The Navier-Stokes Equation

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The Navier-Stokes Equation

• Charity Russell

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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).

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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.

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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 Eulers equations to account for
  viscosity.
• Stokes rediscovered Naviers equations, with
  proper mathematical reasoning.

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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.

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The Equations

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

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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.

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Problems with the Utilization of the
Navier-Stokes Equation

• To date, no one has successfully determined a
  formula that solves the equationor 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.