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

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

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

      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, sans-serif upper D stands for the stretching tensor, defined as

sans-serif upper D equals one half left-bracket nabla bold v plus left-parenthesis nabla bold v right-parenthesis Superscript down-tack Baseline right-bracket period

      With respect to an orthonormal basis, the left-parenthesis i comma j right-parenthesisth 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-parenthesisth entry is partial-differential v Subscript j slash partial-differential x Subscript i.

      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, 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].

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