Unformatted text preview: = ν (∆v)(x, y, z, t) − ∇p(x(t), y (t), z (t), t).
Here the Laplace operator is applied componentwise and ν is a constant, called
the viscosity of the ﬂuid. The equations used by Navier and Stokes to model an
incompressible viscous ﬂuid (with ρ constant) are then
∇ · v = 0, ∂v
1
+ (v · ∇)v = ν ∆v − ∇p.
∂t
ρ It is remarkable that these equations, so easily expressed, are so diﬃcult to
solve. Indeed, the NavierStokes equations form the basis for one of the seven
Millenium Prize Problems, singled out by the Clay Mathematics Institute as
central problems for mathematics at the turn of the century. If you can show
that under reasonable initial conditions, the NavierStokes equations possess a
unique wellbehaved solution, you may be able to win one million dollars. To
ﬁnd more details on the prize oﬀered for a solution, you can consult the web
address: http://www.claymath.org/millennium/
Exercise:
5.4.1. Show that if the tension and density of a membrane are given by variab...
View
Full
Document
This document was uploaded on 01/12/2014.
 Winter '14
 Equations

Click to edit the document details