Numerical Analysis of Unsteady Boundary Layer Flow Over an Impulsively Started Sheet Using the Crank Nicolson Method

Fluid flow over moving surfaces creates complex flow patterns and thermal gradients, making analytical solutions computationally inefficient. Therefore, numerical methods that are stable, accurate, and capable of handling complex nonlinear problems are used. In this study, the Crank-Nicolson numerical technique was applied to investigate the combined effects of variable viscosity and viscous dissipation on unsteady flow over an impulsively moving flat sheet in a quiescent fluid. A two-dimensional laminar boundary layer flow of an incompressible Newtonian fluid, driven by the movement of the surface, with viscosity considered an inverse function of temperature, was analyzed. The effects of various flow parameters on velocity and temperature profiles were determined. The boundary layer partial differential equations governing the flow were transformed into a non-dimensional form and solved numerically. The numerical results demonstrate that an increase in Reynolds number, the variable viscosity parameter, and surface velocity leads to an increase in velocity profiles. Additionally, an increase in Prandtl number, the variable viscosity parameter, and surface velocity results in a decrease in temperature profiles. Finally, an increase in Eckert number was found to increase temperature profiles. Therefore, with suitable flow parameters, the velocity and temperature of the fluid can be regulated. These findings are useful for cooling hot sheets or metallic plates drawn over a quiescent fluid to obtain high-quality final products, as well as in hot rolling during metal processing and in the lubrication of sliding or rotating bearings.