The behavior of hybrid nanofluid and stagnation point flow toward a stretched surface along with melting heat transfer, second-order slip, electric field, and magnetic field effect is investigated in this study. Hybrid nanoparticles alumina and copper (Cu) are considered with the base fluids water . The PDEs with corresponding boundary constraints are transformed into a set of nonlinear ODEs using similarities transformation. The set of nonlinear ODEs are analyzed analytically using semianalytical method HAM in Mathematica software. Dual solution is determined which relaying on the emerging parameters included magnetic field, volume fractions, electric field, dimensionless surface velocity slip factors, Eckert number, and Prandtl number. The results are shown in the velocity and temperature curves as well as skin friction coefficient and local Nusselt number. The analysis shows that velocity profile is an increasing function of slip parameter, electric field, while reducing function of magnetic field. Temperature profile is an increasing function of magnetic field parameter, electric field parameter, volume fraction parameter, and Eckert number, while reducing function of Prandtl number. The main outcomes are as follows that hybrid nanofluids are higher thermal properties as compared to conventional fluids. As a result, hybrid nanofluid has numerous uses in engineering cosmetics, automotive industry, home industry, for cancer treatment, food packaging, pharmaceuticals, fabrics, paper plastics, paints, ceramics, food colorants, and soaps as well.

In this paper, the Darcy–Forchheimer laminar thin film flow with MHD and heat transfer on an unsteady horizontal stretched surface is investigated. The impact of thermal radiation and viscous dissipation is also considered for thin film flow. The analysis of heat source with thermal radiation in a boundary layer flow can play a great role in manufacturing engineering procedures such as the production of electric power, solar energy modernization, and astrophysical flow. By using similarity transformation, the system of PDEs is converted to ODEs. Then HAM is applied for the solution of the problem. Moreover, the velocity and temperature profile for various embedded variables are discussed through graphs while the numerical solution of concerned physical quantities such as skin friction and Nusselt number are discussed through tables. The analysis shows that velocity profile is reducing function of , , , and . The temperature profile is an increasing function and while a reducing function of and . The overall behavior of the proposed system is highlighted analytically.

Theoretical influence of the buoyancy and thermal radiation effects on the MHD (magnetohydrodynamics) flow across a stretchable porous sheet were analyzed in the present study. The Darcy–Forchheimer model and laminar flow were considered for the flow problem that was investigated. The flow was taken to incorporate a temperature-dependent heat source or sink. The study also incorporated the influences of Brownian motion and thermophoresis. The general form of the buoyancy term in the momentum equation for a free convection boundary layer is derived in this study. A favorable comparison with earlier published studies was achieved. Graphs were used to investigate and explain how different physical parameters affect the velocity, the temperature, and the concentration field. Additionally, tables are included in order to discuss the outcomes of the Sherwood number, the Nusselt number, and skin friction. The fundamental governing partial differential equations (PDEs), which are used in the modeling and analysis of the MHD flow problem, were transformed into a collection of ordinary differential equations (ODEs) by utilizing the similarity transformation. A semi-analytical approach homotopy analysis method (HAM) was applied for approximating the solutions of the modeled equations. The model finds several important applications, such as steel rolling, nuclear explosions, cooling of transmission lines, heating of the room by the use of a radiator, cooling the reactor core in nuclear power plants, design of fins, solar power technology, combustion chambers, astrophysical flow, electric transformers, and rectifiers. Among the various outcomes of the study, it was discovered that skin friction surges for 0.3 $\le F_1\le$ 0.6, 0.1 $\le k_1\le$ 0.4 and 0.3 $\le M\le$ 1.0, snf declines for 1.0 $\le G_r\le$ 4.0. Moreover, the Nusselt number augments for 0.5 $\le R\le$ 1.5, 0.2 $\le N_t\le$ 0.8 and 0.3 $\le N_b\le$ 0.9, and declines for 2.5 $\le Pr\le$ 5.5. The Sherwood number increases for 0.2 $\le N_t\le$ 0.8 and 0.3 $\le Sc\le$ 0.9, and decreases for 0.1 $\le N_b\le$ 0.7.

Aggregation operators serve as fundamental mathematical tools for synthesizing multiple inputs into a single representative output. In this study, we introduce a family of aggregation operators (AOs) tailored to the framework of complex Pythagorean fuzzy sets (CPyFSs). Owing to their superior ability to capture both amplitude and phase information, CPyFSs offer enhanced flexibility in modeling uncertainty and vagueness, making them highly suitable for real-world decision-making problems. Motivated by these advantages, this paper develops several novel AOs within the CPyFS environment to address multi-attribute decision-making (MADM) problems more effectively. To further enhance the flexibility and robustness of the proposed operators, Aczél–Alsina (AA) operational laws are incorporated. In particular, we propose the complex Pythagorean fuzzy Aczél–Alsina Heronian mean (CPyFAAHM) operator and the complex Pythagorean fuzzy Aczél–Alsina geometric Heronian mean (CPyFAAGHM) operator, which integrate AA operations with Heronian mean and geometric Heronian mean structures. We investigate the key mathematical properties of the proposed operators, including their structural characteristics and aggregation behavior. Additionally, weighted versions of these operators are developed to accommodate the varying importance of decision attributes. Based on these operators, a MADM methodology is constructed for decision-making under the CPyFS framework. Finally, the effectiveness and practicality of the proposed approach are demonstrated through a numerical example, followed by a comparative analysis with existing MADM methods under complex Pythagorean fuzzy information, highlighting its superiority and applicability.

Complex Fermatean fuzzy sets effectively handle two‐dimensional periodic uncertain information through complex‐valued membership and nonmembership functions. The role of distance measures becomes pivotal when analyzing such sets, yet existing methodologies, despite extensive research, frequently generate illogical outcomes in various scenarios. However, existing distance measures of complex Fermatean fuzzy sets exhibit counterintuitive results and fail to satisfy fundamental axiomatic properties. Addressing this gap, this study introduces a novel distance measure of complex Fermatean fuzzy sets that incorporates membership, nonmembership, and hesitancy grades while considering both amplitude and phase terms. We rigorously prove that the proposed measures satisfy the properties of a distance function. Besides, based on 2D and 3D Jensen–Shannon divergence measures, we further present a normalized distance measure of complex Fermatean fuzzy sets. Moreover, numerical examples verify that normalized distance measures can obtain more reasonable and superior results. A multiattribute decision‐making (MADM) framework is developed to demonstrate practical applicability. Based on the proposed algorithms, a comprehensive methodology involving these proposed measures has been illustrated, along with their implementation in solving different problems in pattern classification and decision‐making under uncertainty with periodic characteristics. Additionally, a comparative analysis has been conducted to provide better clarity and understanding of the effectiveness of these measures. The results indicate that the proposed distance measures offer significant advantages and improved accuracy for pattern classification, the computer‐aided engineering market, and medical diagnostic problems.