Amidst the nationwide flooding triggered by the rainy season compounded by El Niño, this study endeavors to devise a sophisticated mathematical model geared towards enhancing the efficiency of open circular channels. Focusing on key parameters like depth, velocity, and pressure within the context of incompressible, Newtonian, and steady flow dynamics, we aim to meticulously analyze their interplay. By delving into the impact of depth on flow characteristics, the influence of pressure on velocity, and the modulation of Reynolds and Froude numbers on velocity, our investigation unveils crucial insights. Leveraging MATLAB, we meticulously derive numerical solutions and visually represent our findings through insightful graphs. Given the inherent non-linearity of the governing equations; the continuity and momentum equations of motion. We used the finite difference method for the resolution. This robust model holds promise for diverse applications spanning water supply, treatment facilities, and irrigation networks, offering a comprehensive tool for optimizing operational efficiency.

Fluid flow coupled with heat transfer through a vertical cylindrical Collapsible tube in the presence of a magnetic field and an obstacle has been investigated. The governing equations for this flow are the equations of continuity, motion and energy. These are non linear partial differential equations and has been transformed into non-linear ordinary differential equations by introducing a similarity transformation. The resulting equations are solved simultenously in the bvp4c MATLAB library to obtain the profiles and also the rate of heat transfer has been calculated. The effects of varying the Reynolds number, Hartmann number, Eckert number, Unsteadiness parameter and Prandtl number on fluid temperature , fluid velocity and the rate of heat transfer are presented in the form of tables and graphs and has been discussed. Variation in the various parameters is observed to change the fluid primary velocity, temperature and the rate of heat transfer. This kind of result is important due to its widespread application in physical, biological and applied sciences.