Transient dynamics of pressure-driven encroachment in narrow conduits with rate-dependent body force

Abstract

We analytically explore the flow of a Newtonian liquid forced to encroach a narrow tube of uniform cross section, by an unsteady pressure gradient, assisted by an encroachment-rate dependent external force. This novel problem is thought to have interesting implications. For instance in medicine where narrow tubes like syringes and needles are typically used to administer medication and in the printing industry. Using an unsteady eigenfunction expansion, the velocity distribution is accurately defined to yield unsteady profiles, contrasting with the classical Poiseuille parabola. We subsequently used our unsteady spectral decomposition to properly capture the kinematics and dynamics hidden in the models. By a detailed comparison between rectangular and circular channels, we show that such model ducts yield interesting similarities that can inform the choices of channels. Moreover, we obtain short and long-time dynamic behaviors, captured using a robust perturbation scheme that elegantly highlights the early and long-time characteristics. In the end, we present plots for encroachment depth and rate and the early and long-term asymptotic approximations and appropriately their graphical trends.