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Numerical analysis of the effects of roughness on the electro-osmotic laminar flow between two parallel plates

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Abstract

In the present study, the effects of wall roughness on the dynamic of laminar electro-osmotic flow (EOF) between two parallel plates have been investigated numerically. The governing equations of the flow in a system of two-dimensional general coordinates are solved using the finite-volume numerical method in a non-uniform grid with the maximum orthogonality of the grid lines adjacent to the rough boundaries. In the first step, the surface ruggednesses of the wall are divided into two categories: “surface roughness” and “wall blocks”. The wall surface ruggednesses have profiles in the forms of sinuses, saw teeth, and square teeth. Then, the distinction boundary of surface roughness from wall blocks is determined as \({\left.\mathrm{h}/\mathrm{H}\right|}_{cr}=0.078\) by defining and applying two qualitative and quantitative criteria. Finally, the dynamic of EOF near the walls surface roughness with \(0.001\le\upvarepsilon /\mathrm{H}\le 0.05\) is investigated. According to the results, unlike macro-scale pressure-driven flows, the effect of surface roughness is not only non-negligible on the laminar EOF through the microchannels but also disruptive for the ionic and dynamic development of the flow adjacent to the walls. Finally, based on the numerical results, a new correlation has been developed for variations of \(\dot{\mathrm{m}}\) for laminar EOF between two parallel plates in the presence of the surface roughness and its maximum error is 2%.

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Notes

  1. Arithmetical mean deviation of the assessed profile (\(Ra=\frac{1}{n}\sum_{i=1}^{n}\left|{y}_{i}\right|\)).

Abbreviations

\(A\) :

A dimensionless parameter, representing the ratio of the external applied voltage to the base voltage

\(B\) :

A dimensionless parameter, representing the ratio of ionic pressure to dynamic pressure

\(D^{ \pm }\) :

Positive and negative ion diffusion coefficients (m2/s)

\(E\) :

External electric field strength (V/m)

\(e\) :

Elementary charge of an electron (C)

\(F\) :

Force (N)

\(H\) :

Microchannel height (m)

\(h\) :

Height of the surface ruggedness (m)

\(i,j\) :

Node number in \(\xi\) and \(\eta\) direction

\(\hat{i}, \hat{j}\) :

Unit vectors in x and y directions

\(K\) :

Debye-Huckel parameter (1/m)

\(k_{b}\) :

Boltzmann constant (J/K)

\(L\) :

Microchannel length (m)

\(L_{L} , L_{M} , L_{R}\) :

Inlet, Middle and Outlet region length (m)

\(\dot{m}\) :

Mass flow rate per unit depth of channel (kg/s)

\(n\) :

Perpendicular to the wall of the microchannel

\(n_{0}\) :

Bulk concentration of ions in the electrolyte solution per unit volume (ions/m3)

\(n^{ \pm }\) :

Concentration of positive and negative ions (ions/m3)

\(p\) :

Pressure (Pa)

\(Re\) :

Reynolds number

\(Sc^{ \pm }\) :

Schmidt number of ions

\(T\) :

Absolute temperature (K)

\(u, v\) :

X- and y-component of velocity (m/s)

\(\forall\) :

Volume of a computational cell (m3)

\(V_{app}\) :

External applied voltage (V)

\(x, y\) :

Cartesian coordinates

\(z\) :

Charge valence number

\(\varepsilon\) :

Height of the surface roughness (m)

\(\varepsilon_{0}\) :

Vacuum permittivity (C/V m)

\(\varepsilon_{r}\) :

Relative permittivity

\(\zeta\) :

Wall zeta potential (V)

\(\eta , \xi\) :

Perpendicular and Tangential directions of the body-fitted coordinate

\(\kappa\) :

Double-layer thickness parameter

\(\mu\) :

Fluid dynamic viscosity (Pa s)

\(\rho\) :

Fluid density (kg/m3)

\(\rho _{e}\) :

Net electric charge density (C/m3)

\(\varphi\) :

External electric potential field (V)

\(\psi\) :

Electric potential field in EDL (V)

\(1, 2\) :

At the inlet and outlet

\(cr\) :

Critical value

\(e, w, n, s\) :

The eastern, western, northern and southern facets of the control volume

\(i, j\) :

Node number in \(\xi\) and \(\eta\) direction

\(in\) :

Inlet

\(N\) :

Normal

\(ref\) :

Reference value

\(S\) :

Shear

\(x, y\) :

Cartesian coordinates

\(\overline{\,\,}\) :

Bar, for dimensionless parameter

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Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, University of Birjand, Birjand, Iran

    Mohammad Mahdi Fakhari & Seyed Ali Mirbozorgi

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  1. Mohammad Mahdi Fakhari
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  2. Seyed Ali Mirbozorgi
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Correspondence to Seyed Ali Mirbozorgi.

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Fakhari, M.M., Mirbozorgi, S.A. Numerical analysis of the effects of roughness on the electro-osmotic laminar flow between two parallel plates. Meccanica 56, 1025–1045 (2021). https://doi.org/10.1007/s11012-020-01257-4

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