# Electrohydrodynamic (In)Stability of Microfluidic Channel Flows: Analytical Expressions in the Limit of Small Reynolds Number

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## Abstract

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## 1. Introduction

#### Structure of the Article

## 2. Problem Formulation

#### 2.1. General Equations of Motion

#### 2.2. General Boundary Conditions

#### 2.3. Two-Dimensional Flow and Nondimensional Equations

## 3. Linearization in the Interface Displacement $\zeta $

#### 3.1. Velocity Field to Zeroth Order in $\zeta $ (The Unperturbed Flow)

#### 3.2. Electric Potentials to Zeroth Order in $\zeta $ (The Unperturbed Electric Potentials)

#### 3.3. Velocity Field to First Order in $\zeta $

#### 3.4. Electric Potentials to First Order in $\zeta $

#### 3.5. Boundary Conditions

- BC1–BC4: no-slip conditions at the rigid boundaries:$$\begin{array}{cccc}\hfill {\psi}_{1}^{\text{}}\left(1\right)& =0,\hfill & \hfill \phantom{\rule{1.em}{0ex}}{\psi}_{1}^{\prime}\left(1\right)& =0,\hfill \end{array}$$$$\begin{array}{cccc}\hfill {\psi}_{2}^{\text{}}(-h)& =0,\hfill & \hfill \phantom{\rule{1.em}{0ex}}{\psi}_{2}^{\prime}(-h)& =0,\hfill \end{array}$$
- BC5: continuity of W at the interface$${\psi}_{1}^{\text{}}\left(0\right)={\psi}_{2}^{\text{}}\left(0\right),$$
- BC6: continuity of U at the interface$${\psi}_{1}^{\prime}\left(0\right)-{\psi}_{2}^{\prime}\left(0\right)=\frac{{\psi}_{1}^{\text{}}\left(0\right)}{\tilde{c}}(1-\mu ){a}_{2}^{\text{}},$$$${W}^{\left(1\right)}\left(0\right)=\left({\partial}_{t}^{\text{}}+{U}^{\left(0\right)}\left(0\right){\partial}_{x}^{\text{}}\right)\zeta $$$$\zeta =\frac{{\psi}_{1}^{\text{}}\left(0\right)}{\tilde{c}}exp\left[ik(x-ct)\right].$$We note that if $\tilde{c}=0$, a second-order expansion of the kinematic condition is required to avoid problems involving division by $\tilde{c}$. Now, Equation (57b) together with the linearized Equation (13a) for U$$\left[\phantom{\rule{-0.166667em}{0ex}}\left[{\partial}_{z}^{\text{}}{U}^{\left(0\right)}\left(0\right)\zeta +{U}^{\left(1\right)}\left(0\right)\right]\phantom{\rule{-0.166667em}{0ex}}\right]=0,$$
- BC7: continuity of tangential stresses at the interface$${\psi}_{1}^{\u2033}\left(0\right)+{k}^{2}{\psi}_{1}^{\text{}}\left(0\right)=\mu \left({\psi}_{2}^{\u2033}\left(0\right)+{k}^{2}{\psi}_{2}^{\text{}}\left(0\right)\right),$$$$\left[\phantom{\rule{-0.166667em}{0ex}}\left[{T}_{\mathrm{T}\mathrm{k}}^{{M}_{\text{}}}\right]\phantom{\rule{-0.166667em}{0ex}}\right]{n}_{k}^{\text{}}=\left[\phantom{\rule{-0.166667em}{0ex}}\left[\u03f5{E}_{\mathrm{T}}^{\text{}}{E}_{\mathrm{N}}^{\text{}}\right]\phantom{\rule{-0.166667em}{0ex}}\right]={E}_{\mathrm{T}}^{\text{}}\left[\phantom{\rule{-0.166667em}{0ex}}\left[\u03f5{E}_{\mathrm{N}}^{\text{}}\right]\phantom{\rule{-0.166667em}{0ex}}\right]=0,$$$$\left[\phantom{\rule{-0.166667em}{0ex}}\left[\mu \left({\partial}_{z}^{2}{U}^{\left(0\right)}\zeta +{\partial}_{z}^{\text{}}{U}^{\left(1\right)}\left(0\right)+{\partial}_{x}^{\text{}}{W}^{\left(1\right)}\left(0\right)\right)\right]\phantom{\rule{-0.166667em}{0ex}}\right]=0,$$
- BC8: balance of normal stresses$$\begin{array}{cc}\hfill ik\left({k}^{2}S+{T}_{\mathrm{el}}^{\text{}}\right)\frac{{\psi}_{1}^{\text{}}\left(0\right)}{\tilde{c}}& =\mu \left({\psi}_{2}^{\u2034}\left(0\right)-3{k}^{2}{\psi}_{2}^{\prime}\left(0\right)\right)+i\rho kR\left(\tilde{c}\phantom{\rule{0.222222em}{0ex}}{\psi}_{2}^{\prime}\left(0\right)+{a}_{2}^{\text{}}{\psi}_{2}\left(0\right)\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}-\left({\psi}_{1}^{\u2034}\left(0\right)-3{k}^{2}{\psi}_{1}^{\prime}\left(0\right)\right)-ikR\left(\tilde{c}\phantom{\rule{0.222222em}{0ex}}{\psi}_{1}^{\prime}\left(0\right)+{a}_{1}^{\text{}}{\psi}_{1}\left(0\right)\right),\hfill \end{array}$$$$S=\frac{\widehat{\gamma}}{{\widehat{\mu}}_{1}^{\text{}}{\widehat{U}}_{0}^{\text{}}},$$$${T}_{\mathrm{el}}=\left\{\begin{array}{c}-k\frac{{\widehat{\u03f5}}_{2}^{\text{}}{\widehat{V}}_{0}^{2}}{{\widehat{\mu}}_{1}^{\text{}}{\widehat{h}}_{1}^{\text{}}{\widehat{U}}_{0}^{\text{}}}\frac{{(1-\epsilon )}^{2}}{{(h+\epsilon )}^{2}}\frac{1}{\epsilon tanh\left[k\right]+tanh\left[kh\right]}\\ +k\frac{{\widehat{\u03f5}}_{1}^{\text{}}{\widehat{E}}_{0}^{2}{\widehat{h}}_{1}^{\text{}}}{{\widehat{\mu}}_{1}^{\text{}}{\widehat{U}}_{0}^{\text{}}}{(1-\epsilon )}^{2}\frac{1}{\epsilon tanh\left[kh\right]+tanh\left[k\right]},\end{array}\right\},$$$$\begin{array}{cc}\hfill \left[\phantom{\rule{-0.166667em}{0ex}}\left[{T}_{\mathrm{N}\mathrm{k}}^{{M}_{\text{}}}\right]\phantom{\rule{-0.166667em}{0ex}}\right]{n}_{k}^{\text{}}& =\left\{\begin{array}{c}\left[\phantom{\rule{-0.166667em}{0ex}}\left[\u03f5{\partial}_{z}^{\text{}}{\Phi}^{\left(0\right)}\left(0\right){\partial}_{z}^{\text{}}{\Phi}^{\left(1\right)}\left(0\right)\right]\phantom{\rule{-0.166667em}{0ex}}\right]\\ {E}_{0}^{\text{}}\left[\phantom{\rule{-0.166667em}{0ex}}\left[-\u03f5{\partial}_{x}^{\text{}}{\Phi}^{\left(1\right)}\left(0\right)\right]\phantom{\rule{-0.166667em}{0ex}}\right]\end{array}\right\}\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =\left\{\begin{array}{c}-k{\u03f5}_{2}^{\text{}}\frac{{(1-\epsilon )}^{2}}{{(h+\epsilon )}^{2}}\frac{1}{\epsilon tanh\left[k\right]+tanh\left[kh\right]}\phantom{\rule{0.222222em}{0ex}}\zeta \\ +k{\u03f5}_{1}^{\text{}}{E}_{0}^{2}{(1-\epsilon )}^{2}\frac{1}{\epsilon tanh\left[kh\right]+tanh\left[k\right]}\phantom{\rule{0.222222em}{0ex}}\zeta \end{array}\right\}.\hfill \end{array}$$Note that the effective nondimensional values of ${\u03f5}_{2}^{\text{}}$ in the case of the normal E-field, and of ${\u03f5}_{1}^{\text{}}{E}_{0}^{2}$ in the case of the tangential E-field, in dimensional units correspond to ${\widehat{\u03f5}}_{2}^{\text{}}{\widehat{V}}_{0}^{2}/\left({\widehat{\mu}}_{1}^{\text{}}{\widehat{h}}_{1}^{\text{}}{\widehat{U}}_{0}^{\text{}}\right)$ and ${\widehat{\u03f5}}_{1}^{\text{}}{\widehat{E}}_{0}^{2}{\widehat{h}}_{1}^{\text{}}/\left({\widehat{\mu}}_{1}^{\text{}}{\widehat{U}}_{0}^{\text{}}\right)$, respectively. Equations (62) and (63b) were derived here for our bounded system, but recover known results when (one of the) boundaries effectively go to infinity ($kh\gg 1,k\gg 1$).

## 4. Perturbation Expansion in the Reynolds Number $\mathit{R}$ to Zeroth Order

**Figure 3.**For a normal E-field, the imaginary phase velocity ${\tilde{c}}_{i}^{\left[0\right]}$ to zeroth order in R is plotted as function of the wavenumber k, for the viscosity ratio $\mu =0,0.2,1$ and a fixed set of S, ${\u03f5}_{2}^{\text{}}$, h, and $\epsilon $. The instability region $0<k<{k}_{c}^{\left[0\right]}$, for which ${\tilde{c}}_{i}^{\left[0\right]}>0$ is independent of $\mu $, which can be deduced from the impending voltage Equation (78). The increase in $\mu $ decreases the growth rates of unstable waves for small k, but also decreases the damping rates for large k, i.e., makes the short waves relatively less stable, although it does not cause the actual instability. In the limit $k\to \infty $, ${\tilde{c}}_{i}^{\left[0\right]}$ reaches the value $-S/\left[2\right(\mu +1\left)\right]$. Note that in microfluidic systems ${\u03f5}_{2}^{\text{}}\sim 1$, which makes the positive ${\tilde{c}}_{i}^{\left[0\right]}\sim 1$, i.e., ${10}^{3}$ times smaller than depicted; we used a large ${\u03f5}_{2}^{\text{}}$ to better emphasize the trends.

#### 4.1. Validation of Results: Limit of Vanishingly Small Wavenumbers

#### 4.2. Validation of Results: Limit of Large Wavenumbers

#### 4.3. Onset of EHD Instability to Zeroth Order in R

**Figure 4.**Plots of the neutral stability line separating stable ($\mathrm{s}$) and unstable ($\mathrm{u}$) regions in the ${\widehat{V}}_{\mathrm{inst}}^{\left[0\right]}$–$\widehat{k}{\widehat{h}}_{1}^{\text{}}$ plane (with log–log axes) for $h=0,1,4,20$. The increase in h causes lower electric fields and, thus, increases the stable regions and the impending voltages.

## 5. Perturbation Expansion in the Reynolds Number $\mathit{R}$ to First Order

#### 5.1. Pure Shear Stress Instability to First Order in R

**Figure 5.**(

**a**) The first-order-R correction $R{c}_{i}^{\left[1\right]}$ vs. k for the pure shear flow, i.e., without the first-order electric field (voltage). Two sets of flow parameters are featured: for $h=4$, instability happens for $k>{k}_{c}^{\left[1\right]}$, and for $h=1/4$ for all k. The instability is caused by the discontinuity of the slope of the zero-order-$\zeta $ velocity, which happens here due to viscosity stratification $\mu =2\ne 1$ (see text for details). Note the small magnitudes. Compare with Figure 3. (

**b**) The physical voltage correction $R{\widehat{V}}^{\left[1\right]}$ vs. k, corresponding to the cases in (

**a**). The voltages counteract the stability trends due to shear to bring the system to neutral stability to first order in R; for $h=4$, positive values of ${\widehat{V}}^{\left[1\right]}$ are needed to destabilize the stable wavenumbers $k<{k}_{c}^{\left[1\right]}$ of ${c}_{i}^{\left[1\right]}$ and vice versa; for $h=1/4$, the voltages are always negative to dampen the modes unstable for all k. The EHD system is coupled; Equations (94)–(101), and, in general, the extrema of the functions ${c}_{i}^{\left[1\right]}$ and ${\widehat{V}}^{\left[1\right]}$ do not coincide. However, the value ${k}_{c}^{\left[1\right]}(=1.836)$ is the same in the two panels as it should be. Note the very small magnitudes.

#### 5.2. Onset of EHD Instability to First Order in R

**Figure 6.**(

**a**) First-order-R physical voltages $R{\widehat{V}}^{\left[1\right]}$ vs. k for three different flow cases for an EO pump. Magnitudes are small, but larger than in Figure 5b. The increase in h and adverse pressure ${G}_{1}^{\text{}}$ increase the stability by shifting ${k}_{c}^{\left[1\right]}$ to the right (stable k regions enlarged), but the increase in h enhances the voltage corrections around ${k}_{c}^{\left[1\right]}$, whereas the increase in ${G}_{1}^{\text{}}$ dampens them. (

**b**) Neutral stability diagram of the overall impending voltage ${\widehat{V}}_{\mathrm{tot}}^{\text{}}={\widehat{V}}^{\left[0\right]}+R{\widehat{V}}^{\left[1\right]}$ vs. k to both orders, for a set of parameters for the EO pump (updated Figure 4). The small first-order magnitudes show up only when enhanced. EHD stability increases with h, as the impending voltages increase with h. For $h=4$ and $h=20$, stable regions protrude into unstable ones for $k<{k}_{c}^{\left[1\right]}$, and withdraw for $k>{k}_{c}^{\left[1\right]}$; compare with Figure 5b. For $h=1$, the first-order-R voltages are negative for all k, i.e., the system is unstable to first-order-R shear, like $h=1/4$ in Figure 5a. The stability diagrams differ for different flow parameters.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Coefficients for ${\Psi}_{1}^{\left[0\right]}$ and ${\Psi}_{2}^{\left[0\right]}$

## Appendix B. Coefficients for ${\Psi}_{1,\mathrm{P}}^{\left[1\right]}$ and ${\Psi}_{2,\mathrm{P}}^{\left[1\right]}$

## Appendix C. Phase Velocity c^{[1]}

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**Figure 1.**(

**a**) A rectangular microchannel of height ${h}_{1}^{\text{}}+{h}_{2}^{\text{}}$, length L, and width W. We consider the large aspect ratio channels for which ${h}_{1}^{\text{}}+{h}_{2}^{\text{}}\ll W\phantom{\rule{-0.166667em}{0ex}}L$. The two main approximations are negligible gravity and two-dimensional parallel flows, characteristic of microfluidic electroosmotic pumps [20]. (

**b**) Panel (

**a**) is approximated to two streaming viscous dielectrics confined between two infinite, microscopically spaced plates. The liquids differ in mass density, viscosity, and dielectric constants, and occupy different depths of the microchannel. The liquids are, in addition, exposed to an electric field. ${U}_{0}$ is a slip (driving) velocity at the wall, countering an adverse pressure $\Delta P$.

**Figure 2.**(

**a**) The unperturbed flow profiles at fixed $\mu =2$ and $h=4$ for a pure Couette flow, ${G}_{1}^{\text{}}=0$, and an adverse pressure ${G}_{1}^{\text{}}=0.12$ marked by arrows. (

**b**) A list of parameters used in the analysis.

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**MDPI and ACS Style**

Goranović, G.; Sørensen, M.P.; Bruus, H.; Brøns, M.
Electrohydrodynamic (In)Stability of Microfluidic Channel Flows: Analytical Expressions in the Limit of Small Reynolds Number. *Water* **2024**, *16*, 544.
https://doi.org/10.3390/w16040544

**AMA Style**

Goranović G, Sørensen MP, Bruus H, Brøns M.
Electrohydrodynamic (In)Stability of Microfluidic Channel Flows: Analytical Expressions in the Limit of Small Reynolds Number. *Water*. 2024; 16(4):544.
https://doi.org/10.3390/w16040544

**Chicago/Turabian Style**

Goranović, Goran, Mads Peter Sørensen, Henrik Bruus, and Morten Brøns.
2024. "Electrohydrodynamic (In)Stability of Microfluidic Channel Flows: Analytical Expressions in the Limit of Small Reynolds Number" *Water* 16, no. 4: 544.
https://doi.org/10.3390/w16040544