Viscoelastic liquid flow dynamics in small porous-medium channels under pressure and velocity forces

Motivated by flow applications in medicine, biology, and pharmaceuticals, such as intravenous (IV) and gastrointestinal (G) tubes, this study presents a detailed analysis of viscoelastic fluids infiltrating narrow channels, specifically those with cylindrical or rectangular cross sections. The creeping flow is driven by an unsteady decaying pressure gradient and a v-dependent body force accommodating porous channels and media. By employing the Navier–Stokes equation alongside the linear viscoelastic constitutive model, we derive and dimensionalize the governing equations, bringing attention to key parameters which include the Weissenberg number (Wi). The nature of these equations necessitates using the separation of variables technique, where the Sturm–Liouville theorem is applied to achieve a spectral decomposition of the coupled dependent variables. This approach allows us to identify a geometric parameter resulting in temporal Volterra Integro-Differential equations, thus moving beyond the traditional Hagen–Poiseuille profile to accurately capture the unsteady velocity profile. Using a robust and efficient fourth-order Runge–Kutta numerical scheme, we generate, plot, and compare the kinematic characteristics of rectangular and cylindrical ducts, highlighting the similarities and differences between viscoelastic and purely viscous fluids. Additionally, we perform three perturbation analyses: first, treating the Weissenberg number as a small parameter (⁠ $Wi≪1$⁠) to explore the cumulative effects of viscoelasticity on the base purely viscous case; second, examining the long-term approximation by stretching the timescale; and finally, investigating the short-term approximation by compressing the timescale. The plots demonstrate that the short- and long-term approximations offer accurate predictions of the corresponding short- and long-term dynamics of the system. Additionally, the plots reveal that, when comparing a square channel to a circular channel with an identical area-to-perimeter ratio, the kinematics of the square channel generally dominate throughout most of the time evolution, although strong viscoelastic effects intermittently disrupt this trend. These analyses offer a comprehensive insight into the flow's long- and short-term characteristics. Consequently, the interaction between momentum transport, viscous dissipation, fluid porosity, fluid memory effects (Wi), and conduit geometry is emphasized—supported by kinematic plots.