Microfluidics for Biologists Fundamentals and Applications

Volumetric flow rate can be expressed in terms of mass flow rate with the
relation

Q ¼Im=ρ ð 1 : 56 Þ

Replacing (1.56)in(1.55) gives

ΔP¼RhdIm=ρ¼CncηlIm=A^2 :ρ¼ðÞη=ρ: CnclIm=A^2



¼ν:CnclIm=A^2 ð 1 : 57 Þ

where, Imis mass flow rate,νis kinematic viscosity
For circular pipes, (1.57) can be written as

ΔP ¼RQ¼ 8 νl=Πr^4 ð 1 : 58 Þ

An extension to Hagen-Poiseuille law isDarcy–Weisbach equation
Darcy–Weisbach equationrelates head loss or pressure loss due to friction
along a given circular channel and is expressed as

Pressure loss form:

ΔP pressure lossðÞ¼fDlρV^2 =2D ð 1 : 59 Þ

Head lossform:
Replacing (1.6)in(1.59)

ρgΔh¼fDlρV^2 =2D ð 1 : 60 Þ

Δh head lossðÞ¼fDlV^2 =2gD ð 1 : 61 Þ

where,fDis Darcy friction factor from channel wall which is

fD¼ 64 =Re ð 1 : 62 Þ

Fanning equationrelates the ratio of local shear stress to the local fluid kinetic
energy and is expressed as

f ¼τ=Kinetic Energy¼ 2 τ=ρU^2 ¼ 16 =Re ð 1 : 63 Þ

where,
fis fanning friction factor,τis shear stress, Re is Reynolds number.

1 Fundamentals of Fluidics 23