Asymptotic analysis on viscoelastic rheological fluid on gravity driven drainage flow through a vertical heated tube

The study of tank drainage phenomena has significant applications in chemical and petroleum industries, particularly for non-Newtonian fluids like pseudo-plastic materials. Understanding drainage dynamics helps optimize storage tank designs, ensuring efficient emptying while minimizing residue buildup in chemical processing. In the oil industry, accurate drainage models estimate crude oil evacuation times, accounting for viscosity variations during transfer. Similarly, in food processing, shear-thinning fluids like sauces and dairy products require controlled drainage systems to prevent clogging and maintain product consistency. Keeping these applications in view, this paper investigates the tank drainage phenomenon on a cylindrical surfaces, focusing on a non-Newtonian shear-thinning pseudo-plastic fluid model. The study examines unsteady drainage flow and its associated heat transfer effects. Analytical and exact solutions are obtained via the Adomian Decomposition and Homotopy Perturbation methods for the highly nonlinear ordinary differential equations that model pseudo-plastic fluid flow. Key physical quantities such as the velocity profile, volume flux, temperature distribution, and total drainage time are evaluated. The influence of fluid parameters such as the non-Newtonian fluid parameter, Stokes number, tank height, and pipe length on the velocity profile, flow rate, heat dissipation, and drainage time is analyzed graphically. Notably, the study reveals that the time required to drain the tank does not scale linearly with fluid height. While a larger reservoir or a smaller discharge hole increases drainage time, doubling the fluid height does not double the drainage time due to the faster flow rate at greater depths. A key finding with significant industrial implications is the characterization of the non-linear relationship between drainage time and fluid height, which critically informs storage tank design. Doubling the fluid height [Formula: see text] leads to a [Formula: see text] prolongation of the drainage time, while a [Formula: see text] extension of the pipe length [Formula: see text] causes a [Formula: see text] diminution in the flow rate. Furthermore, the velocity profile is enhanced by [Formula: see text] for a [Formula: see text] increase in the Stokes number [Formula: see text], and the temperature distribution rises by [Formula: see text] with an increasing fluid parameter [Formula: see text]. A comparative analysis demonstrates that the solutions from ADM and HPM are in close agreement, with a negligible discrepancy of less than [Formula: see text], confirming the robustness of the applied analytical techniques.