These days, heat transfer plays a significant role in the fields of engineering and energy, particularly in the biological sciences. Ordinary fluid is inadequate to transfer heat in an efficient manner, therefore, several models were considered for the betterment of heat transfer. One of the most prominent models is a single-phase nanofluid model. The present study is devoted to solving the problem of micropolar fluid with a single-phase model in a channel numerically. The governing partial differential equations (PDEs) are converted into nonlinear ordinary differential equations (ODEs) by introducing similarity transformation and then solved numerically by the finite difference method. Response surface methodology (RSM) together with sensitivity analysis are implemented for the optimization analysis. The study reveals that sensitivity of the skin friction coefficient (Cf_x) to the Reynolds number (R) and magnetic parameter (M) is positive (directly proportional) and negative (inversely proportional) for the micropolar parameter.

Studying waste discharge concentration across a convergent/divergent channel is essential in environmental-related applications. Successful environmental administration must understand the behavior and concentration of waste contaminants released into these channels. Analyzing waste discharge concentrations aids in determining the efficacy of treatment techniques and regulatory controls in lowering pollutant scales. Because of this, the current analysis examines the ternary-based nanofluid flow across convergent/divergent channels, including non-uniform heat source/sink and concentration pollutants. The study also concentrates on understanding the movement and heat transmission characteristics in ternary-based nano-liquid systems with divergent and convergent channels and maximizing the ternary nanofluid flow’s effectiveness. The equations representing the flow, temperature, and concentrations are transformed into a system of ODEs (ordinary differential equations) and are obtained by proper similarity variables. Further, solutions of ODEs are gathered by using the Runge Kutta Fehlberg 4-5 (RKF-45) method and shooting procedure. The significant dimensionless constraints and their impacts are discussed using plots. The results mainly focus on improving local and external pollutant source variation will enhance the concentration for the divergent channel while declining for the convergent channel. Adding a solid fraction of nanoparticles will escalate the surface drag force. These findings may enhance heat management, lessen pollutant dispersion, and enhance the circulation of nanofluid systems.

Wastewater discharge plays a vital role in environmental management and various industries. Water pollution control and tracking are critical for conserving water resources and maintaining adherence to environmental standards. Therefore, the present analysis examines the impact of pollutant discharge concentration considering the non-Newtonian nanoliquids over a permeable Riga surface with thermal radiation. The analysis is made using two distinct kinds of non-Newtonian nanoliquids: second-grade and Walter’s liquid B. The governing equations are made using the applications of boundary layer techniques. Utilizing the suitable similarity variable reduces the formulated governing equations into an ordinary differential set of equations. The solutions will be obtained using an efficient numerical technique and the significance of various dimensionless constraints on their individual profiles will be presented using graphical illustrations. A comparative analysis is reported for second-grade and Walter’s liquid B fluids. The results show that the porous factor declines the velocity profile for both fluids. Radiation and external pollutant source variation constraints will improve thermal and concentration profiles. The rate of thermal distribution improved with the rise in radiation and solid volume factors. Further, essential engineering factors are analyzed. The outcomes of the present study will help in making decisions and putting efficient plans in place to reduce pollution and safeguard the environment.

Significance: Bio-nanofluids have achieved rapid attention due to their potential and vital role in various fields like biotechnology and energy, as well as in medicine such as in drug delivery, imaging, providing scaffolds for tissue engineering, and providing suitable environments for cell growth, as well as being used as coolants in various energy systems, wastewater treatment, and delivery of nutrients to plants.Objective: The present study proposes a novel mathematical model for the ternary hybrid cross bio-nanofluid model to analyse the behaviour of blood that passes through a stenosed artery under the influence of an inclined magnetic field. The model considers the effect of expanding/contracting cylinder, infinite shear rate viscosity, and bio-nanofluids.Methodology: The considered model of the problem is bounded in the form of governing equations such as PDEs. These PDEs are transformed into ODEs with the help of similarity transformations and then solved numerically with the help of the bvp4c method.Findings: The results show that the flow rate and velocity decrease as the inclination angle of the magnetic field increases. Additionally, research has found that the presence of nanoparticles in the bio-nanofluid has a significant impact on the velocity and flow rate. Therefore, the flow rate decreases, in general, as the stenosis becomes more severe.Advantages of the study: The results obtained from this study may provide insights into the behaviour of blood flow in stenosed arteries and may be useful in the design of medical devices and therapies for the treatment of cardiovascular diseases.