Channel flow with variable geometry and Navier slip at the walls using high-order lubrication theory

We investigate the effect of Navier type slip for the steady, laminar flow of a highly viscous and incompressible Newtonian fluid in a microfluidic channel with variable geometry for one of the two walls of the channel. A formal perturbation expansion in terms of the square of the aspect ratio of the channel is used and an extended, high-order, lubrication theory is applied. The analysis generalizes the works of Tavakol et al. (2017) and Housiadas and Tsangaris (2022) both of which were studied under the classic no-slip and no-penetration conditions along the solid walls. Analytical expressions in series form for the pressure drop, required to maintain the constant flowrate through the channel, are derived, where the formulas are provided in terms of the function that describes the shape of the wall and the dimensionless slip coefficient that appears in the slip law. Furthermore, the formulas are processed with analytical non-linear techniques that increase the accuracy and extend the domain of convergence of series. It is shown that the slip at the walls decreases substantially the required pressure drop even for small values of the dimensionless slip coefficient, revealing the sensitivity of the results on the boundary conditions. It is also revealed that significant part of the kinetic energy of the flow is transferred along the walls due to slip decreasing the viscous dissipation energy and facilitating the fluid transport through the channel. Finally, it is demonstrated that the higher-order terms in the asymptotic solution provide important information for the field variables and the major quantities of interest (pressure drop, kinetic energy transferred along the walls, and viscous dissipation) for this type of internal and confined flows. (c) 2022 Elsevier Masson SAS. All rights reserved.

Automated summary

In this study, the effect of Navier slip in the steady, laminar flow of a highly viscous and incompressible Newtonian fluid in a microfluidic channel with variable geometry is investigated. Analytical expressions in series form for the pressure drop are derived, and it is shown that slip at the walls substantially reduces the required pressure drop and facilitates fluid transport. Additionally, the higher-order terms in the asymptotic solution provide important information for the field variables and major quantities of interest in this type of internal and confined flows.