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The Navier-Stokes Equation

The Navier-Stokes Equation. Charity Russell. Louis Marie Henri Navier.

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The Navier-Stokes Equation

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  1. The Navier-Stokes Equation Charity Russell

  2. Louis Marie Henri Navier Famous in his time for bridge building; also a government consultant on scientific and technological improvements could be made for the nation. Though primarily an engineer, Navier was also trained as a mathematician. He taught at the Ecole Des Ponts at Chaussées, while there considered mathematics of fluid flow and discovered the now famous equation (1921-1922).

  3. George Gabriel Stokes Was awarded a scholarship from Pembroke College (Cambridge) to carry out research after his graduation there. During this time, he looked to calculus to understand fluid flow and rediscovered the same equations Navier had two decades earlier.

  4. The Development of the Equation • Bernoulli adapted methods of calculus to analyze fluid motion when subjected to various forces. • Euler formulated a set of equations, which combined solutions describe precisely the motion of a viscosity-free fluid. • Navier amended Euler’s equations to account for viscosity. • Stokes rediscovered Navier’s equations, with proper mathematical reasoning.

  5. Methods of Derivation The Navier-Stokes equation of motion was derived by Claude-Louis-Marie Navier in 1827, and independently by Siméon-Denis Poisson in 1831. Their motivations of the stress tensor were based on what amounts to a molecular view of how stresses are exerted by one fluid particle against another. Later, Barré de Saint Venant (in 1843) and George Gabriel Stokes (in 1845) derived the equation starting with the linear stress rate-of-strain argument.

  6. The Equations • The equation models the motion of non-turbulent, with incompressible fluids following a much simplified version. • Basic motion described by the following:

  7. The Equations, Cont. • The Navier-Stokes equation, on the most basic level, is a combination of the fluid kinematics and constitutive relation into the fluid equation of motion. • Also, the normal “time-derivative” of the equation is the materials derivative which helps explain convection within the fluid system.

  8. Problems with the Utilization of the Navier-Stokes Equation • To date, no one has successfully determined a formula that solves the equation—or been able to determine if one should even necessarily exist. • The solution is considered to be one of the most desirable in mathematics and engineering today. It is one of the “Millennium Problems” from the Clay Mathematics Institute.

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