CONTINUUM MECHANICS for ENGINEERS

posed in this section are relevant only for laminar flows. Turbulent flows require
additional considerations.
7.3 Specialized Fluids
Although the study of viscous fluids in the context of the equations presented
in Section 7.2 occupies a major role in fluid mechanics, there is also a number
of specialized situations resulting from simplifying assumptions that provide
us with problems of practical interest. Here, we list some of the assumptions
that are commonly made and consider briefly their meaning with respect to
specific fluids.
(a) Barotropic fluids — If the equation of state happens to be independent
of temperature as expressed by Eq 7.2-9, the changes of state are
termed barotropic, and fluids which obey these conditions are called
barotropic fluids. In particular, we may cite both isothermal changes
(in which the temperature is constant) and adiabatic changes (for
which no heat enters or leaves the fluid) as barotropic changes.
(b) Incompressible fluids — If the density of a fluid is constant, the equation
of state becomes
ρ = constant
(7.3-1)
which describes incompressibility. This implies ˙ρ = 0 and, by the continuity
equation, vi,i = 0 for incompressible flows. Physically, incompressibility
means that the elements of a fluid undergo no change in
density (or volume) when subjected to a change in pressure. For
incompressible flows, the Navier-Stokes equations become
˙ i i , i i, jj
*
ρv = ρ b − p + µ v
(7.3-2)
due to the vi,i = 0 condition. Water and oil, among others, are generally
assumed to be incompressible, whereas most gases are highly
compressible.
(c) Inviscid (frictionless) fluids — A fluid that cannot sustain shear stresses
even when in motion is called an inviscid, or sometimes a perfect fluid.
Clearly, if the coefficients λ * and µ * in Eq 7.1-10 are equal to zero, that
equation describes a perfect fluid and the Navier-Stokes equations
reduce to
˙ ,
ρvi = ρbi
−pi (7.3-3)