What are the differences between Navier and Stokes versions of equation?

Question

What are the main differences between Navier(1822) and Stokes(1845) versions of equation? If I understood correctly, original equations formulated Euler, what did they invent that they deserved to have the equation named after them?

Answer 1

It took some years to get the current form of the Navier$-$Stokes equations, as it usually happens in Science: a well-established model is usually a result of some years of development and improvements.

Development of the governing equations of viscous fluids

Navier and Stokes where 2 of the main contributors to the introduction of the force due to viscosity in the equations governing the motion of fluids.

Navier first introduced in friction in fluid equations in 1822; many other scientists/engineers working on continuum mechanics (Cauchy, Poisson, de Saint-Venant) derived the same equations, whose names are not referred to in these equations; Stokes worked on this topic starting from 1845, 23 years later. Stokes focused on the comparison of the predictions of these equations with experimental results, especially on fluids at low Reynolds number (Stokes equations are a limit case of the Navier$-$Stokes equations with the viscous term dominating convection), like pipe flows and the flow around a sphere, for which he derived some "famous" results, like the expression of the drag of a sphere at very low Reynolds number $\mathbf{F} = 6 \pi \mu R \mathbf{U}$.

References

You can find a summary of the historical development of Navier$-$Stokes equations in the first paragraphs of "200 years of the Navier$-$Stoked equation" publicly available on ArXiv and references therein, as an example O. Darrigol, Between Hydrodynamics and Elasticity Theory: The First Five Births of the Navier-Stokes Equation. Arch Hist Exact Sc. 56, 95–150 (2002)

Navier published in 1822 Sur les lois des mouvements des fluides, en ayant égard à l’adhésion des molecules, Annales de Chimie et de Physique, Paris, XIX, 244-260 (1822), (link), where the contribution of molecular forces of neighboring regions of the fluid appears in the equation as the Laplacian of the velocity field.

Further development in this field were needed to:

• find a complete set of equations and the proper boundary conditions (remember that a differential problem is determined by both differential equations and initial/boundary conditions)
• establish a general relation between stress and velocity field for isotropic fluids

Navier and the development of continuum mechanics

Navier was one of the main contributors in the development of the governing partial differential equations of continuum mechanics that nowadays, after development and contributions from other scientists, involve similar relations between stress and strain in isotropic elastic solids and stress and strain velocity in isotropic fluids, namely:

• linear elastic media with small displacement and strain: Navier$-$Cauchy equations are the formulation of the elastic problem in terms of the displacement field, for a isotropic medium with linear isotropic relation between the stress tensor and the strain tensor $$\mathbb{T} = 2\mu \varepsilon \hspace{-3pt} \varepsilon + \lambda (\nabla \cdot \mathbf{d}) \mathbb{I} \ ,$$

  being $\mathbb{T}$ the stress tensor, $\varepsilon \hspace{-3pt} \varepsilon = \frac{1}{2} \left[ \nabla \mathbf{d} + \nabla^T \mathbf{d} \right]$ the strain tensor, $\mathbf{d}$ the displacement field, so that $\nabla \cdot \mathbf{d} = \text{Tr}(\varepsilon \hspace{-3pt} \varepsilon)$, and $\mu, \lambda$ the Lamé coefficients.

• Newtonian fluids: Navier$-$Stokes equations are the governing equations of a viscous fluid with a linear isotropic relation between viscous stress and strain-rate tensor,

  $$\mathbb{S} = 2 \mu \mathbb{D} + \lambda (\nabla \cdot \mathbf{u}) \mathbb{I}$$

  being $\mathbb{S}$ the viscosity stress tensor, $\mathbb{D} = \frac{1}{2} \left[ \nabla \mathbf{u} + \nabla^T \mathbf{u} \right]$ the strain-rate tensor, $\mathbf{u}$ the velocity field, so that $\nabla \cdot \mathbf{u} = \text{Tr}(\mathbb{D})$, and $\mu, \lambda$ the viscosity coefficients.