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Pure axial flow of viscoelastic fluids in rectangular microchannels under combined effects of electro-osmosis and hydrodynamics

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Abstract

This paper presents an analysis of the combined electro-osmotic and pressure-driven axial flows of viscoelastic fluids in a rectangular microchannel with arbitrary aspect ratios. The rheological behavior of the fluid is described by the complete form of Phan-Thien–Tanner (PTT) model with the Gordon–Schowalter convected derivative which covers the upper convected Maxwell, Johnson–Segalman and FENE-P models. Our numerical simulation is based on the computation of 2D Poisson–Boltzmann, Cauchy momentum and PTT constitutive equations. The solution of these governing nonlinear coupled set of equations is obtained by using the second-order central finite difference method in a non-uniform grid system and is verified against 1D analytical solution of the velocity profile with less than 0.06% relative error. Also, a parametric study is carried out to investigate the effect of channel aspect ratio (width to height), wall zeta potential and the Debye–Hückel parameter on 2D velocity profile, volumetric flow rate and the Poiseuille number in the mixed EO/PD flows of viscoelastic fluids with different Weissenberg numbers. Our results show that, for low channel aspect ratios, the previous 1D analytical models underestimate the velocity profile at the channel half-width centerline in the case of favorable pressure gradients and overestimate it in the case of adverse pressure gradients. The results reveal that the inapplicability of the Debye–Hückel approximation at high zeta potentials is more significant for higher Weissenberg number fluids. Also, it is found that, under the specified values of electrokinetic parameters, there is a threshold for velocity scale ratio in which the Poiseuille number is approximately independent of channel aspect ratio.

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Abbreviations

APG/FPG:

Adverse/favorable pressure gradient

B, D :

Coefficient functions

\(\mathbf{D}\) :

Rate of deformation tensor (\(s^{-1})\)

e :

Elementary charge (\(1.6022\times 10^{-19}C)\)

\({\varvec{E}},E_z \) :

Vector/z-component of electric field (\(Vm^{-1})\)

\(f\left( {\tau _{kk} } \right) \) :

PTT stress coefficient function

\(f_r \) :

Friction factor

\(k_B \) :

Boltzmann constant (\(1.3807\times 10^{-23}JK^{-1})\)

L, 2H, 2W :

Microchannel length/height/width (m)

M :

Coefficients of transformation function

\(n_0 \) :

Ionic number concentration (\(m^{-3})\)

p :

Pressure (Pa)

Q :

Volumetric flow rate \((m^3 s^{-1})\)

\(\hbox {Re}\) :

Reynolds number

t :

Time (s)

\(T_m \) :

Average absolute temperature (K)

\({\varvec{u}},u\) :

Vector/z-component of velocity (\(ms^{-1})\)

\(\hbox {We}_\kappa \) :

Weissenberg number

x, y, z :

Transverse/depth-wise/axial coordinate (m)

\(\mathbb {z}^{\pm }\) :

Valence of ions

\(\alpha \) :

Channel aspect ratio

\(\beta \) :

Stretch parameter

\(\varGamma \) :

Ratio of PD to HS velocities

\(\Delta \) :

Error index

\(\varepsilon \) :

Extensibility parameter

\(\epsilon \) :

Permittivity of the fluid (\(CV^{-1}m^{-1})\)

\(\phi / \varPhi \) :

External/total electrical potential (V)

\(\eta _p \) :

Polymer viscosity coefficient (Pa.s)

\(\kappa \) :

Debye–Hückel parameter (\(m^{-1})\)

K :

Dimensionless Debye–Hückel parameter

\(\lambda \) :

Relaxation time (s)

\(\xi \) :

PTT model parameter

\(\Pi \Pi \) :

Physical domain of the problem (\(m^{2})\)

\(\rho _e \) :

Electric charge density (\(Cm^{-3})\)

\({\varvec{\tau }} ,\tau _{kk} \) :

Polymeric/trace of extra-stress tensor (Pa)

\(\tau _{xz} ,\tau _{yz} \) :

Stream-wise shear stresses (Pa)

\(\tau _{xx} ,\tau _{yy} \) :

Transverse normal stresses (Pa)

\(\tau _{xy} \) :

Transverse shear stress (Pa)

\(\tau _w \) :

Shear stress at the wall (Pa)

\(\tau _{zz} \) :

Stream-wise normal stress (Pa)

\(\psi ,\psi _0 \) :

EDL/wall zeta potential (V)

\(\omega \) :

Parameter to calculate the relative error

i, j, k :

Refer to transverse/depth-wise/axial direction

HS :

Refers to Helmholtz–Smoluchowski

m :

Refers to average of parameter

\(P,\textit{NB}\) :

Refer to central node and neighbor grid point

PD :

Refers to pressure driven

PTT :

Refers to Phan-Thien–Tanner model

sPTT :

Refers to simplified PTT model

\(\psi ,u\) :

Refer to potential and velocity

g :

Relevant to the initial guess

\(\hbox {T}\) :

Transpose of the matrix

−:

Relevant to dimensionless variable/domain

\({\wedge }\) :

Relevant to transformed variable/domain

\(\square \) :

Gordon–Schowalter convected derivative

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Authors and Affiliations

  1. Center of Excellence in Energy Conversion, School of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9567, Tehran, Iran

    Milad Reshadi & Mohammad Hassan Saidi

  2. Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran

    Abbas Ebrahimi

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  1. Milad Reshadi
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Correspondence to Mohammad Hassan Saidi.

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Reshadi, M., Saidi, M.H. & Ebrahimi, A. Pure axial flow of viscoelastic fluids in rectangular microchannels under combined effects of electro-osmosis and hydrodynamics. Theor. Comput. Fluid Dyn. 32, 1–21 (2018). https://doi.org/10.1007/s00162-017-0428-y

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