In the mid-19^{th} century, the theory of porous media were unknown, it was a long vacuum since the development of volume fractions by Woltman in 1794. However, the fundamental discoveries were found by Delesse, Fick and Darcy. Delesse develop the concept of surface fraction in 1848, Fick studied the problem of diffusion through membrane in 1855, and then Darcy in 1856 investigated the permeability of water running through sand as a basic study in multiphase continuum [1].

The Darcy equation actually was derived from the Navier-Stokes equation by assuming stationary, creeping, incompressible flow through porous medium. Since the Darcy’s equation becomes invalid for liquid for high velocity, or gases for very low or very high velocities, the corrections was build by Brinkmann. Brinkman’s filtration equation is usually used to describe the low-Reynolds-number flow in porous media in situations where velocity gradients are non-negligible [2].

References:

- R. de Boer, Theory of porous media: Highlights in the historical development and current state, Springer-Verlag, Berlin-Heidelberg, 2000.
- F. J. Valdes-Parada, J. A. Ochoa-Tapia, and J. Alavarez-Ramirez, On the effective viscosity for the Darcy-Brinkmann equation,
*Physica A*, 2007, **385**, 69-79.

The viscous fluids theory is marked by the Navier-Stokes equations, that’s named after Claude-Louis Navier and George Gabriel Stokes, applying Newton’s idea written in his *Philosophiae naturalis principia mathematica* in which describing the motion of mass point and Euler’s equations in his *Mechanica *in which formulated the axioms of continuum mechanics, that is: the balance of mass, the balance of momentum, the balance of moment of momentum, and the cut principle. The historical development of Navier-Stokes equations was based on the Newton’s hypothesis that notices that the motion of fluid past other bodies is held back by friction. Navier introduces both types of friction into the equations of motions, for the material points inside as well as on the boundary in his *Memoir sur les lois du movement des fluids*. This work was continued by Poisson, De Saint-Venant, and finally by Stokes [1].

Stokes briefly explained his theory of fluid motion in various initials and boundary conditions in *Transactions of the Cambridge Philosophical Society* and* Cambridge Mathematical Journal*, and describe in *On the steady motion of incompressible fluids * in 1842. He also wrote about flow in cylindrical coordinate in *On the motion of a piston and of the air in a cylinder *in 1843 [2]. The development of this equation was built more than 150 years, however a fundamental problem is to decide whether such smooth, physically reasonable solutions exist for the Navier-Stokes equations. It is make the Navier-Stokes equations is one of seven unsolved mathematical problems in this millennium [3].

References:

- R. de Boer, Theory of porous media: Highlights in the historical development and current state, Springer-Verlag, Berlin-Heidelberg, 2000.
- G. G. Stokes, Mathematical and physical papers, Cambridge University Press, Cambridge, 1880.
- K. J. Devlin, The millennium problems: The seven greatest unsolved mathematical puzzles of our time, Basic Books, 2002.