ϕh≤0.050.01\le {\phi }_{{\rm{h}}}\le 0.05 as the dispersal of temperature enhances when the nanoparticle nanofluid constraint is improved.

Since obtaining an analytic solution to some mathematical and physical problems is often very difficult, academics in recent years have focused their efforts on treating these problems using numerical methods. In science and engineering, systems of integral differential equations and their solutions are extremely important. The Taylor collocation method is described as a matrix approach for solving numerically Linear Differential Equations (LDE) by using truncated Taylor series. Integral equations are used to solve problems such as radiative transmission and the oscillation of a string, membrane, or axle. Differential equations can be used to tackle oscillating difficulties. To discover approximate solutions for linear systems of integral differential equations with variable coefficients in terms of Taylor polynomials, the collocation approach, which is offered for differential and integral equation solutions, will be developed. A system of LDE will be translated into matrix equations, and a new matrix equation will be generated in terms of the Taylor coefficients matrix by employing Taylor collocation points. The needed system will be converted to a linear algebraic equation system. Finding the Taylor coefficients will lead to the Taylor series technique.

This theoretical work suggests a novel nonlinear thermal radiation and an applied magnetic feature-based three-dimensional Casson nanomaterial flow. This flow is assumed in the rotating frame design. Gyrotactic microorganisms (GMs) are utilized in the Casson nanofluid to investigate bioconvection applications. The altered Buongiorno thermal nano-model is used to understand the thermophoretic and Brownian mechanisms. Convective boundary conditions must be overcome to solve the flow problem. With suitable variables, the dimensionless pattern of equations is obtained. The solutions to the nonlinear formulations are then obtained using semi-analytical simulations using a homotopy analysis mechanism. It was found that the velocity outline is enhanced with the enhancing estimations of the buoyancy ratio, rotation factor, and Casson parameter while it is reduced with mixed convection, porosity, slippery parameters, and Rayleigh number. The temperature profile is increased with radiation, the temperature ratio, the thermophoretic parameter, the Brownian parameter, and the Biot number. The Brownian parameter reasons an improvement in the concentration outline contrary to the thermophoretic parameter. The concentration of GMs is decreased with the Peclet number inversely to the Lewis number effect, which causes an increase in the microorganisms’ concentration.

There are flow research centers on magnetohydrodynamic (MHD) emission of auxiliary liquid in an extended region. The prevailing model is constrained by attractions/infusion and gooey release. The administering model is based on the Koo–Kleinstreuer–Li nanofluid model in the existence of entropy generation. Final requirements of this model are addressed by implementing the shooting strategy, which incorporates a fourth approach for the Runge–Kutta strategy. Into the bargain, the last adds (in standard ordinary differential equations (ODE) divisions) are obtained from the measurable controls partial differential equations, which were represented toward the start of the overseeing model. The varieties for all boundaries are exhibited through graphical arrangements. It is noticed that expanding the substantial volume portion diminishes speed but builds nuclear power dispersion. Likewise, the classification of mathematical qualities on divider heat move rate and skin contact is introduced. Both Reynolds and Brinkman numbers improve the entropy rate of the thermal system resulting in the growth effects of inertial forces and the surface heat dissipation, respectively.

The mathematical modeling of hybrid nanofluid flow and heat transfer with entropy generation toward parabolic trough surface collector (PTSC) inside the solar-powered ship (SPS) is performed. The mathematical model used non-Newtonian Oldroyd-B model amidst a constant inclined magnetic field influence is being considered. The mathematical model is then reduced by adopting appropriate similarity transformation into a higher-order nonlinear differential equations system. The reduced model is computed using the well-known technique called the Keller Box scheme. Physical parameters effectiveness, for instance, thermal radiation, viscous dissipation, hybrid nanoparticles, and Joule heating, is displayed in graphs. The silver-ethylene glycol (Ag-EG) characteristic performance outperformed the silver-magnetite-ethylene glycol (Ag-Fe3O4/EG). The maximum efficiency of Ag-EG is about 26.3%, while the minimum is at least 5.6%.

The heat transport and entropy formation of an unsteady Sutterby hybrid nanofluid (SBHNF) are investigated in this work. SBHNF’s flowing and thermal transport properties are investigated by exposing the nanofluid to a slippery hot surface. This analysis includes the influences of solid-shaped nanoparticles, porous materials, radiative flux, and viscous dissipative flow. The Galerkin finite element technique (G-FEM) is used to find self-similar solutions to equations that are then transformed into ODEs using appropriate transformations. This research considers two diverse kinds of nanosolid-particles, copper (Cu) and graphene oxide (GO), using non-Newtonian engine-oil (EO) as the working fluid. In the flowing, energy, skin friction, Nusselt number, and entropy production, important findings for the various variables are visually depicted. The most notable finding of the analysis is that when SBHNF (GO–Cu/EO) is compared to a typical nanofluid (Cu–EO), the thermal transmission rate of SBHNF (GO–Cu/EO) gradually increases. Furthermore, heat transfer is greatest for spherical-shaped nanoparticles and lowest for lamina-shaped nanoparticles. The entropy in the model is increased when the size of the nanoparticles ϕ\phi is increased. The comparable impact is noticed once the radiation flowing Nr{N}_{\text{r}} and Deborah number λ\lambda increase.