PROOF COPY [TRIB-08-1155] 017903JTQ
PROOF COPY [TRIB-08-1155] 017903JTQ
Wang-Long Li
Associate Professor
Institute of Nanotechnology and Microsystems
Engineering,
Center for Micro/Nano Science and Technology,
National Cheng Kung University,
No. 1 University Road,
Tainan City 701, Taiwan
e-mail: wlli@mail.ncku.edu.tw
Derivation of Modified Reynolds
Equation: A Porous Media Model
With Effects of Electrokinetics
A lubrication theory that includes the effects of electrokinetics and surface microstructure
is developed. A porous layer attached to the impermeable substrate is used to model the
microstructure on a bearing surface. The Brinkman-extended Darcy equations and Stokes
equations are modified by considering the electrical body force and utilized to model the
flow in porous media and fluid film, respectively. The stress jump boundary conditions on
the porous media/fluid film interface and the effects of viscous shear and electric double
layer (EDL) are also considered when deriving the modified Reynolds equation. Under
the usual assumptions of lubrication and Debye–Hückel approximation for low surface
potential, the velocity distributions, the apparent viscosity, and the modified Reynolds
equation are then derived. The apparent viscosity is expressed explicitly as functions of
the Debye length, the electroviscosity, the charge density, the stress jump parameter, and
the porous parameters (permeability, porosity, and porous film thickness). The consider-
ations of EDL near the interface and the charge density of the flow in the porous media
increase the apparent viscosity. The existence of porous film also increases the apparent
viscosity as well. Both effects are important for flow within microspacing and lubrication
problems. The apparent viscosity and the performance of 1D slider bearings are analyzed
and discussed. The results show that the apparent viscosity and the load capacity in-
crease as the permeability decreases, the stress jump parameter decreases, the charge
density increases, the inverse Debye length decreases, or the porosity decreases.
DOI: 10.1115/1.3140610
Keywords: stress jump, electrokinetics, EDL, Brinkman-extended Darcy’s equation, lu-
brication theory
1 Introduction
Porous materials are widely used in chromatography, chemical
reactions, heat transfer, and analytical chemistry filtering analyte
due to their large surface area and small pore sizes. Based on the
pressure driven and electrokinetic flow 1,2, porous materials in
microfluidic devices also offer a number of advantages, such as
multiwalled microchannel microchannels with multiple open and
porous regions in microfluidic flow sensors 3,4, high-
performance liquid chromatography HPLC5, and microstruc-
tured enzyme reactors composed of many parallel rectangular
channels 6. The modeling of electro-osmotic and pressure driven
flows in porous microfluidic devices is also proposed 7.
In lubrication applications, porous media have been used on
many kinds of bearings 8 for a long time. There are many mod-
els in deriving the lubrication theory of porous bearings: 1 the
Darcy model 9,10, only viscous damping effects Darcy resis-
tance are considered for thick porous film; 2 the Beavers and
Joseph slip flow model 11, slip flow on the porous media/fluid
film interface; 3 the Brinkman-extended Darcy model 12–14,
the viscous shear effects and the viscous damping effects Darcy
resistance are considered for thin porous film; and 4 the
Brinkman-extended Darcy model with stress jump boundary con-
ditions 15–17. The microstructure of lubricating surfaces can be
modeled as thin porous layers attached to impermeable substrates
17–19. Due to the thin porous film, the velocity distribution
inside it is no more uniform. Darcy’s law, which is applicable to
thick porous media, is modified by adding the Brinkman term to
consider the viscous shear effects of thin porous films 13. More-
over, the stress jump boundary conditions on the porous media/
fluid film interface are also considered to account for the excess
viscous stress with stress jump parameters ranging from 1 to
1.47 16. However, the electrokinetic effects are not considered
for the porous lubrication modeling.
Electrokinetic phenomena occur when a charged surface is
brought into contact with an ionic liquid. The counterions in the
liquid are attracted by the charged wall firmly Stern layer.A
diffuse electric double layer EDL forms away from the wall as
the concentration of counterions decreases away from the wall.
Near the Debye screen length, the counterions screen the charged
solid surface. When the distance is beyond the Debye screen
length, the net charge is essentially zero. The effect of EDL as
well as Helmholtz–Smoluchowski’s equation is introduced to de-
rive the modified Reynolds equation in thin film lubrication theory
20–23. A potential field streaming potential is then generated
as the counterions move downstream because of the pressure gra-
dient. Therefore, the existence of electrokinetic force opposing the
primary fluid flow increases the flow resistance, and thus the elec-
troviscous effect greatens. The electroviscosity caused by the ex-
istence of EDL on the liquid/solid interface changes the rheology
of the thin lubricating film. The EDL retards liquid flow in the
microchannel and results in streaming potential. The concept of
apparent viscosity, which is defined as the combination of the
electroviscosity and the viscosity of bulk fluid regarded as an
equivalent or apparent viscosity, is proposed and measured ex-
perimentally in microchannels or capillaries 1,2. The electrovis-
cosity depends on the Debye length, the material of lubricating
surfaces, the ionic concentration, the dielectric constant, and the
conductivity of liquid. The primary nature of surface phenomena
in EDL makes it important as the size of channel or lubrication
Contributed by the Tribology Division of ASME for publication in the JOURNAL OF
TRIBOLOGY. Manuscript received October 12, 2008; final manuscript received April
26, 2009; published online xxxxx-xxxxx-xxxxx. Assoc. Editor: Michel Fillon.
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Journal of Tribology JULY 2009, Vol. 131 / 1-1 Copyright © 2009 by ASME
PROOF COPY [TRIB-08-1155] 017903JTQ