High-order lubrication theory in channels and tubes with variable geometry

Lubrication theory is used widely due to its simplicity and accuracy in many circumstances such as for the modeling of thin fluid films, the motion of particles near surfaces, and the flow in narrow geometries and configurations. Here, we present an extension to the classical lubrication theory to study the laminar flow of an incompressible and highly viscous simple Newtonian fluid in microfluidic channels and tubes with variable geometry using a formal perturbation expansion in terms of the aspect ratio. The analysis generalizes and extends the work of Tavakol et al., Proc. R. Soc. A, 473 (2017) by considering (a) different shapes for the upper and lower walls of the channel, and (b) axisymmetric tubes with variable circular cross section. Analytical expressions in series form for the average pressure drop, required to maintain the constant flowrate through the channel or tube, are derived, where the formulas are provided in terms of the function(s) that describe the shape of the wall(s). Furthermore, the formulas are processed with techniques that increase the accuracy and extend the domain of convergence of series. For symmetric and periodic undulating channels and tubes, the comparison of the analytical results derived here with numerical results from the literature reveals the great accuracy and efficiency of the high-order lubrication theory, as well as its superiority against the well-known domain perturbation method.