A depthwise averaging solution for cross-stream diffusion in a Y-micromixer by considering thick electrical double layers and nonlinear rheology

Abstract

Both nonlinear rheology and finite EDL thickness effects on the mixing process in an electroosmotically actuated Y-sensor are being investigated in this paper, utilizing a depthwise averaging method based on the Taylor dispersion theory. The fluid rheological behavior is assumed to obey the power-law viscosity model. Analytical solutions are obtained assuming a large channel width to depth ratio for which a 1-D profile can efficiently describe the velocity distribution. Full numerical simulations are also performed to determine the applicability range of the analytical model, revealing that it is able to provide accurate results for channel aspect ratios of ten and higher and quite acceptable results for smaller aspect ratios down to four. The model is then used for a complete parametric study to determine the effects of the governing parameters on the mixing performance. It is observed that utilizing a fluid with a higher flow behavior index gives rise to a smaller mixing length. Moreover, whereas a larger EDL thickness is accompanied by an improved mixing efficiency, the opposite is true for the channel aspect ratio. The effective diffusivity is also found to be an increasing function of the EDL extent and a decreasing function of the flow behavior index.

Appendix

The coefficients of the velocity distribution in Eq. (9) are given as

$$a_{1} = \frac{{\frac{{2^{1/n} }}{n + 1} + e^{K/n} - e^{1/n} }}{\gamma },\quad b_{1} = \frac{{\frac{{2^{1/n} }}{n + 1}}}{\gamma }, \quad a_{2} = \frac{{e^{K/n} }}{\gamma }, \quad b_{2} = \frac{1}{\gamma }$$

(27)

in which

$$\gamma = \left( {1 - \frac{n}{K}} \right)e^{K/n} + \frac{n - 1}{K}e^{1/n} + \frac{{2^{1/n} }}{{K\left( {1 + 2n} \right)}}$$

(28)

Also, the coefficients of Eq. (20) are given as below

$$\begin{aligned} {\mathbb{A}} & = \frac{{\left( {a_{1} - 1} \right)^{2} }}{6},\quad {\mathbb{B}} = - b_{1} \left( {a_{1} - 1} \right)\left( {\frac{n}{4n + 1}} \right)\left[ {\frac{{n^{2} }}{{\left( {2n + 1} \right)\left( {3n + 1} \right)}} + \frac{1}{2}} \right]K^{{\left( {n + 1} \right)/n}} \\ {\mathbb{C}} & = \frac{{b_{1}^{2} n^{3} K^{{2\left( {n + 1} \right)/n}} }}{{\left( {2n + 1} \right)\left( {3n + 1} \right)\left( {5n + 2} \right)}},\quad {\mathbb{D}} = \frac{{\left( {a_{2} - 1} \right)^{2} }}{6},\quad {\mathbb{E}} = - \frac{{b_{2} \left( {a_{2} - 1} \right)n^{3} }}{{K^{3} }} \\ {\mathbb{F}} & = - \frac{{b_{2} \left( {a_{2} - 1} \right)n^{3} }}{{2K^{3} }},\quad {\mathbb{G}} = \frac{{b_{2}^{2} n^{3} }}{{2K^{3} }} \\ \end{aligned}$$

(29)

Finally, the coefficients of Eq. (21) are obtained as

$${\mathbb{H}} = \frac{{\left( {a_{3} - 1} \right)^{2} }}{6},\quad {\mathbb{I}} = - \frac{{b_{3} \left( {a_{3} - 1} \right)^{2} }}{{K^{3} }}, \quad {\mathbb{J}} = - \frac{{b_{3} \left( {a_{3} - 1} \right)}}{{2K^{3} }},\quad {\mathbb{K}} = \frac{{b_{3}^{2} }}{{2K^{3} }}$$

(30)

in which $a_{3} = K/\left( {K - \tanh{\textit{}} K} \right)$ and $b_{3} = K/\left( {K\cosh{\textit{}} K - \sinh{\textit{}} K} \right)$.