文档介绍：Revised January 29, 2003 Copyright 2000 David A. Randall
Derivation of the Navier-Stokes
Equation
David A. Randall
Department of Atmospheric Science
Colorado State University, Fort Collins, Colorado 80523
Analysis of the relative motion near a point
Suppose that the velocity of the fluid at position r0()xyz,, and time t is
V()xyzt,,, , and that the simultaneous velocity at a neighboring position r0 + r is
V + dV . Then the velocity at the neighboring position r0 + r relative to the velocity of the
reference position r0 is
ʈu u u
Á˜-----∂-----∂-----∂
∂x ∂y ∂z
ʈdu Á˜Êˆx
Á˜v v v
Á˜dv = Á˜-----∂-----∂-----∂Á˜y . ()
Á˜ x z z Á˜
˯w Á˜∂∂∂˯z
d Á˜w w w
Á˜------∂------∂------∂
˯∂x ∂y ∂z
In vector form, this can be written as
dV = D ◊ r , ()
where D is called the deformation tensor because it describes how the fluid element is
being deformed by the non-uniform motion field. Introducing Cartesian tensor notation,
we can alternatively express () as
∂vi
dvi = -------xj , where i = 123,, . ()
∂xj
We can pose ∂vi § ∂xj into two parts which are symmetric and anti-
symmetric in the suffices i and j . They are
1 ∂vi ∂vj
eij = ---ʈ------- + ------- , ()
2˯∂xj ∂xi
Selected Nuggets 235
236 Derivation of the Navier-Stokes Equation
and
1 ∂vi ∂vj
xij = ---ʈ------- –------- , ()
2˯∂xj ∂xi
respectively. The deformation D can now be expressed by
D = eX+ , ()
where we define two “parts” of D :
ʈexx exy exz ʈ0 xxy xxz
Á˜Á˜
. ()
e ∫∫Á˜eyx eyy eyz , X Á˜xyx 0 xyz
Á˜Á˜
˯ezx ezy ezz ˯xzx xzy 0
We show below that eX is related to the divergence, and is related to the vorticity.
Correspondingly, dV can be divided into two parts, .
()s ()a ()s ()a
dV = dV + dV , where d V i == eijXj,xdVi ijXj . ()
Here the superscripts s and a denote “symmetric” and “antisymmetric,” respectively.
We refer to e as the rate of strain tensor (or strain te