Electrokinetic and aspect ratio effects on secondary flow of viscoelastic fluids in rectangular microchannels

Abstract

The secondary flow of PTT fluids in rectangular cross-sectional plane of microchannels under combined effects of electroosmotic and pressure driving forces is the subject of the present study. Employing second-order central finite difference method in a very refined grid network, we investigate the effect of electrokinetic and geometric parameters on the pattern, strength and the average of the secondary flow. In this regard, we try to illustrate the deformations of recirculating vortices due to change in the dimensionless Debye–Hückel and zeta potential parameters as well as channel aspect ratio. We demonstrate that, in the presence of thick electric double layers, significant alteration occurs in the secondary flow pattern by transition from favorable to adverse pressure gradients. Moreover, it is found that for polymer-electrolyte solutions with large Debye lengths, the secondary flow pattern and the shape of vortices are generally dependent upon the width-to-height ratio of the channel cross section. Also, the inspections of strength and average of secondary flow reveal that the sensitivity of these quantities with respect to the electrokinetic, geometric and rheological parameters increases by increasing the absolute value of velocity scale ratio. In this regard, utilizing the curve fitting of the results, several empirical expressions are presented for the strength and average of the secondary flow under various parametric conditions. The obtained relations with the other predictions for secondary flow are of high practical importance when dealing with the design of microfluidic devices that manipulate viscoelastic fluids.

Appendix

• The coefficients $C_{i}$ s in Eq. (30) are given as:

  $$C_{1} = - \text{Re} \frac{{\partial \bar{\varOmega }}}{{\partial \bar{y}}},\quad C_{2} = \text{Re} \frac{{\partial \bar{\varOmega }}}{{\partial \bar{x}}}$$

  (33)
  $$\begin{aligned} C_{3} & = \left[ {1 - \frac{{{\text{Wi}}_{\kappa } }}{2K}\left( {\xi \bar{\tau }_{yy} + \left( {\xi - 2} \right)\bar{\tau }_{xx} } \right)} \right]\left[ {\frac{{\partial^{2} \varLambda_{3} }}{{\partial \bar{x}^{2} }} - \frac{{\partial^{2} \varLambda_{3} }}{{\partial \bar{y}^{2} }}} \right] + \frac{{{\text{Wi}}_{\kappa } }}{K}\left\{ {\frac{{\partial^{2} \bar{\tau }_{yx} }}{{\partial \bar{x}\partial \bar{y}}}(\xi \varLambda_{1} + (2 - \xi )\varLambda_{2} )} \right. \\ & \quad + \bar{\tau }_{xy} \left[ {\xi \frac{{\partial^{2} \varLambda_{1} }}{{\partial \bar{x}\partial \bar{y}}} + \left( {2 - \xi } \right)\frac{{\partial^{2} \varLambda_{2} }}{{\partial \bar{x}\partial \bar{y}}}} \right] + \frac{{\varLambda_{3} }}{2}\left[ {\xi \frac{{\partial^{2} \bar{\tau }_{yy} }}{{\partial \bar{y}^{2} }} + \left( {2 - \xi } \right)\frac{{\partial^{2} \bar{\tau }_{xx} }}{{\partial \bar{x}^{2} }}} \right]\left. { - \frac{{\varLambda_{3} }}{2}\left[ {\xi \frac{{\partial^{2} \bar{\tau }_{yy} }}{{\partial \bar{x}^{2} }} + \left( {2 - \xi } \right)\frac{{\partial^{2} \bar{\tau }_{xx} }}{{\partial \bar{y}^{2} }}} \right]} \right\} \\ \end{aligned}$$

  (34)
  $$\begin{aligned} C_{4} & = \left[ {1 - \frac{{{\text{Wi}}_{\kappa } }}{2K}\left( {\xi \bar{\tau }_{xx} + (\xi - 2)\bar{\tau }_{yy} } \right)} \right]\left[ {\frac{{\partial^{2} \varLambda_{3} }}{{\partial \bar{y}^{2} }} - \frac{{\partial^{2} \varLambda_{3} }}{{\partial \bar{x}^{2} }}} \right] + \frac{{{\text{Wi}}_{\kappa } }}{K}\left\{ {\frac{{\partial^{2} \bar{\tau }_{yx} }}{{\partial \bar{x}\partial \bar{y}}}\left( {\xi \varLambda_{2} + (2 - \xi) \varLambda_{1} } \right)} \right. \\ & \quad + \bar{\tau }_{xy} \left[ {\xi \frac{{\partial^{2} \varLambda_{2} }}{{\partial \bar{x}\partial \bar{y}}} + (2 - \xi )\frac{{\partial^{2} \varLambda_{1} }}{{\partial \bar{x}\partial \bar{y}}}} \right] + \frac{{\varLambda_{3} }}{2}\left[ {\xi \frac{{\partial^{2} \bar{\tau }_{xx} }}{{\partial \bar{x}^{2} }} + (2 - \xi )\frac{{\partial^{2} \bar{\tau }_{yy} }}{{\partial \bar{y}^{2} }}} \right]\left. { - \frac{{\varLambda_{3} }}{2}\left[ {\xi \frac{{\partial^{2} \bar{\tau }_{xx} }}{{\partial \bar{y}^{2} }} + (2 - \xi )\frac{{\partial^{2} \bar{\tau }_{yy} }}{{\partial \bar{x}^{2} }}} \right]} \right\} \\ \end{aligned}$$

  (35)
  $$C_{5} = 2\frac{{\partial^{2} \varLambda_{1} }}{{\partial \bar{x}\partial \bar{y}}} + 2\frac{{\partial^{2} \varLambda_{2} }}{{\partial \bar{x}\partial \bar{y}}}$$

  (36)
  $$C_{6} = \left[ {1 - \frac{{{\text{Wi}}_{\kappa } }}{2K}(\xi \bar{\tau }_{yy} + (\xi - 2)\bar{\tau }_{xx} )} \right]\varLambda_{3}$$

  (37)
  $$C_{7} = \left[ {1 - \frac{{{\text{Wi}}_{\kappa } }}{2K}\left( {\xi \bar{\tau }_{xx} + (\xi - 2)\bar{\tau }_{yy} } \right)} \right]\varLambda_{3}$$

  (38)
  $$C_{8} = 2(\varLambda_{1} + \varLambda_{2} ) - 2\left[ {1 - \frac{{{\text{Wi}}_{\kappa } }}{2K}\left( {\bar{\tau }_{xx} + \bar{\tau }_{yy} } \right)(\xi - 1)} \right]\varLambda_{3}$$

  (39)
  $$C_{9} = \frac{{{\text{Wi}}_{\kappa } }}{K}\bar{\tau }_{xy} \left( {\xi \varLambda_{1} + (2 - \xi )\varLambda_{2} } \right)$$

  (40)
  $$C_{10} = \frac{{{\text{Wi}}_{\kappa } }}{K}\bar{\tau }_{xy} \left( {\xi \varLambda_{2} + (2 - \xi )\varLambda_{1} } \right)$$

  (41)
  $$C_{11} = \frac{{\partial^{2} I_{1} }}{{\partial \bar{x}\partial \bar{y}}} - \frac{{\partial^{2} I_{2} }}{{\partial \bar{x}\partial \bar{y}}} - \frac{{\partial^{2} I_{3} }}{{\partial \bar{y}^{2} }} + \frac{{\partial^{2} I_{3} }}{{\partial \bar{x}^{2} }}$$

  (42)

• The coefficients $F_{i}$ s in Eq. (31) are given as:

  $$\begin{aligned} F_{1} & = - {\rm{Re}}\frac{{\partial \bar{\varphi }}}{{\partial \bar{y}}} + \frac{{{\text{Wi}}_{\kappa } }}{2K}\left[ {\varLambda_{4} \left( {\frac{{\partial \bar{\tau }_{xx} }}{{\partial \bar{x}}}(2 - \xi ) - \frac{{\partial \bar{\tau }_{zz} }}{{\partial \bar{x}}}\xi } \right) + \frac{{\partial \varLambda_{5} }}{{\partial \bar{y}}}\bar{\tau }_{xy} (2 - \xi ) + \varLambda_{5} \frac{{\partial \bar{\tau }_{xy} }}{{\partial \bar{y}}}(2 - \xi )} \right] \\ & \quad + \frac{{\partial \varLambda_{4} }}{{\partial \bar{x}}}\left[ {1 - \frac{{{\text{Wi}}_{\kappa } }}{2K}\left( {\bar{\tau }_{xx} (\xi - 2) + \bar{\tau }_{zz} \xi } \right)} \right] \\ \end{aligned}$$

  (43)
  $$\begin{aligned} F_{2} & = {\rm{Re}}\frac{{\partial \bar{\varphi }}}{{\partial \bar{x}}} + \frac{{{\text{Wi}}_{\kappa } }}{2K}\left[ {\varLambda_{5} \left( {\frac{{\partial \bar{\tau }_{yy} }}{{\partial \bar{y}}}(2 - \xi ) - \frac{{\partial \bar{\tau }_{zz} }}{{\partial \bar{y}}}\xi } \right) + \frac{{\partial \varLambda_{4} }}{{\partial \bar{x}}}\bar{\tau }_{xy} (2 - \xi ) + \varLambda_{4} \frac{{\partial \bar{\tau }_{xy} }}{{\partial \bar{x}}}(2 - \xi )} \right] \\ & \quad + \frac{{\partial \varLambda_{5} }}{{\partial \bar{y}}}\left[ {1 - \frac{{{\text{Wi}}_{\kappa } }}{2K}\left( {\bar{\tau }_{yy} (\xi - 2) + \bar{\tau }_{zz} \xi } \right)} \right] \\ \end{aligned}$$

  (44)
  $$F_{3} = \varLambda_{4} \left[ {1 - \frac{{{\text{Wi}}_{\kappa } }}{2K}\left( {\bar{\tau }_{xx} (\xi - 2) + \bar{\tau }_{zz} \xi } \right)} \right]$$

  (45)
  $$F_{4} = \varLambda_{5} \left[ {1 - \frac{{{\text{Wi}}_{\kappa } }}{2K}\left( {\bar{\tau }_{yy} (\xi - 2) + \bar{\tau }_{zz} \xi } \right)} \right]$$

  (46)
  $$F_{5} = \frac{{{\text{Wi}}_{\kappa } }}{2K}\bar{\tau }_{xy} (2 - \xi )(\varLambda_{4} + \varLambda_{5} )$$

  (47)
  $$F_{6} = - \varGamma + \frac{{\partial I_{4} }}{{\partial \bar{x}}} + \frac{{\partial I_{5} }}{{\partial \bar{y}}} + \frac{{K^{2} }}{{\bar{\psi }_{0} }}\sinh \bar{\psi }$$

  (48)