Hydrodynamic bifurcation in electro-osmotically driven periodic flows

In this paper, we report an inertial instability that occurs in electro-osmotically driven channel flows. We assume that the charge motion under the influence of an externally applied electric field is confined to a small vicinity of the channel walls that, effectively, drives a bulk flow through a prescribed slip velocity at the boundaries. Here, we study spatially periodic wall velocity modulations in a two-dimensional straight channel numerically. At low slip velocities, the bulk flow consists of a set of vortices along each wall that are left-right symmetric, while at sufficiently high slip velocities, this flow loses its stability through a supercritical bifurcation. Surprisingly, the flow state that bifurcates from a left-right symmetric base flow has a rather strong mean component along the channel, which is similar to pressure-driven velocity profiles. The instability sets in at rather small Reynolds numbers of about 20–30, and we discuss its potential applications in microfluidic devices.

Figure 1

Electro-osmotically driven periodic flow; after Ajdari [7].

Figure 2

Velocity profiles given by Eq. (8) for $\tilde{k}=\pi$. Vertical axes are positions in the gap $(y$ coordinates), and the horizontal axes give the distance along the channel in units of $\tilde{k}x$. The top row corresponds to the symmetric boundary conditions $\left(\tilde{v}=1\right)$, while the bottom row shows the effect of their asymmetry $\left(\tilde{v}=1/3\right)$. The phase shift $\varphi$ is (left) 0, (middle) $\pi /2$, and (right) $\pi$.

Figure 3

The ratio of the kinetic energies of the inertial and Stokes solutions for $\tilde{v}=1,\varphi =0$, and $\tilde{k}=\pi$ as a function of the Reynolds number $\text{Re}$. The solid line is well approximated by $1+{(\text{Re}/151.41)}^2$.

Figure 4

Velocity profile at $\text{Re}=40$ for $\tilde{v}=1,\varphi =0$, and $\tilde{k}=\pi$. (Left) The velocity field in the channel without its mean profile (zeroth Fourier harmonics). (Right) The mean profile $U\left(y\right)$.

Figure 5

(Left) The bifurcation diagram, $\chi$ vs $\text{Re}$, for various values of $\varphi$ at $\tilde{k}=\pi$. (Right) The critical Reynolds number ${\text{Re}}_{\mathrm{crit}}$ as a function of $\tilde{k}$. For $\varphi =0$, the minimal value of ${\text{Re}}_{\mathrm{crit}}$ is ${\text{Re}}_{\mathrm{crit}}^{\min }=33.31$ at $\tilde{k}=3.3$, while for $\varphi =\pi ,{\text{Re}}_{\mathrm{crit}}^{\min }=21.2$ at $\tilde{k}=2.6$. In both plots $\tilde{v}=1$.

Figure 6

The bifurcation diagram for the modified boundary conditions with a bias, Eq. (14) for $\varphi =\pi$ and $\tilde{k}=2.6$. The strength of the bias, ${\text{Re}}_c$, is fixed and the strength of the spatially oscillatory component, $\text{Re}$, is varied.