Slippage effect on the dispersion coefficient of a passive solute in a pulsatile electro-osmotic flow in a microcapillary

The hydrodynamic dispersion of a neutral solute released into a pulsatile electro-osmotic flow in a microcapillary that is affected by slippage at the wall (modeled by the Navier slip condition) is studied theoretically. The long-time Taylor dispersion is analytically derived using the homogenization method with multiple scales. The results indicate that the effective dispersion coefficient depends on a dimensionless slip length, an angular Reynolds number, the amplitude of the oscillatory component of the external electric field, and an electrokinetic parameter that relates the radius of the microcapillary with the Debye length. Our results suggest that in the presence of the Navier slip condition, the dispersivity is maximized by up to two orders of magnitude compared with that obtained through the classical no-slip condition.

Figure 1

Schematic diagram of the physical model under study.

Figure 2

Dimensionless velocity profiles caused by the PEOF in the periodic stage for $0\le t\le 2\pi$ [Eq. (3)] for $R_{\omega }$ ($=0.1$, 10, and 100) with $\xi =1.0$, $\delta =0.1$, and two EDL thicknesses quantified by (a) $\kappa =10$ and (b) $\kappa =100$. Schemes I and IV correspond to $R_{\omega }=0.1$, II and V to $R_{\omega }=10.0$, and III and VI to $R_{\omega }=100$.

Figure 3

Dimensionless velocity profiles caused by the PEOF in the periodic stage for $0\le t\le 2\pi$ [Eq. (3)]. Red lines represent the case of no slippage with $\delta =0$ and blue lines denote the case of slippage with $\delta =0.1$. Both cases are plotted for $R_{\omega }=100$ and (a) $\kappa =10$ and (b) $\kappa =100$.

Figure 4

Effective dispersion coefficient $\mathcal{D}=(\mathcal{E}-1)/{\mathrm{Pe}}^2$ versus the electrokinetic parameter $\kappa$ evaluated for four different values of the dimensionless frequency $R_{\omega }$ ($=0.1$, 1, 10, and 100). The relative amplitude of the oscillatory electric field is fixed at $\xi =1.0$, considering (a) the no-slip condition $\delta =0$ and (b) the Navier slip condition for $\delta =0.1$. The curve corresponding to $R_{\omega }=1.0$ has been plotted using Eq. (50).

Figure 5

Effective PEOF dispersion coefficient $\mathcal{D}=(\mathcal{E}-1)/{\text{Pe}}^2$ as a function of the slip length, which varies from $\delta =0$ (no-slip condition) to $\delta =0.1$, calculated for different values of $R_{\omega }$ ($=0.1$, 1, 10, and 100) with a relative amplitude of the oscillatory electric field of 1.0 and (a) $\kappa =10$ and (b) $\kappa =100$. The curve corresponding to $R_{\omega }=1.0$ has been plotted using Eq. (50).

Figure 6

Effective PEOF dispersion coefficient $\mathcal{D}=(\mathcal{E}-1)/{\text{Pe}}^2$ as a function of $R_{\omega }$, considering also three different slip lengths at the microcapillary wall $\delta$ ($=0$, 0.05, and 0.1) with $\xi =1$ and (a) $\kappa =10$ and (b) $\kappa =100$. For each curve, the effective dispersion coefficient corresponding to $R_{\omega }=1.0$ has been calculated using Eq. (50).

Figure 7

Effective PEOF dispersion coefficient $\mathcal{D}=(\mathcal{E}-1)/{\text{Pe}}^2$ as a function of $R_{\omega }$ with (a) the no-slip condition and (b) the Navier slip condition, both with $\kappa$ ($=10$, 50, and 100) and a relative amplitude of the oscillatory electric field $\xi =1$. For each curve, the effective dispersion coefficient corresponding to $R_{\omega }=1.0$ has been calculated using Eq. (50).

Figure 8

Effective PEOF dispersion coefficient $\mathcal{D}=(\mathcal{E}-1)/{\text{Pe}}^2$ as a function of $R_{\omega }$. The variation in the electric field amplitude is analyzed considering $\xi =0$, 0.5, and 1, with the slip length at the microcapillary wall fixed at $\delta =0.1$ for (a) $\kappa =10$ and (b) $\kappa =100$. For each curve, the effective dispersion coefficient corresponding to $R_{\omega }=1.0$ has been calculated using Eq. (50).

Figure 9

Effective PEOF dispersion coefficient $\mathcal{D}$ as a function of $\mathrm{\Omega }$ for Sc ($=100$, 500, and 2000) with $\kappa =10$, $\xi =1.0$, and (a) the no-slip condition $\delta =0$ and (b) the slip condition $\delta =0.1$. The points where $\text{Sc}=\mathrm{\Omega }$ were calculated considering Eq. (50) for the oscillatory dispersion component.

Figure 10

Effective PEOF dispersion coefficient $\mathcal{D}$ as a function of $\mathrm{\Omega }$ for Sc ($=100$, 500, and 2000) with $\kappa =100$, $\xi =1$, and (a) the no-slip condition $\delta =0$ and (b) the slip condition $\delta =0.1$. The points where $\text{Sc}=\mathrm{\Omega }$ were calculated considering Eq. (50) for the oscillatory dispersion component.

Figure 11

Relative dispersion coefficient for a ${\text{Sc}}_{\text{slow}}$ substance (sucrose) with respect to a ${\text{Sc}}_{\text{fast}}$ substance (ethanol), under the no-slip condition $\delta =0$ (blue curve) and with the slip condition $\delta =0.1$ (black curve) with $\xi =1.0$ and (a) $\kappa =10$ and (b) $\kappa =100$.