Compressibility Effect in Microscale Flows

Compressibility is a measure of the relative change in volume of a fluid as a response to pressure change, which is defined as 

For incompressible approximation, is very small, that is, the density chainges following a fluid particle is very small. Note that density may change from one particle to another without violating the incompressible approximation, that is, variable density/temperature/salinity flow is often treated as incompressible. 

Let us consider the state principle of thermodynamics, where one can express the density changes of a simple system in terms of the change in pressure and temperature as 

This expression indicates that the flow should be treated as compressible if pressure and temperature changes are sufficiently strong. Note the first term in the right-hand side can 

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