Unsteady Electroosmotic Flow of Jeffrey Fluid in a Circular Microchannel under the Combined Action of Vertical Magnetic Field, External Electric Field, and Pressure at High Zeta Potential

Abstract

The unsteady electroosmotic flow (EOF) for one kind of linear viscoelastic fluid, which is Jeffrey type fluid, is investigated under the common impact of vertical magnetic field, external electric field, and pressure at high Zeta potential in a circular microchannel. The numerical solutions of the potential and velocity distributions are obtained by solving the nonlinear Poisson-Boltzmann equation, the constitutive equation of the Jeffrey fluid, and the Cauchy momentum equation applying the Chebyshev spectral method and the finite difference method. By contrast, the Chebyshev spectral method has higher accuracy and less computation. The flow characteristics of Jeffrey fluid at high Zeta potential are analyzed with the numerical solution obtained by the Chebyshev spectral method. The results show that the velocity of Jeffrey fluid increases with the increase of the wall Zeta potential and electric width. The oscillation amplitude of velocity distribution increases with the increase of relaxation time but decreases with the increase of retardation time. When the Hartmann number is smaller, the increase of relaxation time leads to the increase of velocity; when the Hartmann number is larger, the increase of relaxation time leads to the decrease of velocity. No matter what the Hartmann number is, the velocity always decreases with the increase of the retardation time. The velocity tends to be stable gradually with the increase of time.

Transforming Navier‐stokes equation to a fourth order nonlinear differential equation for the study of electrolyte flow in a circular microchannel

“…From mathematical Handbook [42], the modified Bessel functions I 0 and I 1 have the following relationships: $$$$\begin{equation}I_0^{\prime} = {I_1} \to \frac{d}{{dr}}\;\left[ {{I_0}\left( {\frac{r}{{{\lambda _D}}}} \right)} \right] = \frac{1}{{{\lambda _D}}}{I_1}\left( {\frac{r}{{{\lambda _D}}}} \right)\end{equation}$$$$…”

“…From mathematical Handbook [42], the modified Bessel functions 𝐼 0 and 𝐼 1 have the following relationships:…”

“…Substitution of functions G and H and the following series solution for $${I_0}( {\frac{r}{{{\lambda _D}}}} )$$in the following form [42]: $$$$\begin{equation}{I_0}\left( {\frac{r}{{{\lambda _D}}}} \right) = 1 + \frac{{{{\left( {\frac{r}{{{\lambda _D}}}} \right)}^2}}}{{{2^2}}} + \frac{{{{\left( {\frac{r}{{{\lambda _D}}}} \right)}^4}}}{{{2^2}{{.4}^2}}} + \frac{{{{\left( {\frac{r}{{{\lambda _D}}}} \right)}^6}}}{{{2^2}{{.4}^2}{{.6}^2}}} + \frac{{{{\left( {\frac{r}{{{\lambda _D}}}} \right)}^8}}}{{{2^2}{{.4}^2}{{.6}^2}{{.8}^2}}} + \frac{{{{\left( {\frac{r}{{{\lambda _D}}}} \right)}^{10}}}}{{{2^2}{{.4}^2}{{.6}^2}{{.8}^2}{{.10}^2}}} + \cdots \end{equation}$$$$into Equation () and integration from $$z = 0$$ to $$z = L$$ gives: …”

“…Furthermore, when the slip length increases or the pressure-gradient-to-external-electric-field ratio decreases, the accuracy of this method increases. Ren et al [42] numerically studied the unsteady electroosmotic flow for a linear viscoelastic Jeffrey type fluid under the common impact of external electric field, vertical magnetic field, and pressure in a circular microchannel at high zeta potential. The nonlinear Poisson-Boltzmann equation, the Cauchy momentum equation, and the constitutive equation were solved using finite difference and Chebyshev spectral methods.…”