In May 2002, the Clay Mathematics Institute (CMI) of Cambridge, Massachusetts, in an initiative to further the study of mathematics, allocated a $7m prize fund for the solution of seven Millennium Problems, 'focusing on important classic questions that have resisted solution over the years'. One of the $1m problems stands out for its massive practical importance: the solution of the Navier-Stokes equations (NSEs) for fluid flow.

Although there are many named variants and special cases, the fundamental equations are the incompressible Navier-Stokes for Newtonian fluids. In their most compact form, they comprise a pair of vector partial differential equations (PDEs): one expresses the forces acting (pressure, viscosity and body forces); the other is the continuity equation, which says that divergence of the velocity field is zero for an incompressible fluid (that is, 'what comes in, goes out').

The NSEs are among the most-studied partial differential systems, the subject of around 15-20 published papers a week. Nevertheless, they're among the least understood at a theoretical level. Charles Fefferman writes in the problem statement paper, Existence and Smoothness of the Navier-Stokes equation, 'Fluids are important and hard to understand. There are many fascinating problems and conjectures about the behaviour of solutions of the Navier-Stokes equations. Since we don't even know whether these solutions exist, our understanding is at a very primitive level.'

To the many workers who find the NSEs problematic, it may be consolation that so did Navier, who produced the viscous fluid equations in 1821 by a faulty derivation that ignored shear stress. The association with Stokes comes from the latter's more rigorous independent derivation in 1845. (By historical accident, Saint-Venant, who published two years before Stokes, isn't commemorated in the name).

Since then, in common with other physically important systems, exact solutions have been found for some simplified forms of the NSE. For instance, the dynamics have been solved for very slow flow past a sphere (the well-known Stokes' Law), fully developed laminar flow (Hagen-Poiseuille flow) and Couette flow (fluid in an annulus between two concentric spinning cylinders). However, the nonlinearity of the equations means that further solutions can't be developed because the principle of superposition does not hold, and so numerical methods are needed for faster flow and other shapes. Despite some promising alternatives, such as Stephen Wolfram's hexagonal-grid CA fluid models, the dominant practical approach is the finite element method (FEM). Even so, the FEM attacks the problem by solving a 'weak' transformed version of the equations, without any proof that the 'strong' form (the underlying PDEs) have smooth, physically reasonable solutions. Finding such a proof - or conversely, proof of conditions where the NSE fall over - is the Clay Mathematics Institute's challenge.

The Reynolds number (Re) - a dimensionless quantity based on the ratio of inertial to viscous forces - is a good measure of the difficulty of FEM solution. For small Re (i.e. low flow velocities) there's a regime of stable laminar flow where solutions easily converge but, as Re increases, fluids develop vortices, with fine-scale time-dependent turbulence at the edges and in mid-stream, the flow structure ultimately going mathematically chaotic. With gases (also fluids in the NSE context) any significant compression requires a shift to the compressible NSEs, or to related models such as the Euler equations. These are very unstable numerically, equation stiffness becoming a problem as flow rates increase.

When velocities exceed the speed of sound in the medium, solvers also have to manage the strong velocity discontinuities of shock waves. Handling gas turbulence correctly is a problem at any speed. In aircraft aerodynamics, for example, laminar flow generally holds for all but a turbulent 'skin' around the craft, and it's usually easier to model the general airflow with the NSE, and ignore the detail in the turbulent region, treating it as a semi-empirical drag effect. Problems in calculating turbulence contributed also to the long-standing myth that 'aerodynamics proves that a bumblebee can't fly'. This, one hopes, has been dispelled finally by the work of Cornell physicist Z. Jane Wang, whose supercomputer NSE models of unsteady flow demonstrate, at least in 2D, the role of vortex-shedding in creating lift for insects.

Another basic modification to the NSE is to introduce variable viscosity for fluids such as polymers or blood which, unlike Newtonian fluids such as water, have a non-linear relationship between stress and strain. Bread dough is classically non-Newtonian. Fluid dynamics specialists CD adapco recently published a case study of work by a team at the University of Padova, Italy, which used its Star-CD software to optimise pasta moulding and extrusion. This is a typical mainstream commercial use of the NSE in Computational Fluid Dynamics (CFD) directed at areas such as industrial processing of gases and liquids, aerodynamics of aircraft and vehicles, and maritime uses.

Star-CD is one of a strong core of CFD packages associated with engineering design. Others include CFdesign, fluid and thermal simulation software for use with Autodesk's MCAD; FLOTRAN from the computer-aided engineering specialists ANSYS; and AEA Technology's CFX range (currently under acquisition by ANSYS). Apart from these, many general-purpose FEM programs handle fluid dynamics as part of a 'multiphysics' repertoire; one of these, Comsol's Femlab, is described in the panel opposite. Freeware also exists, such as CFDLIB, the Los Alamos Library of computer codes for solving CFD problems in 2D and 3D.

The CFD field contains many sub-specialisms where companies have invested time in the mathematics for solving computationally difficult aspects of CFD. Phoenics, a Fortran-based package from the Wimbledon-based CHAM, specialises in chemically awkward flows. For instance, a coal-fired furnace simulation is a turbulent multi-phase flow, with multiple chemical reactions in fluids of complex composition. Many uncertainties apply - but CHAM points out that 'CFD is nevertheless used because the uncertainties resulting from its non-use are even greater.'

Free surfaces also need special handling. R&D consultancy DynaFlow Inc., alongside more general CFD products, specialises in the fluid dynamics of cavitation, bubbles and waves; its 2DynaFS and 3DynaFS programs use Boundary Element Methods to model such effects as underwater explosions, surface waves against objects and beaches, bubble interactions, and propeller cavitation. FLOW-3D from Flow Science also specialises in accurate simulation of free surface flows, using the recently developed Volume of Fluid (VOF) technique to model crucial boundary conditions at the gas-fluid interface.

One of the most striking aspects of the Navier-Stokes models is the unity of phenomena across different types of fluid. Lucretius, writing his On the Nature of the Universe in the first century BCE, commented on the 'swiftly circling vortex' and other ways that winds 'in their actions and behaviour are found to rival great rivers'. I think he would have been delighted by the 'von Karman vortex street', a downstream chain of spiral eddies generated by an obstacle at the onset of turbulence. First described in atmospheric observations by Theodore von Karman, a co-founder of NASA's Jet Propulsion Laboratory, vortex streets can be seen at all scales, from 100-mile cloud patterns downwind of islands, large enough to be visible from orbit, to rivers and experimental laboratory flows in soap films.

This ubiquity extends to the Navier-Stokes equations themselves, which are remarkable in their breadth of scientific application. They can be applied to: arterial flow; fluid-dynamical models for traffic flow and crowd flow analogous to compressible NSE; hydrodynamic models of astrophysical phenomena; as one component of the multiphysics models used in weather forecasting; and even digital image processing. In a collaboration funded by the US Office of Naval Research (ONR), Andrea Bertozzi of Duke University and Guillermo Sapiro of the University of Minnesota have used the NSE for digital inpainting. Treating image intensity as a 'stream function' for a 2D incompressible flow, structures in an image can be computed to 'flow under' missing or obscured regions. This allows automatic restoration of digital images, such as a scan of a photograph that has been scratched or written on. The naval appeal is a possible use for the enhancement of surveillance images, or to fill in gaps in video transmissions sent under noisy or low-bandwith conditions.

This ability of the Navier-Stokes to 'cross-fertilise' different scientific disciplines is, I think, reason for hope that significant insights will continue to surface, even if the underlying proofs remain elusive. Charles Fefferman concludes: 'Standard methods from PDE appear inadequate to settle the problem. Instead, we probably need some deep, new ideas.' If you have any, that $1 million prize is waiting.

## Fluid dynamics: a multiphysics approach

Femlab, from Stockholm-based Comsol AB, is effectively a Matlab toolbox for modelling physical systems based on 1D, 2D or 3D partial differential equations (including 'multiphysics' systems where multiple coupled equations apply).

Script-driven by a programming language that extends the Matlab language, it provides all stages of modelling - CAD design, meshing, solution, and visualisation - via control-panel operation. Windows, Unix/Linux and MacOS versions are available.

The Navier-Stokes equations represent one of the preset physics models you can load from Femlab's Model Navigator, either in PDE mode as raw equations, or in Physics Mode where the PDE coefficients are provided as physical units such as viscosity and density.

Each solution shell is tailored to its particular area of physics: for instance, with fluid models, the output graphs have predefined plot variables for x or velocity, pressure, velocity field, or vorticity.

Femlab's Model Library also contains specific models of several classic fluid flow problems, which can be adapted as needed. Flow over a backward step in a tube is a useful benchmark problem, as experimental results have been published despite the lack of an exact solution.

The basic approach is to solve the coupled incompressible Navier-Stokes equations. Alternatives are an equivalent stream function/vorticity formulation that outputs streamlines directly, and a solver using Argyris elements (which match derivatives stringently between mesh triangles) guaranteeing zero divergence and so dispensing with the continuity equation.

Other models include flow past a cylinder, demonstrating Von Karman vortices; flow of paper pulp, a non-Newtonian liquid; waves in a pool using the Saint-Venant equations (3D Navier-Stokes simplified to 2D); and a shock wave in gas modelled by the Euler equations.

In the latter case, the numerical instability requires the use of 'streamline diffusion stabilisation', an algorithmic option that reduces error propagation.

Exploring these models gives good experience of how the pitfalls increase with rising Reynolds number.

The solver for the non-linear Navier-Stokes equations must be fed initial conditions. To ensure convergence at higher Reynolds numbers, these are best provided by a trial run at a lower value. The mesh also needs refining as turbulent structures develop.

Femlab's Chemical Engineering Module explores further what happens in the fully turbulent regime often used in industrial reaction and mixing equipment.

Computing a detailed flow pattern would need extremely fine grid resolution, and give no clear picture of the overall liquid movement, so models such as the 'K-epsilon' are useful - though limited in accuracy - in averaging the mass transport in turbulent regions.

The K-epsilon is the basis of several turbulent models in this module, including axisymmetric and 2D jets, a static mixer, and a backstep with high Reynolds number flow. Other models include flow in porous media, non-Newtonian polymer flow, and supersonic gas flow (Mach 1.4) exhibiting shock waves.

The Navier-Stokes equations also feature in an advanced 'microphysics' model developed by Dr Jordan Macinnes at the University of Sheffield. Here, the NSEs are coupled with electrostatics to model electrokinetic flow in a microlaboratory for DNA analysis.

By applying charges to the walls of a channel, solutions moving in laminar flow can be channelled and mixed precisely, without the need for moving mechanical parts - a reminder that fluid dynamics is not just confined to chunky industrial applications.