.
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
With respect to an orthonormal basis, the ![left-parenthesis i comma j right-parenthesis]()
th entry of the matrix representation of ![nabla bold v]()
is ![partial-differential v Subscript i slash partial-differential x Subscript j]()
, and ![left-parenthesis nabla bold v right-parenthesis Superscript down-tack]()
denotes the transpose of ![nabla bold v]()
, whose ![left-parenthesis i comma j right-parenthesis]()
th entry is ![partial-differential v Subscript j slash partial-differential x Subscript i]()
.
The coefficient ![mu]()
appearing in Eq. (2.14) is the dynamic viscosity, a nonnegative function of space and time having dimension ![normal upper M normal upper L Superscript negative 1 Baseline normal upper T Superscript negative 1]()
. A common unit for measuring dynamic viscosity is the centipoise, abbreviated cP and named after the French physicist Jean Léonard Marie Poiseuille. In SI units, ![1 cP equals 1 0 Superscript negative 3]()
![kg normal m Superscript negative 1 Baseline normal s Superscript negative 1]()
, which is approximately the viscosity of water at a temperature of ![20 Superscript ring]()
C and a pressure of 1 atm. For comparison, the viscosity of air at these conditions is ![1.516 times 1 0 Superscript negative 2]()
cP.
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:
(2.15)![StartFraction partial-differential bold v Over partial-differential t EndFraction plus left-parenthesis bold v dot nabla right-parenthesis bold v equals minus StartFraction 1 Over rho EndFraction nabla p plus nu nabla squared bold v period]()
Here, ![nu equals mu slash rho]()
is the kinematic viscosity, having dimension ![normal upper L squared normal upper T Superscript negative 1]()
, and ![nabla squared bold v]()
has the following representation with respect to an orthonormal basis:
![sigma-summation Underscript i equals 1 Overscript 3 Endscripts Start 3 By 1 Matrix 1st Row partial-differential squared v 1 slash partial-differential x Subscript i Superscript 2 Baseline 2nd Row partial-differential squared v 2 slash partial-differential x Subscript i Superscript 2 Baseline 3rd Row partial-differential squared v 3 slash partial-differential x Subscript i Superscript 2 Baseline EndMatrix period]()
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 ![upper R]()
, as drawn in Figure 2.9. We examine a simplified version of this flow, called the Stokes problem, in Section 2.4. Assume that, as distance from the sphere increases, ![bold v right-arrow v Subscript infinity Baseline bold e 1]()
. Using the radius ![upper R]()
and the
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