A study on mixed electro-osmotic/pressure-driven microchannel flows of a generalised Phan-Thien–Tanner fluid

Abstract

This work presents new semi-analytical solutions for the combined fully developed electro-osmotic pressure-driven flow in microchannels of viscoelastic fluids, described by the generalised Phan-Thien–Tanner model (gPTT) recently proposed by Ferrás et al. (Journal of Non-Newtonian Fluid Mechanics, 269:88–99, 2019). This generalised version of the PTT model presents a new function for the trace of the stress tensor—the Mittag–Leffler function—where one or two new fitting constants are considered in order to obtain additional fitting flexibility. The semi-analytical solution is obtained under sufficiently weak electric potential that allows the Debye–Hückel approximation for the electrokinetic fields and for thin electric double layers. Based on the solution, the effects of the various relevant dimensionless numbers are assessed and discussed, such as the influence of $\varepsilon Wi^2$, of the parameters $\alpha$ and $\beta$ of the gPTT model, and also of ${\bar{\kappa }}$, the dimensionless Debye–Hückel parameter. We conclude that the new model characteristics enhance the effects of both $\varepsilon Wi^2$ and ${\bar{\kappa }}$ on the velocity distribution across the microchannels. The effects of a high zeta potential and of the finite size of ions are also studied numerically.

Appendix A: Finite-sized ionic species

The Boltzmann distribution breaks down when taking into account finite-sized ionic species. Therefore, a modified Poisson–Boltzmann equation for the ionic distribution that takes these effects into account leads to [24, 26, 27]:

$$\begin{aligned} \frac{\mathrm {d}^2\psi }{\mathrm {d}y^2}=\frac{2n_{0}ez}{\epsilon }\frac{\sinh \left( \frac{ez}{k_\mathrm{B}T}\psi \right) }{1-\varTheta +\varTheta \cosh \left( \frac{ez}{k_\mathrm{B}T}\psi \right) }, \end{aligned}$$

(A.1)

where $\varTheta$ is the steric factor, representing the excluded volume effects owing to the finite size of the ionic species. This is a non-linear differential equation, and in order to obtain the induced potential distribution, the procedure used in [24] was followed.

The shear rate can be obtained as a function of the zeta potential, recalling that $\tau _{xy}= \epsilon E_x \frac{\mathrm{d}\psi }{\mathrm{d}y}$ [24]. Considering Eq. (20) for $\xi =0$, it leads to

$$\begin{aligned} \frac{\mathrm {d}\bar{u}}{\mathrm {d}\bar{y}}=\frac{-\varGamma (\beta )}{{\bar{\psi }}_{0}}\frac{\mathrm {d}{\bar{\psi }}}{\mathrm {d}\bar{y}}E_{\alpha ,\beta }\left[ \frac{2\varepsilon Wi^{2}}{{\bar{\kappa }}^2{\bar{\psi }}_{0}^{2}}\left( \frac{\mathrm {d}{\bar{\psi }}}{\mathrm {d}\bar{y}}\right) ^{2}\right] . \end{aligned}$$

(A.2)

For a low zeta potential, the size of the ionic species is negligible, the Debye–Hückel approximation applies and we obtain the solutions derived previously. For a high zeta potential [24, 25], Eq. (A.1) in non-dimensional form simplifies to

$$\begin{aligned} \frac{\mathrm {d}^{2}{\bar{\psi }}}{\mathrm {d}\bar{y}^{2}}=\frac{{\bar{\kappa }}^2\sinh ({\bar{\psi }})}{1-\varTheta +\varTheta \cosh ({\bar{\psi }})}\approx \frac{{\bar{\kappa }}^2}{\varTheta }, \end{aligned}$$

(A.3)

which together with the boundary conditions $\left. ({\mathrm {d}{\bar{\psi }}}/{\mathrm {d}\bar{y}})\right| _{\bar{y}=0}=0$ and ${\bar{\psi }}(1)={\bar{\psi }}_{0}$, lead to

$$\begin{aligned} {\bar{\psi }}={\bar{\psi }}_0+\frac{{\bar{\kappa }}^2(\bar{y}^2-1)}{2\varTheta }. \end{aligned}$$

(A.4)

The corresponding velocity profile can be obtained numerically, for example, using Simpson’s rule for $\bar{y}\in [0,1]$:

$$\begin{aligned} \bar{u}(\bar{y})=\int _{\bar{y}}^{1}\frac{\varGamma (\beta )}{{\bar{\psi }}_{0}}\frac{{\bar{\kappa }}^{2}\bar{z}}{\varTheta }E_{\alpha ,\beta }\left[ \frac{2\varepsilon Wi^{2}}{{\bar{\psi }}_{0}^{2}}\left( \frac{{\bar{\kappa }}\bar{z}}{\varTheta }\right) ^{2}\right] \mathrm {d}\bar{z}. \end{aligned}$$

(A.5)

The higher the steric factor, the lower the induced transverse EDL field will be, and a lower volumetric flow rate will be obtained. This effect is shown in Fig. 11, where the velocity profiles for both the exponential and gPTT models are plotted (considering $\varTheta =0.2$ and 0.25 and a high zeta potential ${\bar{\psi }}_0=4$).

Fig. 11

Velocity profiles for the exponential and gPTT models considering $\varTheta =0.2$ and 0.25 and a high zeta potential (${\bar{\psi }}_0=4$). Dashed line: pure EO flow with $\beta =1$, $\alpha =0.8$; full line: pure EO flow with $\beta =1$, $\alpha =0.8$. ${\bar{\kappa }}=1$ and $\varepsilon Wi^2=0.5$

As expected, the gPTT model allows one to obtain a higher flow rate due to the higher rate of destruction of junctions in the polymer entanglements when $\alpha$ decreases. Note also the non-linear increase of the flow rate with decreasing $\alpha$ at constant $\varTheta$, showing the complex interaction between the rate of destruction of junctions and the steric effect.