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The co-doping of vanadium pentoxide (V2O5) with rare-earth (RE) elements, namely 1.5 % holmium (Ho) and 1.5 % ytterbium (Yb) has been conducted using an eco-friendly, straightforward hydrothermal approach to assess the combined effects on structural, optical, and photocatalytic properties. The application of the density functional theory (DFT) approach effectively examined the impact of RE ions on the photocatalytic efficiency of co-doped V2O5. The stable orthorhombic crystal structure of co-doped V2O5 has been confirmed using DFT and X-ray diffraction without a secondary phase. It appears that homogeneous nucleation occurs while heterogeneous nucleation slows down in co-doped samples, as evidenced by the larger crystallite sizes in co-doped samples compared to doped ones. It means a result, the co-doped samples exhibit photodegrades more quickly and have a higher rate constant than the doped samples. This is because they have less dislocation density (4.26 × 10−3 nm−2) and internal micro-strain (4.93 × 10−3). The bandgap and degradation efficiency are determined by the UV–vis spectroscopy and found to be 2.33 eV and 95 %, respectively, at the optimal pH of 7 in the visible range. The co-doped sample has a rate constant of 24 × 10−3 min−1, which is the highest in the RE-doped V2O5 system. This is a good reason to think of co-doped V2O5 as a possible catalyst.

Nowadays, the advancement of heat transmission for the heat exchanger device is an important field of research for many researchers. In this work, a numerical study has been conducted to investigate the thermal performance of a mixed convective flow through the octagonal heat exchanger covered by hybrid nanofluid (Cu-TiO2-H2O). A magnetic field has been introduced inside the cavity to investigate the mixed convective hydrodynamics heat flow characteristics. The nanofluid cores absorb/release energy to manage heat transmission by increasing or decreasing inside the cavity domain as the host fluid and dispersed hybrid nanofluid circulate within the cavity. After transforming the governing equations into a generalized, non-dimensional formulation, the finite element approach is utilized to solve the associated equations. Additionally, response surface methodology is also applied to test the responses of the associated factors. Heat transport was examined in relation to the effects of nanofluids fusion temperature, boundary wall properties, Reynolds number, Hartmann number and nanoparticle volume fractions. The outcomes of this study are analysed by measuring streamline profiles, isotherms, average Nusselt number, velocity profile, and 2D and 3D response surfaces of the computational domain. The underlying flow controlling parameters for instance Reynolds number (10 ≤ Re ≤ 200), Hartmann number (0 ≤ Ha ≤100), and nanoparticle volume fractions (0 ≤ ϕ ≤ 0.1), the influences have been considered. The findings also reveal that the thermal performance is being boosted due to augmentation of Re and ϕ, but reverse behavior is noticed for Ha. Furthermore, the response surfaces obtained from response surfaces methodology express that the Re and ϕ have shown positive influence, and Ha has shown negative influence on Nuav. Utilizing a hybrid nanofluid of Cu-TiO2-H2O increases the heat transfer capacity of water to 25.75 %. Moreover, the findings could guide to design of a mixed convective heat exchanger for industrial purposes.

Mixed convective nanofluid flow has substantial importance in improvement of thermal performance, and thermal engineering to meet the global energy crisis. In this study, mixed convective nanofluid flow in a porous-wavy channel with an inner heated triangular obstacle under magnetic field effect is numerically examined. Nanofluid within the channel is heated and cooled from its bottom and top wavy-surfaces. A heated triangular cylinder is located at the centerline of the wavy-channel. Finite element method is utilized to solve the non-dimensional governing equations. The code is validated comparing present results with published numerical and experimental results. The response surface method is also implemented to analyze the obtained results and its sensitivity. The numerical results indicate that strength of flow velocity is accelerated with rising Reynolds number, Darcy numbers and inlet-outlet ports length but declined for Hartmann number and volume fraction. Heat transferring rate and heat transfer irreversibility are substantially increased for higher values of Reynolds number, inlet-outlet ports length, Darcy number and nanoparticle volume fraction but a reverse trend is occurred for magnetic field effect. The thermal performance is found significantly improved with simultaneous increment in Re, ϕ, Da and decrement in Ha. Positive sensitivity is achieved for input factors Re, ϕ, Da in computing Nuav while negative sensitivity to Ha. Heat transfer rate is found more sensitive to the impact of Re and ϕ compared to Da and Ha. 45.59 % more heat transmission potentiality is developed for using Al2O3–H2O nanofluid (vol.5 %) instead of using base fluid water. Heat transfer enhancement rate is decreased by 36.22 % due to impact of magnetic field strength. In addition, 84.12 % more heat transferring rate is recorded in presence of triangular obstacle. Moreover, irreversibility components are influenced significantly for the presence of heated triangular obstacle. Bejan number is also found declined for increasing physical parameters. The findings of this investigation may offer a guideline for finding experimental results to design high-performance convective heat exchangers.

Sturm-Liouville problems have yielded the biggest achievement in the spectral theory of ordinary differential operators. Sturm-Liouville boundary value issues appear in many key applications in natural sciences. All the eigenvalues for the standard Sturm-Liouville problem are guaranteed to be real and simple, and the related eigenfunctions form a basis in a suitable Hilbert space. This article uses the weighted residual collocation technique to numerically compute the eigenpairs of both regular and singular Strum Liouville problems. Bernstein polynomials over [0,1] has been used to develop a weighted residual collocation approach to achieve an improved accuracy. The properties of Bernstein polynomials and the differentiation formula based on the Bernstein operational matrix are used to simplify the given singular boundary value problems into a matrix-based linear algebraic system. Keeping this fact in mind such a polynomial with space defined collocation scheme has been studied for Strum Liouville problems. The main reasons to use the collocation technique are its affordability, ease of use, well-conditioned matrices, and flexibility. The weighted residual collocation method is found to be more appealing because Bernstein polynomials vanish at the two interval ends, providing better versatility. A multitude of test problems are offered along with computation errors to demonstrate how the suggested method behaves. The numerical algorithm and its applicability to particular situations are described in detail, along with the convergence behavior and precision of the current technique.

The slip flow and thermal transfer inside the boundary layer are extremely significant for various problems in aerodynamics, wing stall, skin friction drag on an entity, high-level velocity aircraft, etc. The current research investigated the effect of the slip factor and shape factor on the axisymmetric bullet-shaped object by taking the viscous dissipation parameter and location parameter. The analysis is conducted for both fixed and moving bullet-shaped objects due to thinner and thicker surfaces. The governing equations are transformed into a system of ordinary differential equations using suitable local axisymmetric similarity transformations and solved by applying the spectral quasi-linearization method. A new correlation analysis is made for velocity and temperature gradients. It is observed that the boundary layer structure has no defined shape due to a thicker bullet-shaped object instead it forms a steep angle with the axis which is contradictory to the formation of the boundary layer. A negative correlation is observed for the parameters M, Ec, Q*, and s but a positive correlation is observed for the parameters such as Pr, P, λ, and ε. The surface thickness and stretching ratio significantly affect the fluid flow and heat transfer processes. It is also noticed that the thinner bullet-shaped object performs as a better cooling conductor compared to a thicker one. The skin friction is reduced in the case of a thinner bullet-shaped object compared to a thicker one. The present analysis reveals that the heat transfer rate and the friction factor may be helpful in industrial sectors for controlling the cooling rate and quality of the final product. This research brings forward to increase in the rate of heat transfer inside the boundary layer region. The result may help to design the various types of moving objects in the automobile engineering sector when the objects pass through the fluid.