One of the great achievements of the nineteenth century was the discovery of the Navier-Stokes equation that describes the motion of ordinary fluids. Before its advent, the behavior of fluids could be catalogued but not predicted. Afterwards, predicting the motion of fluids became a matter of solving the equation with the proper boundary and initial conditions. This took hydrodynamics from being a collection of observations to a science with the power of prediction.
By ?ordinary fluid? we mean fluids that are described by a constant viscosity, i.e. Newtonian fluids. Fluids that do not possess this property, i.e. their viscosity depends on the forces acting on them, are non-Newtonian fluids. There is only one way a fluid may be Newtonian, but there are many ways for it to be non-Newtonian.
Equations governing the motion of non-Newtonian fluids, i.e., equations corresponding to the Navier-Stokes equations for Newtonian fluids have since long been constructed. However, these equations are much tougher than the Navier-Stokes equation to deal with as they become highly non-linear, so that there are many unanswered questions lingering on.
At close range porous media consist of connected pores through which fluid may flow. The fluid is governed by the same equations of motion as fluids in general, with the pore walls providing the boundary conditions. When dealing with the flow on the scale of kilometers, the ability to solve the flow equations on the pore scale, typically on the micron scale, is useless. Effective flow equations are then needed that treat the porous medium as a continuum. In the case of Newtonian fluids, this is done by combining the Darcy equation with equations ensuring mass conservation. When the fluid is non-Newtonian, it is only recently that the search of these equations has started. Our goal is to construct such equations.

Three laboratories, SFF PoreLab (NTNU/UiO), Laboratoire FAST (U. Paris-Sud, France) and Complex Systems Group (U. Fed. Ceara, Brazil) will collaborate to address a central outstanding problem in porous media.
Porous media are all around us. We find them in biology. Fluids need to be transported in orgranisms. In geology and/or geophysics, water moves in soils, e.g. replinishing aquifers. Frost heave is the result of water transport through clay. They at the heart of oil production and CO2 sequestration. Industrial filters and fluidized bed reactors belong to chemical engineering and constitute examples of porous media. Hydrogen fuel cells rely on the simultaneous transport of water, hydrogen and oxygen through porous solid electrolytes to work. In materials science, impregnation processes, e.g. wood impregnation, rely on the transport of fluids into porous media.
We will address the problem of non-Newtonian flow in porous media. Non-Newtonian fluids have viscosities that depend on their state of motion. Mixtures of immiscible fluids moving simultaneously in porous media behave effectively as a single fluid with such properties, making the potential use of the results that will emerge vast.
Porous media typically span orders of magnitude in length scales: The pore scale may be in the micrometer range whereas the largest scale is in the kilometer range. At the macroscopic scale, the porous medium is seen as a continuum and its flow properties are described by phenomenological differential equations. On the pore scale, these differential equations are irrelevant. No one has succeeded in deriving reliable large-scale differential equations from the small-scale physics. The up-scaling problem remains unsolved. We will attempt to solve it for non-Newtonian fluids in porous media.
We will do this through a combination of theoretical and numerical approaches which already have yielded promising results. We will do this in collaboration with experimentalists.