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Try to find a way to treat wastewater and achieve its purification from suspended waste, which was removed by examining the magneto-Williamson fluid on a horizontal cylindrical tube while taking advantage of solar radiation and nanotechnology.

The effect of Cattaneo–Christoph law of heat transfer, solar radiation, oblique magnetic field, porosity and internal heat generation on the flow was studied. The control system was solved by the numerical technique of Chebyshev pseudospectrum (CPS) with the help of the program MATHEMATICA 12. The tables comparing the published data results with the existing numerical calculation match exactly.

The tables comparing the published data results with the existing numerical calculation match exactly. The current simulation results indicate that when using variable viscosity, the Nusselt number and surface friction decrease significantly compared to their value in the case of constant viscosity, and variable viscosity has a significant effect on flow, which reduces speed. Curves and increasing temperature profiles.

Developing a theoretical framework for the problem of sewage turbidity in a healthier and less costly way, by studying the flow of Williamson fluid with variable viscosity (to describe the intensity of sewage turbidity) on a horizontal cylindrical tube, and taking advantage of nanotechnology, solar radiation, Christoph’s thermal law and internal heat generation to reach water free of impurities. Inclined magnetic force and porous force were used, both of which played an effective role in the purification process.

Studying the shear stress and pressure resulting on the walls of blood vessels, especially during high-pressure cases, which may lead to the explosion or rupture of these vessels, can also lead to the death of many patients. Therefore, it was necessary to try to control the shear and normal stresses on these veins through nanoparticles in the presence of some external forces, such as exposure to some electromagnetic shocks, to reduce the risk of high pressure and stress on those blood vessels. This study aims to examines the shear and normal stresses of electroosmotic-magnetized Sutterby Buongiorno’s nanofluid in a symmetric peristaltic channel with a moderate Reynolds number and curvature. The production of thermal radiation is also considered. Sutterby nanofluids equations of motion, energy equation, nanoparticles concentration, induced magnetic field and electric potential are calculated without approximation using small and long wavelengths with moderate Reynolds numbers.

The Adomian decomposition method solves the nonlinear partial differential equations with related boundary conditions. Graphs and tables show flow features and biophysical factors like shear and normal stresses.

This study found that when curvature and a moderate Reynolds number are present, the non-Newtonian Sutterby fluid raises shear stress across all domains due to velocity decay, resulting in high shear stress. Additionally, modest mobility increases shear stress across all channel domains. The Sutterby parameter causes fluid motion resistance, which results in low energy generation and a decrease in the temperature distribution.

Equations of motion, energy equation, nanoparticle concentration, induced magnetic field and electric potential for Sutterby nano-fluids are obtained without any approximation i.e. the authors take small and long wavelengths and also moderate Reynolds numbers.