Microfluidics for Biologists Fundamentals and Applications

ΔE¼constant ð 1 : 40 Þ

The simplest example of energy conservation is Bernoulli’s equation where fluid
flowing through a pipe having different radius as depicted in Fig.1.7.
Bernoulli’s equationfor per unit volume for a given area is written as

Pressure energyþkinetic energyþpotential energy¼constant ð 1 : 41 Þ

Pþ mU^2



= 2 þmgh¼Pþ ρU^2



= 2 þρgh¼constant ð 1 : 42 Þ

Now, from Fig.1.7, at two different points with different cross-section areas,

Equation (1.42) can be written as

P 1 þ ρU^21



= 2 þρgh 1 ¼P 2 þ ρU^22



= 2 þρgh 2 ð 1 : 43 Þ

Important Concepts

• Stokes law: Friction and Drag on spherical particles

Frictional force due to viscosity, which is also known asStokes dragis given by

Fd¼ 6 ΠηrU ð 1 : 44 Þ

where,
Fdis stokes drag,ηis dynamic viscosity, r is hydrodynamic radius of the
spherical particle, U is the flow velocity around the particle. 6Πηr is together is
calleddrag coefficientζ.

A 1 >A 2 , r 1 > r 2 , P 1 < P 2 , U 1 < U 2

Q (Volumetric flow rate) = U.A; Q 1 = Q 2

A 2 , r 2 , h 2 , P 2 ,U 2

A 1 , r 1 , h 1 , P 1 ,U 1

Q 1

Q 2

Fig. 1.7 According to
conservation law,
volumetric flow rate in the
pipe sections depicted as
red circlesmust remain
same. However, channel
diameters, cross-section
areas, and flow velocities of
the highlighted sections
different. Thus, to satisfy
Q 1 ¼Q 2 , flow velocity of
the smaller diameter>the
flow velocity of higher
diameter

20 C.K. Dixit

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