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The evolvement of Navier-Stokes Equation: Newton to Prandtl

The Navier-Stokes equation evolved over centuries of what we see today. Following is the chronological list of how the Navier-Stokes equation evolved in time:

  • 17th century – 1687 – Newton (Mechanics, 3 laws of motion, Calculus(helped in the early concept of inviscid flow)
  • 1738 – Bernoulli’s Equation – Pressure differential is the implied acceleration.
  • 1740’s – Euler’s equation (closely resembles the N-s equation)
  • 1758 – D’Alembert ( Paradox, drag zero on a sphere)
  • 1822- Claude Louis Navier (Extra term in Euler’s equation)
  • 1828 – Cauchy
  • 1829 – Poisson
  • 1843  – Saint Venant
  • 1845 – George Stokes (mathematical rigorous of NS equation)
  • 1934 – Ludwig Prandtl (Most widely used form as we see today)

 

Tensor – Basic Notation

Tensor calculus is an organized expression, which contains sophisticated geometric insights.

  • Combines geometric and analytical perspectives.
  • Enables use of the co-ordinate system without the loss of geometric insight.
  • Provides a framework for establishing equations valid in all coordinate systems.
  • Algorithmic.
  • Provides a language that is concise and powerful.
  • Based on a handful of operations.

Summation convention in Tensor algebra plays a crucial role. The convention is such that any lower-case alphabetic subscript that appears exactly twice in any term of an expression is understood to be summed over all the values that a subscript in that position can take.

The subscripted quantities may appear in the numerator and/or denominator of a term in an expression.

Subscripts that are summed over are called dummy subscripts, and others are free subscripts.

While introducing a dummy subscript into an expression, care needs to be taken not to use which is already present, either as a free / dummy subscript.

Example

aijbjkckl   cannot and must not, be replaced by

aijbjjcjl   -NO

ailblkckl  – NO

aimbmkckl  – YES

aimbmncnl – YES