Instability of electro-osmotic channel flow with streamwise conductivity gradients

This work considers the stability of an electro-osmotic microchannel flow with streamwise electrical conductivity gradients, a configuration common in microfluidic applications such as field amplified sample stacking. Previous work on such flows has focused on how streamwise conductivity gradients set a nonuniform electro-osmotic velocity which results in dispersion of the conductivity field. However, it has been known for many years that electric fields can couple with conductivity gradients to generate unstable flows. This work demonstrates that at high electric fields such an electrohydrodynamic instability arises in this configuration and the basic mechanisms are explored through numerical simulations. The instability is unique in that the nonuniform electro-osmotic flow sets the shape of the underlying conductivity field in a way that makes it susceptible to instability. While nonuniform electro-osmotic flow sets the stage, the instability is ultimately the result of electric body forces due to slight departure from electroneutrality in the fluid bulk. A simple stability map is created where two dimensionless numbers can predict system stability reasonably well, even though the system formally depends on six dimensionless groups.

Figure 1

(Color online) (Top) Schematic of the geometry of interest for this problem. Low conductivity fluid is surrounded by high conductivity buffer. (Middle) When an electric field is applied across the length of this channel, the field is highest in the low conductivity region due to Ohm’s law. The result is that the sample ions have a higher electrophoretic velocity in the low conductivity region. All sample ions quickly leave the low conductivity region and stack at the interface with the high conductivity region. (Bottom) The electro-osmotic velocity is highest in the low conductivity sample region. The centerline velocity must then vary inversely with the slip velocity for mass conservation to be obeyed.

Figure 2

(Color online) Snapshots of the conductivity field during FASS instability. Parameters are: $\mathrm{Ra}=2160$, $R_v=0.056$, $\mathrm{Re}=1.08$, $\gamma =10$, and $\delta =0.1$. In time, the snapshots move from left to right, then down. The dimensionless parameters correspond to $E_o=25000\mathrm{V}∕\mathrm{m}$, $H=50\mu \mathrm{m}$, $\zeta =0.07\mathrm{V}$, and $D=5\times {10}^{-10}$. The other physical parameters are taken as typical of an aqueous solution. The color bar is set such that red corresponds to $\sigma =10$ and blue corresponds to $\sigma =1$. The light colored regions correspond to intermediate values of the conductivity and can be used to easily visualize the interface between high and low conductivity fluids. Given the high Rayleigh number, the conductivity field acts as a tracer to visualize the flow.

Figure 3

(Color online) Snapshots of the conductivity field during FASS instability for different electric fields. The parameters are the same as Fig. 2 only the electric field is varied from $E_o=20000\mathrm{V}∕\mathrm{m}\text{to}{E}_o=35000\mathrm{V}∕\mathrm{m}$ from top to bottom. The applied electric field in kV/m is denoted in the figure. The snapshots are taken at the same dimensionless time, $t=50$.

Figure 4

(Color online) (a) Maximum dimensionless vertical velocity along the channel centerline as a function of time for different electric fields; $E_0=10000$, 15 000, 17 000, 20 000, and $25000\mathrm{V}∕\mathrm{m}$ from bottom to top. (b) Maximum dimensionless centerline vertical velocity achieved over the course of the run as a function of electric field. The other dimensional parameters are the same as in Fig. 2.

Figure 5

(Color online) Maximum dimensionless vertical velocity along the channel centerline as a function of time with (upper curve) and without (lower curve) the body force term in the equations. The inset snapshots of the conductivity field are taken where the dot is located, $t=75$. Parameters in both cases are: $\mathrm{Ra}=880$, $R_v=0.0311$, $\mathrm{Re}=0.88$, $\gamma =10$, and $\delta =0.1$.

Figure 6

(Color online) Maximum dimensionless vertical velocity along the channel centerline as a function of time for the case with no electro-osmotic flow. Parameters are the same as Fig. 2 only the electric field is reduced to $E_o=10000\mathrm{V}∕\mathrm{m}$. The inset snapshots show the conductivity field (top) and vertical velocity field (bottom) at $t=75$.

Figure 7

(Color online) Snapshots of the conductivity field with a 3D simulation in a square channel. The parameters are $E_o=35000\mathrm{V}∕\mathrm{m}$, $H=50\mu \mathrm{m}$, $\zeta =0.07\mathrm{V}$, $D=5\times {10}^{-10}$, $\gamma =10$, and $\delta =0.1$. In time, the snapshots move from left to right, then down.

Figure 8

(Color online) (a) Maximum vertical velocity along the channel centerline during the entire run as a function of $\mathrm{Ra}(\gamma -1)$. The dots correspond to 80 different runs with different parameters except all runs have $\zeta =0.07\mathrm{V}$. The different runs varied the electric field, channel height, conductivity ratio, and diffusivity of the electrolyte. The inset shows the same data as a function of the dimensional electric field. (b) Phase diagram showing stable points as blue dots [defined as $\max \left(V\right)<{10}^{-3}$] and unstable points as red circles. There are 110 runs included in this figure.

Figure 9

(Color online) Phase diagram showing stable points as blue dots [defined as $\max \left(V\right)<{10}^{-3}$] and unstable points as red circles. The data are the same as in Fig. 8, only with different parameters selected for the $x$ and $y$ axis. The experimentally measured critical Rayleigh number of 205 as determined by Posner and Santiago is shown as the dashed line [17].