Electroosmotically driven creeping flows in a wavy microchannel

Abstract

We report our investigation on electroosmotic flow (EOF) in a wavy channel between a plane wall and a sinusoidal wall. An exact solution is obtained by using complex function formulation and boundary integral method. The effects of the channel width and wave amplitude on the electric field, streamline pattern, and flow field are studied. When a pressure gradient of sufficient strength in the opposite direction is added to an EOF in the wavy channel, various patterns of recirculation regions are observed. Experimental results are presented to validate qualitatively the theoretical description. The solution is further exploited to determine the onset condition of flow recirculation and the size of the recirculation region. It is found that they are dependent on one dimensionless parameter related to forces (K, the ratio of the pressure force to the electrokinetic force) and two dimensionless parameters related to the channel geometry (α, the ratio of the wave amplitude to the wavelength, and h, the ratio of the channel width to the wavelength).

Appendix: Solution of electric field and stream function

This Appendix describes the details in solving the periodic functions $G^\varphi (x), G^\beta (x),R^\varphi (x), R^\beta (x),Q^\varphi (x),$ and Q ^β(x) that are used for calculating G(ζ), R(ζ) and Q(ζ). The symmetry of the channel geometry ensures that for Stokes flow $\overline {\Theta (x)} =\Theta (-x),$ so that $\Re [\Theta (x)]=\frac{\Theta (x)+\Theta (-x)}{2}$ and $\Im [\Theta (x)]=\frac{\Theta (x)-\Theta (-x)}{2i}.$ Hence Eqs. (29), (30) can be reduced to

$$m\left({g_m^\varphi +g_{-m}^\varphi} \right)=0$$

(55)

and

$$m\left({g_m^\beta +g_{-m}^\beta} \right)=\frac{\alpha }{4\pi}\left({\delta_{m,1} -\delta_{m,-1}} \right).$$

(56)

Similarly, Eq. (33) becomes

$$\exp (nh)g_n^\varphi -\sum\limits_{m=-\infty}^{+\infty } {g_m^\beta E_{m,n}} =0,$$

(57)

where $E_{m,n} =\left\{{\begin{array}{l} \delta_{m,0} -\frac{\alpha}{2}\left({\delta_{m,1} -\delta_{m,-1}} \right),n=0 \\ \frac{m}{n}I_{n-m} (-n\alpha),n\ne 0 \\ \end{array}} \right.,$ and δ is Kronecker delta, and I _n (z) is the nth order modified Bessel function of the first kind, $I_n \left(z \right)=\frac{1}{2\pi}\int\nolimits_{\theta =-\pi}^\pi {\exp (z\cos \theta)\cos n\theta {\rm d}\theta}.$ These three Eqs. (55)–(57) form a complete set for coefficients $\left\{{g_m^\varphi } \right\}$ and $\left\{{g_m^\beta} \right\}, m\in \left\{{{\mathbb{Z}}\backslash \{0\}} \right\}.$

Since the electric field is $\vec {E}=-\nabla \phi,$ as a result, the tangential electric field strength along the walls [Eqs. (34, 35)] can be evaluated using:

$$E_t^\varphi (x)=\sum\limits_{m=-\infty}^{+\infty} {2mg_m^\varphi \exp (-imx)} -\frac{1}{2\pi}$$

(58)

and

$$E_t^\beta (x)=\frac{\sum\nolimits_{m=-\infty}^{+\infty} {\left[ {m\left({g_m^\beta -g_{-m}^\beta} \right)} \right]\exp (-imx)} -\frac{1}{2\pi}}{\sqrt {1+\alpha^2\sin^2(x)}}.$$

(59)

For the stream function, Eqs. (46)–(49) give

$$m\left({r_m^\varphi +r_{-m}^\varphi} \right)+mh\left({q_m^\varphi +q_{-m}^\varphi} \right)=0,$$

(60)

$$\begin{aligned} \,&2h\delta_{m,0} B-m\left({r_m^\varphi -r_{-m}^\varphi} \right)+\left({q_m^\varphi +q_{-m}^\varphi} \right)-mh\left({q_m^\varphi -q_{-m}^\varphi} \right)\\ \,&\quad=\frac{Kh^2}{4\pi}\delta_{m,0} +m\left({g_m^\varphi -g_{-m}^\varphi} \right)-\frac{1}{2\pi}\delta_{m,0}, \end{aligned}$$

(61)

$$\begin{aligned} \,&\frac{\alpha^2}{4}\left({2\delta_{m,0} +\delta_{m,2} +\delta _{m,-2}} \right)B+\left({r_m^\beta +r_{-m}^\beta} \right)-\frac{\alpha}{2}\left({q_{m+1}^\beta +q_{m-1}^\beta +q_{-m+1}^\beta +q_{-m-1}^\beta} \right)\\ \,&\quad=-\frac{K\alpha^3}{96\pi}\left({3\delta_{m,1} +3\delta _{m,-1} +\delta_{m,3} +\delta_{m,-3}} \right),\end{aligned}$$

(62)

and

$$\begin{aligned} \,& -\alpha \left({\delta_{m,1} +\delta_{m,-1}} \right)B-m\left({r_m^\beta -r_{-m}^\beta} \right)+\left({q_m^\beta +q_{-m}^\beta} \right)+\frac{\alpha}{2}\left[ {\left({m+1} \right)\left({q_{m+1}^\beta -q_{-m-1}^\beta} \right)+\left({m-1} \right)\left({q_{m-1}^\beta -q_{-m+1}^\beta} \right)} \right] \\ \,&\quad =\frac{K\alpha^2}{16\pi}\left({2\delta_{m,0} +\delta_{m,2} +\delta _{m,-2}} \right)+m\left({g_m^\beta -g_{-m}^\beta} \right)-\frac{1}{2\pi}\delta_{m,0}. \end{aligned}$$

(63)

From Eqs. (50), (51) we obtain

$$\exp (nh)r_n^\varphi -\sum\limits_{m=-\infty}^{+\infty } {r_m^\beta E_{m,n}} =0$$

(64)

and

$$\exp (nh)q_n^\varphi -\sum\limits_{m=-\infty}^{+\infty } {q_m^\beta E_{m,n}} =0.$$

(65)

Equations (60)–(65) can be further simplified by eliminating $\left\{{r_m^\varphi} \right\}$ and $\left\{{q_m^\varphi } \right\}$ from the equation set first. We take index n as a positive integer only in the analysis.

Rewrite Eq. (65) as

$$q_n^\varphi =\frac{1}{n}\exp (-nh)\sum\limits_{m=-\infty}^{+\infty} {mI_{n-m}^- q_m^\beta},$$

(66)

and take the negative indices,

$$q_{-n}^\varphi =-\frac{1}{n}\exp (nh)\sum\limits_{m=-\infty}^{+\infty} {mI_{n+m}^+ q_m^\beta}.$$

(67)

where the (n−m)th order modified Bessel function of the first kind for argument −nα, I _n-m (−nα ), has been written as I ⁻ _n–m for simplicity. Similarly, I ⁺ _n+m denotes the (n + m)th order modified Bessel function of the first kind for argument nα, I _n+m (nα ). The same convention is used hereinafter in this appendix. Equation (64) is rewritten as

$$r_n^\varphi =\frac{1}{n}\exp (-nh)\sum\limits_{m=-\infty}^{+\infty} {mI_{n-m}^- r_m^\beta}$$

(68)

and

$$r_{-n}^\varphi =-\frac{1}{n}\exp (nh)\sum\limits_{m=-\infty}^{+\infty} {mI_{n+m}^+ r_m^\beta}.$$

(69)

Changing the dummy variable in Eqs. (60) and (61) from m to n, and using the expressions in Eqs. (66)–(69) to eliminate r ^φ _n , r ^φ_−n , q ^φ _n and q ^φ_−n in Eqs. (60) and (61), we obtain

$$\sum\limits_{m=-\infty}^{+\infty} {m\left[ {\frac{I_{n-m}^-}{\exp (2nh)}-I_{n+m}^+} \right]r_m^\beta} +\sum\limits_{m=-\infty}^{+\infty} {mh\left[ {\frac{I_{n-m}^-}{\exp (2nh)}-I_{n+m}^+} \right]q_m^\beta} =0$$

(70)

and

$$\sum\limits_{m=-\infty}^{+\infty} {m\left[ {\frac{I_{n-m}^-}{\exp (2nh)}+I_{n+m}^+} \right]r_m^\beta} -\sum\limits_{m=-\infty}^{+\infty} {m\left[ {\left({\frac{1}{n}-h} \right)\frac{I_{n-m}^-}{\exp (2nh)}-\left({\frac{1}{n}+h} \right)I_{n+m}^+} \right]q_m^\beta} =-\frac{n\left[ {g_n^\varphi -g_{-n}^\varphi} \right]}{\exp (nh)}.$$

(71)

Next, we multiply I _n-m (−nα) on both sides of Eq. (62) and take summation from m = −∞ to m =  +∞ to obtain

$$\begin{aligned}\,& \frac{\alpha^2}{4}\left[ {2I_n^- +I_{n+2}^- +I_{{n-2}}^-} \right]B+\sum\limits_{m=-\infty}^{+\infty} {\left[ {I_{n-m}^- +I_{n+m}^-} \right]r_m^\beta} -\sum\limits_{m=-\infty}^{+\infty} {\frac{\alpha}{2}\left[ {I_{n+m-1}^- +I_{n+m+1}^- +I_{n-m-1}^- +I_{n-m+1}^-} \right]q_m^\beta} \\ \,&\quad =-\frac{K\alpha^3}{96\pi}\left[ {I_{n-3}^- +I_{n+3}^- +3I_{n-1}^- +3I_{n+1}^-} \right]. \end{aligned}$$

(72)

A similar procedure is applied to Eq. (63), and we have

$$\begin{aligned} \,& -\alpha \left[ {I_{n-1}^- +I_{n+1}^-} \right]B-\sum\limits_{m=-\infty}^{+\infty} {m(I_{n-m}^- +I_{n+m}^-)r_m^\beta} +\sum\limits_{m=-\infty}^{+\infty} {\left[ {\left({I_{n-m}^- +I_{n+m}^-} \right)+\frac{\alpha}{2}m\left({I_{n-m-1}^- +I_{n-m+1}^- +I_{n+m-1}^- +I_{n+m+1}^-} \right)} \right]q_m^\beta} \\ \,&\quad =\frac{K\alpha^2}{16\pi}\left[ {2I_n^- +I_{n-2}^- +I_{n+2}^- } \right]+\sum\limits_{m=-\infty}^\infty {m\left[ {I_{n-m}^- +I_{n+m}^-} \right]g_m^\beta} -\frac{1}{2\pi}I_n^-. \end{aligned}$$

(73)

Taking the zeroth order of Eqs. (61), (62), (63), and (65), we obtain

$$q_0^\varphi +Bh=\frac{Kh^2}{8\pi}-\frac{1}{4\pi},$$

(74)

$$r_0^\beta -\frac{\alpha}{2}\left({q_1^\beta +q_{-1}^\beta} \right)+\frac{\alpha^2}{4}B=0,$$

(75)

$$q_0^\beta +\frac{\alpha}{2}\left({q_1^\beta -q_{-1}^\beta} \right)=\frac{K\alpha^2}{16\pi}-\frac{1}{4\pi},$$

(76)

and

$$q_0^\varphi =q_0^\beta -\frac{\alpha}{2}\left({q_1^\beta -q_{-1}^\beta} \right).$$

(77)

Equations (70)–(73), supplemented by Eqs. (74)–(77), are sufficient to solve for the Fourier coefficient sets $\left\{{r_m^\beta} \right\}$ and $\left\{{q_m^\beta} \right\}.$ Once they are solved, q ^φ _n and r ^φ _n are determined from Eqs. (66) and (68); q ^φ_−n and r ^φ_−n are solved from Eqs. (60) and (61), as

$$q_{-n}^\varphi =\left({2nh-1} \right)q_n^\varphi +2nr_n^\varphi +n\left({g_n^\varphi -g_{-n}^\varphi} \right),$$

(78)

and

$$ r_{-n}^\varphi =-h(q_n^\varphi +q_{-n}^\varphi)-r_n^\varphi.$$

(79)

The zeroth term of $\left\{{r_m^\varphi} \right\}$ and $\left\{{q_m^\varphi} \right\}$ are obtained from Eqs. (64) and (65) at n = 0, namely, Eq. (77) and

$$r_0^\varphi =r_0^\beta -\frac{\alpha}{2}\left({r_1^\beta -r_{-1}^\beta} \right).$$

(80)

In practical calculation, the infinite series in Eqs. (70)–(73) are truncated to only N terms. Together with Eqs. (74)–(77), there are a total of 4N + 4 equations to solve for B, q ^φ₀ , $\left\{{r_m^\beta} \right\}$ and $\left\{{q_m^\beta} \right\},$ $m\in \left[ {-N,N} \right].$ Subsequently, $\left\{{r_m^\varphi} \right\}$ and $\left\{{q_m^\varphi} \right\}$ are solved from Eqs. (66), (68), and (77)–(80). Finally $\left\{{r_m^\beta} \right\},$ $\left\{{r_m^\varphi} \right\},$ $\left\{{q_m^\varphi} \right\}$ and $\left\{{q_m^\beta} \right\}$ are used to construct R(ζ) and Q(ζ) in Eqs. (44) and (45), and they are used to obtain ψ(x,y) in Eq. (40).