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A study on mixed electro-osmotic/pressure-driven microchannel flows of a generalised Phan-Thien–Tanner fluid

  • Ribau, A. M.1
  • Ferrás, L. L.2
  • Morgado, M. L.3
  • Rebelo, M.4
  • Alves, M. A.1
  • Pinho, F. T.1
  • Afonso, A. M.1
  • 1 Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, s/n, Porto, 4200-465, Portugal , Porto (Portugal)
  • 2 Universidade do Minho, Campus de Azurém, Guimarães, 4800-058, Portugal , Guimarães (Portugal)
  • 3 University of Trás-os-Montes e Alto Douro, UTAD, Vila Real, 5001-801, Portugal , Vila Real (Portugal)
  • 4 Universidade NOVA de Lisboa, Quinta da Torre, Caparica, 2829-516, Portugal , Quinta da Torre (Portugal)
Published Article
Journal of Engineering Mathematics
Springer Netherlands
Publication Date
Mar 03, 2021
DOI: 10.1007/s10665-020-10071-6
Springer Nature


This work presents new semi-analytical solutions for the combined fully developed electro-osmotic pressure-driven flow in microchannels of viscoelastic fluids, described by the generalised Phan-Thien–Tanner model (gPTT) recently proposed by Ferrás et al. (Journal of Non-Newtonian Fluid Mechanics, 269:88–99, 2019). This generalised version of the PTT model presents a new function for the trace of the stress tensor—the Mittag–Leffler function—where one or two new fitting constants are considered in order to obtain additional fitting flexibility. The semi-analytical solution is obtained under sufficiently weak electric potential that allows the Debye–Hückel approximation for the electrokinetic fields and for thin electric double layers. Based on the solution, the effects of the various relevant dimensionless numbers are assessed and discussed, such as the influence of εWi2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon Wi^2$$\end{document}, of the parameters α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} and β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} of the gPTT model, and also of κ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{\kappa }}$$\end{document}, the dimensionless Debye–Hückel parameter. We conclude that the new model characteristics enhance the effects of both εWi2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon Wi^2$$\end{document} and κ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{\kappa }}$$\end{document} on the velocity distribution across the microchannels. The effects of a high zeta potential and of the finite size of ions are also studied numerically.

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