Melting heat transmission with erratic movement in a nanofluid having outer velocity across a stretching sheet

A study was conducted investigating the effects of melting heat and mass transfer in a nanofluid flow over an unsteady moving surface. The outer velocity is assumed to act perpendicular to the stretching surface. This study also discusses the consequence of Brownian motion and thermophoresis parameter. The governing partial dofferential equations is change into a nonlinear ordinary differential equation utilizing similarity variables and the transformed nonlinear differential was calculated the Runge-Kutta Fehlberg method in MATLAB. Effects of relevant parameters Prandtl number, unsteadiness, Lewis number, melting parameter on velocity, temperature and concentration contour are well tabulated and discussed. The primary finding of this study is that the temperature rises with an increase in the melting parameter, while the outer velocity leads to a reduction in temperature. The interplay between both factors allows for the regulation of the heat transfer rate, which proves advantageous for industrial applications. The novelty of this research lies in its ability to control the rate of heat transfer through the dual influence of the stretching rate and outer velocity, an aspect that has not been thoroughly explored in prior studies. The findings provide valuable insights into optimizing heat transfer in industrial processes involving nanofluids, where precise control over temperature and concentration is crucial for enhancing system efficiency. This approach opens up new possibilities for engineering applications, where both surface dynamics and external fluid conditions significantly impact the performance of thermal systems.