Abstract
In most microfluidic applications, pressure-driven Poiseuille flow in a contained cross-section with no-slip boundary conditions is the underlying fluid-mechanical model. Solutions for this problem exist for many known cross-sections. We have recently demonstrated a simple method to solve the relevant Poisson equation using a finite difference scheme in a spreadsheet analysis tool such as Microsoft Excel. The numerical solutions obtained from such a spreadsheet are close-to-exact to the analytical solutions with errors on the order of only a few percent. However, there are numerous applications in microfluidics for which the no-slip boundary condition is not valid. Examples include drag-reducing air-retaining surfaces as well as open-channel flow. For these scenarios few to no analytical models exist. In this paper, we derive an analytical model for mixed boundary conditions (slip/no-slip) in two dimensions in a rectangular channel cross-section. We also demonstrate that the equivalent numerical solution can be derived conveniently by adaption of the spreadsheet. In general, mixed boundary-type flow scenarios are especially difficult to solve analytically whereas numerical solutions can be derived using Microsoft Excel within seconds.
Highlights
We have shown that the numerical solutions derived compare well to the analytical solutions for various flow cases including the flow in circular cross-sections (Hagen-Poiseuille flow) as well as in rectangular channel profiles
We extended the concept of using a spreadsheet analysis tool such as Microsoft Excel to solve the fundamental equation for Poiseuille flow, i.e., the simplified Navier-Stokes equation in arbitrary cross-sections in one and two dimensions implementing different boundary conditions
Besides zero-value and fixed-value Dirichlet boundary conditions, we showed that simple modifications to the spreadsheet are sufficient to implement Neumann boundary conditions which set the gradient of the velocity profile to fixed values
Summary
Deriving analytical solutions for flow cases exhibiting Neumann-type boundary conditions or even mixed boundary conditions is significantly more difficult. These cases are usually studied numerical solver packages [8], lattice-Boltzmann or molecular dynamics simulations [9]. This paper will derive an analytical solution to mixed slip/no-slip boundary conditions in two dimensions in rectangular channel cross-sections This case is the most common case in microfluidic systems. We will show that the Microsoft Excel spreadsheet developed for solving no-slip boundary flow scenarios can be adapted to derive the same solution within seconds This allows deriving solutions to mixed Neumann/Dirichlet boundary condition flow scenarios for a wide variety of cross-sections
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