Microfluidics for Biologists Fundamentals and Applications

Δp¼ðÞρ
ρvghρgh ð 2 : 2 Þ

From (2.1) and (2.2):

)h¼

2 γLVcosθ

rρg

ð 2 : 3 Þ

WhereγLV¼interfacial surface tension of the liquid–vapor,γSV¼interfacial
surface tension between solid-vapor,γSL¼interfacial surface tension between
solid–liquid,ρ&ρvare the densities of liquid and the vapor andθis the contact
angle.
The transport of fluid on paper channel in 1-D can be estimated using the Lucas–
Washburn equation [ 76 ]:

L¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

γLVDtcosθ

4 μ

s

ð 2 : 4 Þ

Where L is the distance moved by the fluid front, t is time, D is the average pore
diameter,γLVis the surface tension, andμis viscosity.
The above formula is useful in designing of paper channel. The length of paper
channel is directly proportional to the square root of time. The volumetric flow rate
(Q) of the fluid in a paper channel of constant width can be described by the Darcy’s
law:

Q¼

kA

μL

ΔP ð 2 : 5 Þ

Fig. 2.23 Capillary rise in
the tube

2 Microfluidics Overview 61