In this study, we analyzed the three-dimensional flow of Williamson (pseudoplastic) fluids upon a linear porous stretching sheet. The thermal radiation impact was taken into account. The transformed non-linear equations were solved by the homotopy analysis method (HAM). The influence of the embedded parameters tretching parameter, Williamson parameter, porosity parameter, thermal radiation parameter, thermophoresis parameter, Brownian motion parameter, Prandtl number and Biot number are presented on velocity, temperature and concentration functions in the graphs and explained in detail. The velocity function along the x-direction reduces with the impact of the stretching, porosity and Williamson parameters. Velocity along the y-direction increases with the stretching parameter, while it reduces with the porosity and Williamson parameters. The effect of Skin friction, heat transfer and mass transfer are shown numerically. The numerical values of surface drag force and the impact of different parameters are calculated and it is observed that increasing the stretching parameter and the porosity parameter reduces the surface drag force, while increasing the Williamson parameter augments the surface drag force. Higher values of the stretching parameter, the Prandtl number and the radiation parameter enhance the heat transfer rate, while the augmented value of the thermophoresis and Brownian motion parameters reduces the heat transfer rate, where higher values of the stretching parameter, thermophoresis and Brownian motion parameters enhance the mass transfer rate.

This study presents the magnetized hybrid nanofluid flow with heat source/sink over an exponentially stretching/shrinking sheet. Slip conditions are implemented to analyze the hybrid nanofluid flow for both slip and no-slip conditions. Additionally, the hybrid nanofluid of alumina and copper (hybrid nanoparticles) with blood (base fluid) has been considered and discussed with both suction and injection parameters. The appropriate similarity variables are used to convert partial differential equations (PDEs) into ordinary differential equations (ODEs) and solved analytically with the help of the homotopy analysis method (HAM). The impact of different embedded parameters has been shown in the form of graphs and tables. The numerical values of skin friction and Nusselt number are presented in the form of Tables for both slip and no-slip cases. It is summarized that the upsurge of the velocity slip parameter and magnetic parameter increases the skin friction, while the rising of the thermal slip parameter and heat generation parameter decreases the Nusselt number.

The current article discussed the heat transfer and thermal radioactive of the thin liquid flow of Sisko fluid on unsteady stretching sheet with constant magnetic field (MHD). Here the thin liquid fluid flow is assumed in two dimensions. The governing time-dependent equations of Sisko fluid are modeled and reduced to Ordinary differential equations (ODEs) by use of Similarity transformation with unsteadiness non-dimensionless parameter $St$ . To solve the model problem, we used analytical and numerical techniques. The convergence of the problem has been shown numerically and graphically using Homotopy Analysis Method (HAM). The obtained numerical result shows that the HAM estimates of the structures is closed with this result. The Comparison of these two methods (HAM and numerical) has been shown graphically and numerically. The impact of the thermal radiation $Rd$ and unsteadiness parameter $St$ over thin liquid flow is discovered analytically. Moreover, to know the physical representation of the embedded parameters, like $\beta$ , magnetic parameter M, stretching parameter $\xi$ , and Sisko fluid parameters $\epsilon$ have been plotted graphically and discussed.

In this study paper, we examined the magnetohydrodynamic (MHD) flow of three combined fluids, Maxwell, Jeffry, and Oldroyed- B fluids, with variable heat transmission under the influence of thermal radiation embedded in a permeable medium over a time-dependent stretching sheet. The fluid flow of liquid films was assumed in two dimensions. The fundamental leading equations were changed to a set of differential nonlinear and coupled equations. For this conversion, suitable similarity variables were used. An optimal tactic was used to acquire the solution of the modeled problems. The convergence of the technique has been shown numerically. The obtained analytical and numerical consequences are associated graphically and tabulated. An excellent agreement was obtained between the homotropy analysis method (HAM) and numerical methods. The variation of the skin friction and Nusslet number and their influence on the temperature and concentration profiles were scrutinized. The influence of the thermal radiation, unsteadiness effect, and MHD were the main focus of this study. Furthermore, for conception to be physically demonstrated, the entrenched parameters are discussed graphically in detail along with their effect on liquid film flow.

The combination of q-spherical fuzzy sets and rough sets has emerged as a useful paradigm for fuzzy mathematics and decision-making. The hybrid structure of q-spherical fuzzy sets and rough sets has shown to be useful in the fields of fuzzy mathematics and decision-making. The primary goal of this research is to present Einstein’s operational principles for q-spherical rough numbers (q-SFRNs). The fundamental goal of this research is to develop geometric aggregation operators (AOs), such as q-spherical fuzzy rough Einstein weighted geometric (q-SFREWG) and q-spherical fuzzy rough Einstein ordered weighted geometric (q-SFREOWG) operators. We will look at the idempotency, boundedness, and other theorems linked with the suggested AOs. Recognizing the importance of multi-criteria decision-making (MCDM) in dealing with real-world difficulties, it is important to note that traditional MCDM procedures sometimes provide contradicting outcomes. Using the proposed AOs, this study presents a robust MCDM approach designed to address picture understanding and interpretation issues inside the q-SFRS framework. In addition, a complete comparative study is carried out to assess the suggested method’s efficacy and value in comparison to existing procedures. The findings from these comparison investigations show that our developed technique outperforms current approaches. The study emphasizes the expanded capabilities of the suggested technique in resolving the complexities of picture perception and interpretation within the q-SFRS environment, bringing a potential addition to the field of decision-making and fuzzy mathematics.

Multi-criteria decision-making (MCDM) approaches prove to be effective and reliable in addressing problems under uncertain conditions. The q-spherical fuzzy rough set (q-SFRS) represents the latest advancement in fuzzy set theory. This article aims to introduce a novel approach, q-spherical fuzzy rough Combinative Distance-based Assessment (q-SFR-CODAS), by integrating CODAS with q-SFR set to address MCDM problems. The method utilizes the Hamming distance as the primary measure and the Euclidean distance as the secondary measure to assess the desirability of alternatives, calculated concerning the negative-ideal solution. Additionally, an illustrative example is presented to demonstrate the applicability of the proposed methodology. A comprehensive sensitivity analysis is conducted to validate the results of q-SFR-CODAS, comparing them with existing MCDM methods.