alt="1.01325 times 1 0 Superscript 5"/> Pa, which is the definition of a common unit of measurement, 1 atmosphere, abbreviated as 1 atm.
An extension of the ideal fluid stress provides a more realistic constitutive relationship for many fluids. An incompressible Newtonian fluid is a material for which
Here,
With respect to an orthonormal basis, the
The coefficient
Exercise 2.6 Find the correct pronunciation of “Poiseuille.”
2.3.3 The Navier–Stokes Equation
Exercise 2.7 Substitute the constitutive relationship (2.14) into the momentum balance and assume that gravity is negligible (for example, in a shallow horizontal flow) to derive the Navier–Stokes equation:
Here,
Sir George Gabriel Stokes was an Irish‐born Cambridge professor who made extraordinary contributions to mathematical physics. Claude‐Louis Navier was a French mechanical engineer and professor of mathematics in the early nineteenth century.
Exercise 2.8 Find the correct pronunciation of “Navier.”
Owing largely to mathematical difficulties associated with the inertial terms, the Navier–Stokes equation remains a source of some of the most refractory unsolved problems in mathematics. Proving the existence and smoothness of solutions under general conditions remains one of six unsolved Millennial Prize Problems identified in 2000 by the Clay Institute for Mathematics [79].
To gauge the importance of inertial effects in specific problems, it is useful to cast Eq. (2.15) in terms of dimensionless variables—that is, variables having physical dimension 1. This technique filters out subjective effects associated with the analyst's choice of measurement units, mentioned in Section 1.3.
For concreteness, consider the flow of an incompressible Newtonian fluid in an infinite spatial domain surrounding a solid sphere having radius
By the chain
