Time-dependent analysis of electroosmotic fluid flow in a microchannel

V. K. Narla; Dharmendra Tripathi; G. P. Raja Sekhar

doi:10.1007/s10665-019-09988-4

Journal of Engineering Mathematics

pp 1–20 | Cite as

Time-dependent analysis of electroosmotic fluid flow in a microchannel

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Authors and affiliations

V. K. Narla
Dharmendra Tripathi
G. P. Raja Sekhar

Article

First Online: 07 February 2019

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Abstract

The present work models electroosmotic induced fluid flow in a microfluidic device. For theoretical analysis, the geometry of this device is considered as an asymmetric, narrow, wavy channel with charged surface. It is assumed that the length of the channel is finite and the characteristic wavelength is very large compared to the half width of the channel. The flow is assumed to be governed by Navier–Stokes equations augmented with electric body force. A transient two-dimensional flow analysis is presented by employing lubrication theory. Debye–Hückel linearization is adopted to obtain a general solution of Poisson–Boltzmann equation. The flow rate, velocity profile, pressure distribution, and wall shear stress are analyzed as functions of various parameters involved like zeta-potential ratio, Debye–Hückel parameter, etc. It is noted that the fluctuations in pressure and shear stress increase with the increasing zeta-potential ratio when the Helmholtz–Smoluchowski velocity is positive. The streamline pattern and the particle trajectories are analyzed to understand the trapping phenomenon and the retrograde motion. It is observed that the electroosmosis phenomenon drastically modulates the fluid flow in microchannels. Although the asymmetric nature of the wavy channel does not support active particle transport, the applied electric field enhances the particle motion favoring optimal conditions. It is observed that the extreme asymmetry of the wall motility reduces the net flow rate. Further, it is noticed that the asymmetry reduces the amplitudes of the pressure. This model can help toward designing artificial organs based on microfluidic devices which can also be applicable to analyze lumenal flow inside arteries and flow inside intrauterine system, and to implant the embryo at the best location in the uterus for human-assisted reproduction.

Keywords

Electroosmosis Particle retrograde motion Peristalsis Trapping

Mathematics Subject Classification

35Q35 74M25 76Z05

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Notes

Acknowledgements

The first author gratefully acknowledges the support by Centre for Theoretical Studies (CTS), Indian Institute of Technology Kharagpur, India under a visitors’ program.

Appendix A: Navier–Stokes equations in terms of stream function

The incompressibility condition $\nabla \cdot {\mathbf {V}} = 0$ allows us to introduce the stream function as $u=\frac{\partial \psi }{\partial y}$, $v=-\frac{\partial \psi }{\partial x}$. Correspondingly, pressure can be eliminated from the momentum equation (2) to obtain an equivalent form of the Navier–Stokes equations (after dropping primes for simplicity) in terms of stream function given by

$$\begin{aligned} \begin{aligned}&Re\;\delta \Big [\psi _{tyy}+\psi _y\psi _{xyy}-\psi _x\psi _{yyy}+\delta ^2\Big (\psi _{txx}+\psi _y\psi _{xxx}-\psi _x\psi _{xxy}\Big )\Big ]=\psi _{yyyy}+2\delta ^2\psi _{xxyy} \\&\quad +\delta ^4\psi _{xxxx}+U_\mathrm{{HS}}\Big [\delta ^2\varPhi _{xxy}+\varPhi _{yyy}+\delta ^2\Big (\delta ^2\varPhi _{xxx}+\varPhi _{yyx}\Big )\Big ], \end{aligned} \end{aligned}$$

(20)

where the subscripts x and y denote partial differentiations with respect to those variables. The incompressibility of the electric field $\nabla \cdot {\mathbf {E}}=0$ due to the above non-dimensionalization ensures the growth relation $E_y\sim \delta E_x$, where $E_x$ and $E_y$ are the axial and transverse components of the electric field, respectively.

Appendix C: Averaged flow rate

The transformation of the coordinates from the laboratory frame (x, y) (the fixed coordinates) to the wave frame (X, Y) (the moving coordinates) is related by $X=x-ct$, $Y=y$, and the corresponding velocity components by $U=u-1$ and $V=v$. The non-dimensional-averaged flow rate in the wave frame is defined as

$$\begin{aligned} Q(x,t)=q(x)+h_2(x,t)-h_1(x,t), \end{aligned}$$

(21)

and the time-averaged flow rate over a period T yields

$$\begin{aligned} Q_T(x)=q(x)+2\epsilon +A+\frac{A}{\pi }\frac{\cos [\pi (2x-T)+\phi ]\cos [\phi ]\sin [\pi T]}{T}. \end{aligned}$$

(22)

From Eqs. (21) and (22), the axial pressure gradient in terms of time-averaged flow rate can be presented in the form:

$$\begin{aligned} \frac{\partial p}{\partial x}=\frac{q(x)+(h_2-h_1)+U_\mathrm{{HS}}\zeta _1(1+R_\zeta )\Big [\frac{h_1-h_2}{2}-\frac{1}{\kappa }\tanh \Big [\frac{\kappa (h_1-h_2)}{2}\Big ]\Big ]}{\frac{1}{12}(h_1-h_2)^3}. \end{aligned}$$

(23)

The pressure difference between the inlet and the outlet of the channel is

$$\begin{aligned} \varDelta P_L=\int \limits _{0}^L\frac{q(x)+(h_2-h_1)+U_\mathrm{{HS}}\zeta _1(1+R_\zeta )\Big [\frac{h_1-h_2}{2}-\frac{1}{\kappa }\tanh \Big [\frac{\kappa (h_1-h_2)}{2}\Big ]\Big ]}{\frac{1}{12}(h_1-h_2)^3}\mathrm{{d}}x. \end{aligned}$$

(24)

The maximum average flow rate $Q_{T,\mathrm{{Max}}}$ is obtained by taking $\varDelta P_L=0$ in Eq. (24), so that

$$\begin{aligned} Q_{T,Max}=\frac{\int \nolimits _{0}^L\frac{(2\epsilon +A)+(h_1-h_2)-U_\mathrm{{HS}}\zeta _1(1+R_\zeta )\Big [\frac{h_1-h_2}{2} -\frac{1}{\kappa }\tanh \Big [\frac{\kappa (h_1-h_2)}{2}\Big ]\Big ]}{(h_1-h_2)^3}\mathrm{{d}}x}{\int \nolimits _{0}^L\frac{1}{(h_1-h_2)^3}\mathrm{{d}}x}. \end{aligned}$$

(25)

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Authors and Affiliations

V. K. Narla
- 1
Dharmendra Tripathi
- 2
Email author
G. P. Raja Sekhar
- 3

1.Department of MathematicsGITAM School of TechnologyHyderabadIndia
2.Department of Sciences and HumanitiesNational Institute of TechnologySrinagarIndia
3.Department of MathematicsIndian Institute of TechnologyKharagpurIndia

Time-dependent analysis of electroosmotic fluid flow in a microchannel

Abstract

Keywords

Mathematics Subject Classification

Notes

Acknowledgements

References

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