Newtonian Flow in Converging-Diverging Capillaries

TL;DR

This paper derives and validates analytical expressions for Newtonian fluid flow in converging-diverging capillaries using Navier-Stokes equations, confirming their accuracy against lubrication approximation and numerical solutions.

Contribution

It provides a unified analytical framework for flow in various converging-diverging geometries, validated against numerical methods and previous approximations.

Findings

Analytical expressions match lubrication approximation results.

Derived formulas are validated by numerical solutions.

Flow expressions converge to straight geometry limits.

Abstract

The one-dimensional Navier-Stokes equations are used to derive analytical expressions for the relation between pressure and volumetric flow rate in capillaries of five different converging-diverging axisymmetric geometries for Newtonian fluids. The results are compared to previously-derived expressions for the same geometries using the lubrication approximation. The results of the one-dimensional Navier-Stokes are identical to those obtained from the lubrication approximation within a non-dimensional numerical factor. The derived flow expressions have also been validated by comparison to numerical solutions obtained from discretization with numerical integration. Moreover, they have been certified by testing the convergence of solutions as the converging-diverging geometries approach the limiting straight geometry.