Microfluidics for Biologists Fundamentals and Applications

For circular channelsthe relation is expressed as

R¼ðP 1 P 2 Þ=Q¼ΔP=Q¼ 8 ηl=Πr^4 ð 1 : 51 Þ

The (1.51) can be written as

ΔP¼RQ¼ 8 ηlQ=Πr^4 ¼ 32 ηlU=r^2 ð 1 : 52 Þ

where,ΔP is pressure drop, R is the fluidic resistance, and Q is the volumetric flow
rate through a cross-section area‘A’(Q¼U:A)
For rectangular channelsthe relation in (1.51) is modified to

Rhd¼Cncηl=A^2 ð 1 : 53 Þ

where, Cncis numerical coefficient and is given as

Cnc¼ 8 ðÞAþ 12 =A ð 1 : 54 Þ

where,Ais aspect ratio¼height of the channel (h)/width of the channel (w)
Replacing (1.53) and (1.54)in(1.52)

ΔP¼RhdQ¼CncηlQ=A^2 ¼ 8 ðÞAþ 12 =A

hi

ηlQ=A^2 ð 1 : 55 Þ

Table 1.6 Volumetric flow rates for common geometries

Shape of cross-section Volumetric flow rate (Q) Fabrication approach

Cylindrical Πr^4 ΔP/8ηL Isotropic wet etching, Ball-end milling

Rectangular ΔPw=½ 8 ðAþ 1 Þ^2 =AŠη Photolithogrgaphy

Triangular ΔP(3a^4 )1/2/320ηl Anisotropic wet etching

Pressure drop along L = P 2 − P 1

L

P 2

r

R

Flow Resistance

P 1

Fig. 1.8 Illustration of the

pressure drop‘ΔP’along

the channel length‘L’due

to the flow resistance

offered by the fluid to its

motion from point 1 to point

2

22 C.K. Dixit

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