Electrohydrodynamic instabilities in microchannels with time periodic forcing

In microfluidic applications it has been observed that flows with spatial gradients in electrical conductivity are unstable under the application of sufficiently strong electric fields. These electrohydrodynamic instabilities can drive a nonlinear flow despite the low Reynolds number. Such flows hold promise as a simple mechanism for mixing fluids. In this work, the effect of a time periodic electric field on the instability is explored. The case where an electric field is applied across a diffuse interface of two fluids with varying electrical conductivity is considered. Frequency-dependent behavior is found only in the regime where the instability growth rates are very slow and cannot outpace mixing by molecular diffusion. Improving mixing by modulation of the electric body force is not a viable strategy in this geometry.

Figure 1

Basic configuration of interest to this study. We are interested in two physical geometries: an infinitely deep 2D fluid layer and a shallow channel. For modeling flow in the shallow channel we will use a set of depth-averaged equations that were developed in previous work [24]. The initial assumed conductivity profile is shown as a high conductivity fluid above a low conductivity fluid.

Figure 2

(Color online) Linear stability diagram with the 2D equation set and a conductivity ratio of $10:1$. Solid contours of $s\text{Ra}$ are shown for values of 0 and 250 computed with a steady base state. Two dashed contours show results from the simulation with a transient base state. The curves shown are for amplification factors of $A=10$ and $A=10000$. Snapshots of the conductivity field computed from nonlinear simulations are shown at values of $\text{Ra}=4000$, 7500, and 15 000 in (a), (b), and (c), respectively. The three dotted lines on the contour plot show these values of Ra. We can use this figure to classify different regimes of the flow. For $\text{Ra}\gg 6000$ we observe the flow to be unstable and therefore to have strong mixing. For $\text{Ra}\sim 6000$ we observe unstable waves in the conductivity field but not strong folding and mixing. For $4000<\text{Ra}<6000$ we observe some flow with perhaps only slightly observable waves. For $\text{Ra}<4000$ we find the perturbation to the initial state will only grow by a factor of 10 before the conductivity field simply diffuses away.

Figure 3

(Color online) Linear stability diagram with the depth-averaged equation set and a conductivity ratio of $10:1$. Solid contours of $s\text{Ra}$ are shown for values of 0 and 250 computed with a steady base state. The two dashed contours show results from simulations with a transient base state. The curves shown are for amplification factors of $A=10$ and $A=10000$. Snapshots of the conductivity field computed from nonlinear simulations are shown at values of $\text{Ra}=2000$, 1000, and 500 in (a), (b), and (c), respectively. The dotted lines on the contour plot indicate these values of Ra. We can use this figure to classify different regimes of the flow. For $\text{Ra}{\delta }^2\gg 800$ we observe the flow to be unstable and therefore to have strong mixing. For $\text{Ra}{\delta }^2\sim 800$ we observe unstable waves in the conductivity field but not strong folding and mixing. For $400<\text{Ra}{\delta }^2<800$ we observe some flow with perhaps slightly observable waves. For $\text{Ra}{\delta }^2<400$ we find the perturbation to the initial state will only grow by a factor of 10 before the conductivity field simply diffuses away.

Figure 4

Stability behavior of the $k=2.5$ mode for a conductivity ratio of $10:1$ with the 2D equations. In (a) we show contour lines of growth rate equal to 0, 0.015, and 0.03. We also show the boundary where the flow is unstable in a dc field accounting for a transient base state (dashed line) and the curve above which the flow is unstable in a single forcing cycle. In (b) we show growth rate as a function of frequency at selected Ra.

Figure 5

Stability behavior of the $k=2.5$ mode for a conductivity ratio of $100:1$ with the 2D equations. In (a) we show contour lines of growth rate equal to 0, 0.06, and 0.12. We also show the boundary where the flow is unstable in a dc field accounting for a transient base state (dashed line) and the curve above which the flow is unstable in a single forcing cycle. In (b) we show growth rate as a function of frequency at selected Ra.

Figure 6

Stability behavior of the $k=1.5$ mode for a conductivity ratio of $10:1$ with the depth-averaged equations. In (a) we show contour lines of growth rate equal to 0, 0.05, and 0.07. We also show the boundary where the flow is unstable in a dc field (dashed line) accounting for a transient base state and the curve above which the flow is unstable in a single forcing cycle. In (b) we show growth rate as a function of frequency at selected Ra.