Electrokinetic instability in microchannel ferrofluid/water co-flows

Electrokinetic instability refers to unstable electric field-driven disturbance to fluid flows, which can be harnessed to promote mixing for various electrokinetic microfluidic applications. This work presents a combined numerical and experimental study of electrokinetic ferrofluid/water co-flows in microchannels of various depths. Instability waves are observed at the ferrofluid and water interface when the applied DC electric field is beyond a threshold value. They are generated by the electric body force that acts on the free charge induced by the mismatch of ferrofluid and water electric conductivities. A nonlinear depth-averaged numerical model is developed to understand and simulate the interfacial electrokinetic behaviors. It considers the top and bottom channel walls’ stabilizing effects on electrokinetic flow through the depth averaging of three-dimensional transport equations in a second-order asymptotic analysis. This model is found accurate to predict both the observed electrokinetic instability patterns and the measured threshold electric fields for ferrofluids of different concentrations in shallow microchannels.


Depth-Averaged Asymptotic Analysis
Under the assumptions of rapid charge relaxation, electro-neutrality, and a thin electrical double layer, we have ∇ • ( ) = 0 (S1) Where is the charge density, = ( , , ) is the three-dimensional velocity vector. Eq. (S1) and (S2) are used to calculate the electric field, Eq. (S3) and (S4) are used to study the flow field, and the Eq. (S5) based on the species conservation is used for observing the dynamic changing process of concentration.
The boundary conditions at the channel wall are listed below, To non-dimensionalize our governing equations, following scales which are similar with those in Lin et. al.
[1] are used here Where and denote the half-width (in the y-direction) and half-depth (in the z-direction) of channel, respectively, 0 , 0 , 0 and 0 are the reference electric conductivity, concentration, viscosity and zeta potential of fluid, the characteristic electric field 0 is taken to be the applied electric field in the channel, and is the electroviscous velocity.
After the non-dimensionalization, the governing equations then be written as 2 ∇ • ( ∇ ) + ( ) = 0 (S10) Where ≡ / is a smallness parameter, = ( , ) is two-dimensional velocity vector, ≡ / and ≡ / 0 are the Peclet and Reynolds numbers, respectively, and ∇ is the two-dimensional (x, y) differential operator defined below, Except for the electroosmotic velocity, boundary conditions keep the same form after the nondimensionalization. Considering the dissimilar boundary conditions of top and bottom wall, we have, Here ′ is the zeta potential of top wall, ′′ is the zeta potential of bottom wall, ′ and ′′ are the electroosmotic velocity at top and bottom wall, respectively.
Under the assumption of δ ≪ 1 , the asymptotic analysis introduced in Lin et al.
[1] is performed here for our study. The depth-averaged functions are defined as Where the subscript stands for the corresponding order variables. In this paper, we perform the depth-averaged analysis up to the second order in .

Electric field equation
Performing 0 order balance for Eq. (S10), we get Considering 0 = 0 ( , , ) only, we obtain Applying the insulating boundary conditions 0 = 0, at z = ±1, Performing a 1 order on Eq. (S10) balance and performing the same steps as before, we get Similarly for a 2 order balance: Applying the insulating boundary conditions at the top and bottom surfaces for 2 and simplifying, From Eq. (S24) and (S25), we obtain, For 3 order balance of Eq. (S10): Applying boundary conditions for 3 , Performing (S25) + ×(S28), we obtain the final depth-averaged equation for electric field,

Concentration equation
At the 0 -order, we have 2 0 Applying the boundary condition 0 = 0 at z = ±1 At the 1 -order, we have Depth-averaging and applying the boundary condition, Eq. (S60) can also be integrated in the z-direction, Where 3 is constant of integration.
Based on Eq. (S62) and applying the boundary condition, we obtain