Cross-stream diffusion under pressure-driven flow in microchannels with arbitrary aspect ratios: a phase diagram study using a three-dimensional analytical model

Abstract

This article presents a three-dimensional analytical model to investigate cross-stream diffusion transport in rectangular microchannels with arbitrary aspect ratios under pressure-driven flow. The Fourier series solution to the three-dimensional convection–diffusion equation is obtained using a double integral transformation method and associated eigensystem calculation. A phase diagram derived from the dimensional analysis is presented to thoroughly interrogate the characteristics in various transport regimes and examine the validity of the model. The analytical model is verified against both experimental and numerical models in terms of the concentration profile, diffusion scaling law, and mixing efficiency with excellent agreement (with <0.5% relative error). Quantitative comparison against other prior analytical models in extensive parameter space is also performed, which demonstrates that the present model accommodates much broader transport regimes with significantly enhanced applicability.

Appendix

By substituting the expression of $U(\tilde{x},\tilde{y}),$ we have

$$ M_{mnij} = \kappa \int\limits_{ - 1/2}^{1/2} {\int\limits_{ - 1/2}^{1/2} {\left[ {(1 - 4\tilde{y}^{2} ) + 4\sum\limits_{s = 1}^{\infty } {\frac{{( - 1)^{s} }}{{\varepsilon_{s}^{3} \cosh (\varepsilon_{s} W/H)}}} \cosh (2\varepsilon_{s} \gamma \tilde{x})\cos (2\varepsilon_{s} \tilde{y})} \right]\phi_{m} (\tilde{x})\phi_{i} (\tilde{x})\varphi_{n} (\tilde{y})\varphi_{j} (\tilde{y})} {\text{d}}\tilde{x}{\text{d}}\tilde{y}} $$

(25)

The above equation involves four kinds of integrations in terms of $\tilde{x}$and $\tilde{y},$ shown as:

$$ P_{m,i} = \int\limits_{ - 1/2}^{1/2} {\phi_{m} (\tilde{x})\phi_{i} (\tilde{x})} {\text{d}}\tilde{x} $$

(26)

$$ Q_{n,j} = \int\limits_{ - 1/2}^{1/2} {(1 - 4\tilde{y}^{2} )\varphi_{n} (\tilde{y})\varphi_{j} (\tilde{y}){\text{d}}\tilde{y}} $$

(27)

$$ R_{m,i,s} = \int\limits_{ - 1/2}^{1/2} {\cosh (2\varepsilon_{s} \gamma \tilde{x})\phi_{m} (\tilde{x})\phi_{i} (\tilde{x}){\text{d}}\tilde{x}} $$

(28)

$$ S_{n,j,s} = \int\limits_{ - 1/2}^{1/2} {\cos (2\varepsilon_{s} \tilde{y})\varphi_{n} (\tilde{y})\varphi_{j} (\tilde{y}){\text{d}}\tilde{y}} $$

(29)