Microfluidics and Nanofluidics

, Volume 10, Issue 2, pp 287–300

When MHD-based microfluidics is equivalent to pressure-driven flow

Research Paper

DOI: 10.1007/s10404-010-0668-2

Cite this article as:
Qin, M. & Bau, H.H. Microfluid Nanofluid (2011) 10: 287. doi:10.1007/s10404-010-0668-2

Abstract

Magneto-hydrodynamics (MHD) provides a convenient, programmable means for propelling liquids and controlling fluid flow in microfluidic devices without a need for mechanical pumps and valves. When the magnetic field is uniform and the electric field in the electrolyte solution is confined to a plane that is perpendicular to the direction of the magnetic field, the Lorentz body force is irrotational and one can define a “Lorentz” potential. Since the MHD-induced flow field under these circumstances is identical to that of pressure-driven flow, one can utilize the large available body of knowledge about pressure-driven flows to predict MHD flows and infer MHD flow patterns. In this note, we prove the equivalence between MHD flows and pressure-driven flows under certain conditions other than flow in straight conduits with rectangular cross sections. We determine the velocity profile and the efficiency of MHD pumps, accounting for current transport in the electrolyte solutions. Then, we demonstrate how data available for pressure-driven flow can be utilized to study various MHD flows, in particular, in a conduit patterned with pillars such as may be useful for liquid chromatography and chemical reactors. In addition, we examine the effect of interior obstacles on the electric current flow in the conduit and show the existence of a particular pillar geometry that maximizes the current.

Keywords

Magneto-hydrodynamicsMicrofluidicsLiquid chromatographyLorentz forceNernst–Planck equations

List of symbols

b

Magnetic field (Tesla)

ci

Concentration of species i (mol/m3)

Cp

Specific heat (J/kg K)

Di

Diffusion coefficient (m2/s)

E

Electric field (V/m)

eff

Pumping efficiency

f∇B

Magnetophoretic body force (N/m3)

fE

Electrostatic body force (N/m3)

fL

Lorentz body force (N/m3)

H

Conduit height (m)

h

Convective heat transfer coefficient (W/m2 K)

Ha

Hartmann number

I

Electric current (A)

j

Electric current density (A/m2)

je

Exchange current density (A/m2)

j0

Current density scale (A/m2)

K

Lorentz force to viscous force ratio

k

Thermal conductivity (W/Km)

L

Conduit length (m)

m

Integration constant

Ni

Ion flux of species i (mol/m2 s)

p

Pressure (Pa)

Q

Flow rate (m3/s)

R

Gas constant (J/K mol)

RH

Hydraulic resistance (Pa s/m4)

Re

Reynolds number

Rem

Magnetic Reynolds number

W

Conduit width (m)

α

Charge transfer coefficient

υ

Magnetic diffusivity (m2/s)

μ

Fluid viscosity (Pa s)

λ

Drag coefficient

η

Overpotential (V)

Δpstall

Stall pressure (Pa)

T

Temperature (K)

u

Velocity field (m/s)

\( \bar{u} \)

Average fluid velocity (m/s)

zi

Valence

ϕ

Electric potential (V)

εs

Dielectric permittivity (F/m)

νi

Ion mobility (m2/V s)

σionic

Ionic conductivity (ohm−1 m−1)

ς0

Magnetic permeability (N/A2)

Ξ

Magnetic potential (N/m2)

χm

Molar magnetic susceptibility (m3/mol)

Vext

Externally applied electric potential (V)

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Mechanical Engineering and Applied MechanicsUniversity of PennsylvaniaPhiladelphiaUSA