Qualitative analysis on the electrohydrodynamic flow equation

In this paper, we present a comprehensive analysis of the lower and upper bounds of solutions for a nonlinear second-order ordinary differential equation governing the electrohydrodynamic flow of a conducting fluid in cylindrical conduits. The equation describes the radial distribution of the flow velocity in an "ion drag" configuration, which is affected by an externally applied electric field. Our study involves the establishment of rigorous analytical bounds on the radial distribution, taking into account the Hartmann number $H$ and a parameter $\alpha .$ An analytic approximate solution is obtained as an improvement of the a priori estimates and it is found to exhibit strong agreement with numerical solutions, particularly when considering small Hartmann numbers. Further, estimates for the central velocity $w(0)$ of the fluid occurring at the center of the cylindrical conduit were also established, and some interesting relationships were found between $H$ and $\alpha .$ These findings establish a framework that illuminates the potential range of values for the physical parameter within the conduit.