Myron B. Allen, III - The Mathematics of Fluid Flow Through Porous Media

In problems for which inertial terms are negligible, for example when the fluid is at rest, this equation reduces to

(2.12)

Exercise 2.5 Integrate the third component of Eq . (2.12) to obtain the hydrostatic equation ,

(2.13)

Thus pressure increases linearly with depth in an ideal fluid at rest .

Equation (2.13)closely models the pressure of Earth's atmosphere. At sea level, the pressure of the atmosphere fluctuates around 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 fluidis a material for which

(2.14)

Here, stands for the stretching tensor, defined as

With respect to an orthonormal basis, the th entry of the matrix representation of is , and denotes the transposeof , whose th entry is .

The coefficient appearing in Eq. (2.14)is the dynamic viscosity, a nonnegative function of space and time having dimension . 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, , which is approximately the viscosity of water at a temperature of C and a pressure of 1 atm. For comparison, the viscosity of air at these conditions is cP.

Exercise 2.6 Find the correct pronunciation of “Poiseuille.”

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)

Here , is the kinematic viscosity, having dimension , and has the following representation with respect to an orthonormal basis :

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 , 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, . Using the radius and the far‐field fluid speed as scaling parameters, define the following dimensionless variables: