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In fluid dynamics, the Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes are a set of nonlinear partial differential equations that describe the flow of fluids such as liquids and gases. For example, they model weather or the movement of air in the atmosphere, ocean currents, water flow in a pipe, as well as many other fluid flow phenomena.
It is a famous open question whether smooth initial conditions always lead to smooth solutions for all times; a $1,000,000 prize was offered in May 2000 by the Clay Mathematics Institute for the answer to this question.
NS equations assume that the fluid is a continuum. They are conservation equations over a control volume which we will call Ω, which keeps the portion of the fluid which is studied. Unfortunately, Ω depends on time, and the conservation laws need then to be modified so as to be valid on arbitrary control volumes.
The substantive derivative
main article: substantive derivative.
Before going into the details of the Navier-Stokes equations, one must first define an operator:
This is the usual time derivative, but when following a particle of the fluid. The full form is:
The integral form of a conservation law L over the control volume Ω is:
Using the particular derivative, one can swap the and operators:
As this expression is valid for all Ω, it simplifies to:
Main article: conservation laws.
Over a control volume, using the transformation described above, the following quantities are deemed conserved:
Equation of continuity
Conservation of mass is written:
- ρ is the density of the fluid.
In the case of an incompressible fluid ρ is not a function of time; the equation is reduced to:
Conservation of momentum
Conservation of momentum is written:
We can simplify it further, this becomes:
In which we recognise the usual F=ma.
The form of the equations
The general form of the Navier-Stokes equations is:
For the conservation of momentum. The tensor represents the surface forces applied on a fluid particle. In general, we have the form:
To which we add the continuity equation:
The nature of the diagonal of is known, it is the gradient of pressure, thus:
- p is the pressure
finally, we have:
where the components of are the τ of .
The closure problem
Those equations are incomplete. To complete them, one must make hypotheses on the form of . In the case of an perfect fluid τ components are nil, for example.
So-called non-Newtonian fluids are simply fluids where this tensor has no special properties allowing for special solutions of the equations.
Those are certain usual simplifications of the problem for which sometimes solutions are known.
In Newtonian fluids the following assumption holds:
- μ is the viscosity of the fluid.
In Bingham fluids, we have something slightly different:
Those are fluids capable of bearing some shear before they start flowing. An example is tooth paste.
The Navier-Stokes equations are
for momentum conservation and
for conservation of mass.
- ρ is the density,
- ui (i = 1,2,3) the three components of velocity,
- fi body forces (such as gravity),
- p the pressure,
- μ the dynamic viscosity, of the fluid at a point;
If μ is uniform over the fluid, the momentum equation above simplifies to
If now in addition ρ is assumed to be constant:
Continuity equation (assuming incompressibility):
- Simplified version of the N-S equations. Adapted from Incompressible Flow, second edition by Ronald Panton
The equations are derived by considering the mass, momentum, and energy balances for an infinitesimal control volume. The Navier-Stokes equations need to be augmented by an equation of state for compressible flows. The variables to be solved for are the velocity components, the fluid density, static pressure, and temperature. The flow is assumed to be differentiable and continuous, allowing these balances to be expressed as partial differential equations. The equations can be converted to Wilkinson equations for the secondary variables vorticity and stream function. Solution depends on the fluid properties (such as viscosity, specific heats, and thermal conductivity), and on the boundary conditions of the domain of study. For a derivation of the Navier-Stokes equations, see some of the external links listed below.
Note that the Navier-Stokes equations can only describe fluid flow approximately and that, at very small scales or under extreme conditions, real fluids made out of mixtures of discrete molecules and other material, such as suspended particles and dissolved gases, will produce different results from the continuous and homogeneous fluids modelled by the Navier-Stokes equations. Depending on the Knudsen number of the problem, statistical mechanics may be a more appropriate approach. However, the Navier-Stokes equations are useful for a wide range of practical problems, providing their limitations are borne in mind.
Although the full, unsteady Navier-Stokes equations correctly describe nearly all flows of practical interest, they are too complex for practical solution in many cases and a special "reduced" form of the full equations is often used instead — these are the Reynolds-averaged Navier-Stokes (RANS) equations. The solution of the full steady Navier-Stokes equations is sufficiently accurate alone for cases where the fluid flow is laminar. For turbulent flows the Reynolds-averaged form of the equations are most commonly used. The RANS form of the equations introduce new terms that reflect the additional modelling of the small turbulent motions.
Solution of flow equations by numerical methods is called computational fluid dynamics.
- Inge L. Rhyming Dynamique des fluides, 1991 PPUR
- The Clay Mathematics Institute's Navier-Stokes equation prize
- A good derivation of the Navier-Stokes equations
- Derivation of the Navier-Stokes equations
- NASA page on the Navier-Stokes equations
- Navier-Stokes equations (exact solutions) at EqWorld: The World of Mathematical Equations
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