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 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.
- 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.
aijbjkckl cannot and must not, be replaced by
ailblkckl – NO
aimbmkckl – YES
aimbmncnl – YES