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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

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.

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.

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.

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 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.

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.