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290 Polynomial Approximation of Differential Equations
n pn(0,0)
2 1.570796326
4 1.856395658
6 1.851713702
8 1.851563407
10 1.851562860
12 1.851563065
Table 13.3.1 - Approximations of U(0,0).
13.4 The incompressible Navier-Stokes equations
The Navier-Stokes equations describe, with a good degree of accuracy, the motion of a
viscous incompressible fluid in a region Ω (see for instance batchelor(1967)). They
find applications in aeronautics, meteorology, plasma physics, and in other sciences.
In two dimensions Ω is an open set of R 2 . The unknowns, depending on the space
variables (x,y) ∈ ¯ Ω and the time variable t ∈ [0,T], T > 0, are the two components
of the velocity vector field V ≡ (V1,V2), Vi : Ω × [0,T] → R, 1 ≤ i ≤ 2, and the
pressure P : Ω×]0,T] → R. Assuming that the density of the fluid is constant, after
dimensional scaling the equations become
(13.4.1)
(13.4.2)
(13.4.3)
∂V1
∂t
∂V2
∂t
∂V1
+ V1
∂x
∂V2
+ V1
∂x
∂V1
∂x
+ ∂V2
∂y
∂V1
+ V2
∂y − ν ∆V1 = − ∂P
∂x
∂V2
+ V2
∂y − ν ∆V2 = − ∂P
∂y
= 0 in Ω×]0,T],
in Ω×]0,T],
in Ω×]0,T],