Background: An infinite vertical sheet is the medium of inquiry for studying the copper- water nanofluid behavior with an inclined magnetic field and Cogley radiation. The research contains significant aspects. The use of this nanofluid is of great practical importance, especially in the areas of energy efficiency, thermal management, radiative cooling for space technology, environmental impact reduction, and improving heat transfer efficiency in electronic device cooling operations. Moreover, the research enhances the understanding and manipulation of sophisticated materials. The uttermost significance of this resides in its capacity to uncover issues inherent in the stages of planning, developing, analyzing, and improving manufacturing processes, possibly securing new solutions via patent protection. Objective: This research aims to provide a comprehensive analysis of the unsteady natural convection of a nanofluid along an infinite vertical sheet. The study focuses on understanding how an inclined magnetic field, Cogley radiation, heat source, and varying nanoparticle volume concentrations impact the velocity, thermal, and concentration profiles of the nanofluid. Additionally, the investigation seeks to evaluate the Nusselt number, Sherwood number, and skin friction coefficient across different concentration profiles to derive meaningful insights. Methods: The Laplace transform (L.T.) method, in combination with the MATLAB symbolic computing program, is used to develop numerical solutions for the main boundary value issues. The Laplace transform technique is used to clarify dimensionless partial differential equations (PDEs) and their limiting situations. An in-depth study is conducted on the profiles of momentum, energy, and concentration for various kinds of nanofluids with changing volume concentrations of nanoparticles. This research provides a comprehensive examination and also identifies prospective prospects in this field that might be protected by patents. Result: The work provides an in-depth understanding of how inclined magnetic fields, Cogley radiation, heat generation, and nanoparticle volume concentration affect the velocity, energy, and mass profiles of the nanofluid. Graphical representations depict these effects, offering a visual comprehension of the system's behavior. The Nusselt number and skin friction coefficient are determined by analytical and numerical methods. These findings demonstrate that the Nusselt number and skin friction coefficient have an inverse relationship with changes in concentration profiles. Conclusion: The work improves our comprehension of the fluctuating natural convection of nanofluids across an unbounded vertical surface, taking into account variables such as an inclined magnetic field, heat source, Cogley radiation, and nanoparticle volume concentration. The practical implications of the discoveries, together with the analytical and numerical results, enhance our comprehension of the dynamics of nanofluid systems. Ensuring the protection and widespread distribution of patented technology is essential for promoting sustainable advancements.
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