Microfluidic two-phase interactions under variable liquid to cross-flow gas momentum flux ratios

  • Abbas Ghasemi
  • Xianguo LiEmail author
Research Paper


Volume of fluid (VOF) and large eddy simulations (LES) are coupled to investigate the microfluidic two-phase interactions during the liquid emergence into the cross-flow gas in a super-hydrophobic micro-channel. Spatio-temporal evolution of the gas/liquid interface is presented for nine different cases of the liquid to gas momentum flux ratios, gas/liquid Reynolds numbers and gas/liquid Weber numbers. With increased momentum of the gas flow, the liquid topology is found deflected towards the downstream. Under variable gas resistance effects, the liquid flow emerging through the square pore may or may not develop a circular cross-section governed by the axis-switching phenomenon. At strong gas inertia, vortex shedding in the downstream of the liquid generates vorticular ligaments in the wake region. Shearing effects on the liquid surface are increased at higher liquid injection velocities and/or gas densities. Depending on the competing effects of the viscous diffusion versus gas/liquid inertia, different combinations of the interactions among the three building blocks of the fluid flow problems (boundary layer, shear layer and wake) are described in microfluidics scales. The complexity of the liquid topology is found correlated with the occurrence of the phenomena such as the Kelvin–Helmholtz (KH) instability, the horseshoe vortex system, stationary/shedding vortices in the wake of the liquid topology as well as their interaction with the micro-channel wall boundary layers.


Microfluidics Droplets Vorticity 

List of symbols

\(Bo={\varDelta }\rho g D^{2}/\sigma\)

Eotvos or Bond (Bo) number

\(\mathrm{CFL}_{u}= U_\mathrm{max} \delta t/ {\varDelta }\)

CFL number


Width of the square pore


Pressure force


Gravitational acceleration

\(Kn = \lambda /w\)

Knudsen number


Length of the micro-channel



\(q= \rho _\mathrm{l} {U_\mathrm{l}}^{2}/\rho _\mathrm{g} {U_\mathrm{g}}^{2}\)

Liquid to gas momentum flux ratio

\(Re_\mathrm{g}= DU_\mathrm{g}/\nu _\mathrm{g}\)

Gas Reynolds number

\(Re_\mathrm{l}= DU_\mathrm{l}/\nu _\mathrm{l}\)

Liquid Reynolds number

\(T^{*} = U_\mathrm{l}t/D\)

Dimensionless time


Dimensionless time


Gas temperature


Liquid temperature

\(\mathbf {u} \,(u, v, w)\)

Velocity vector


Gas inlet velocity


Liquid inlet velocity


Width of the square micro-channel

\(We_\mathrm{g}= \rho _\mathrm{g} {U_\mathrm{g}}^{2} D/\sigma\)

Gas Weber number

\(We_\mathrm{l}= \rho _\mathrm{l} {U_\mathrm{l}}^{2} D/\sigma\)

Liquid Weber number


Void fraction

\({\varDelta }\)

LES filter width

\(\delta t\)

Time step


Curvature of the gas/liquid interface


Mean free path of the gas molecules


Dynamic viscosity

\(\mu _\mathrm{g}\)

Gas phase dynamic viscosity

\(\mu _\mathrm{l}\)

Liquid phase dynamic viscosity

\(\nu _\mathrm{g}\)

Gas phase kinematic viscosity

\(\nu _\mathrm{l}\)

Liquid phase kinematic viscosity



\(\rho _\mathrm{g}\)

Gas phase density

\(\rho _\mathrm{l}\)

Liquid phase density

\({\varOmega }^{*} = {\varOmega }D/U_\mathrm{g}\)

Dimensionless vorticity magnitude

\({\omega _{y}}^{*} = \omega _{y} D/U_\mathrm{g}\)

Dimensionless y-vorticity

\({\omega _{z}}^{*} = \omega _{z} D/U_\mathrm{g}\)

Dimensionless z-vorticity


Surface tension



This research is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) via a Discovery Grant, and Ontario Centres of Excellence (OCE) TalentEdge Internship Program (TIP) under the Project Numbers 25398 and 25657.

Supplementary material

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical and Mechatronics EngineeringUniversity of WaterlooWaterlooCanada

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