Steady laminar fluid flow through spiral ducts based on circle involutes

Abstract

We revisit the analysis of steady laminar incompressible Newtonian fluid flow through (planar) spiral duct geometries. Spiral ducts are utilised in a number of fluid flow applications, largely due to their ability to incorporate a large flow distance into a relatively small volume. Archimedean spirals are an attractive design choice in many application areas, including microfluidic devices, due to the simplicity of their description and a constant spacing between turns. Herein we examine fluid flow through a spiral duct geometry based on the involute of a circle. These involute based spiral duct geometries retain some key features of Archimedean spirals while possessing several desirable design properties. Moreover, a curvilinear coordinate system constructed around an involute spiral naturally leads to a straightforward orthogonal coordinate system which facilitates a detailed analysis of fluid flow through the spiral duct geometry. We derive the Navier–Stokes equations using this coordinate system and analyse the fluid flow through the spiral duct geometries with a focus on making detailed comparisons with the flow through axis-symmetric curved ducts. Our analysis ultimately provides an efficient means for estimating spiral duct flow, a robust justification for the use of curved duct flow as an approximation for spiral duct flow in certain regimes and reveals some of the specific differences between the spiral and axis-symmetric curved ducts without being obfuscated by a curvature parameter perturbation. Our methodology is readily applied to study spiral duct of any cross-section shape, including the rectangular cross-sections which are common in a variety of applications.