Microfluidics for Biologists Fundamentals and Applications

AinAout¼Asystem ð 1 : 21 Þ

Expression (1.21) stands true only if the extensive properties participating in
conservation laws are considered, such as mass, volume, length etc. but not for
intensive properties, which include pressure or temperature. The extensive proper-
ties are those which changes with the change in the amount of fluid; while intensive
properties are those that do not change when an amount of fluid changes in the
system.

Equations of Motion

A fluid in motion experience following forces

• Fg—gravity force


• Fp—pressure force


• Fv—Force due to viscosity


• Ft—Force due to turbulence


• Fc—Force due to compressibility

Therefore, according to the Newton’s 2nd law, the net force (Fx) on a fluid with
mass‘m’and acceleration‘a’in x-direction can be expressed as

Fx ¼FgþFpþFvþFtþFc ð 1 : 22 Þ

Recalling the previous assumptions we discussed in fluid compressibility, the
liquids with limited compressibility are considered incompressible and Fc becomes
negligible.
Rewriting (1.22) with Fc¼ 0

Fx ¼Fg þFpþFvþFt ð 1 : 23 Þ

Equation (1.23) is calledReynold’s equation of motion.
If the system is not turbulent then Ft is negligible and the resultant eq is known as
Navier–Stokes equation of motionand is expressed as

Fx ¼FgþFpþFvþFc ð 1 : 24 Þ

And can be written as,

PðÞ¼∂U=∂tþU:¼ΔU Δ:σ ¼ΔPþηΔ^2 Uþf ð 1 : 25 Þ

For real fluids with negligible viscosity, Fv is 0, the eq. is known asEuler’s
equation of motionand is expressed as

1 Fundamentals of Fluidics 17