fluid_mechanics

308 Chapter 6 ■ Differential Analysis of Fluid Flow
1z direction2
The Navier–Stokes
equations are the
basic differential
equations describing
the flow of
Newtonian fluids.
z
(6.127c)
where u, v, and w are the x, y, and z components of velocity as shown in the figure in the margin
of the previous page. We have rearranged the equations so the acceleration terms are on the left
side and the force terms are on the right. These equations are commonly called the Navier–Stokes
equations, named in honor of the French mathematician L. M. H. Navier 11785–18362 and the
English mechanician Sir G. G. Stokes 11819–19032, who were responsible for their formulation.
These three equations of motion, when combined with the conservation of mass equation 1Eq. 6.312,
provide a complete mathematical description of the flow of incompressible Newtonian fluids. We
have four equations and four unknowns 1u, v, w, and and therefore the problem is “well-posed”
in mathematical terms. Unfortunately, because of the general complexity of the Navier–Stokes
equations 1they are nonlinear, second-order, partial differential equations2, they are not amenable
to exact mathematical solutions except in a few instances. However, in those few instances in which
solutions have been obtained and compared with experimental results, the results have been in close
agreement. Thus, the Navier–Stokes equations are considered to be the governing differential
equations of motion for incompressible Newtonian fluids.
In terms of cylindrical polar coordinates 1see the figure in the margin2, the Navier–Stokes
equations can be written as
1r direction2
r a 0v r
0t
1u
direction2
r a 0w
0t u 0w
0x v 0w
0y w 0w
0z b 0p 0z rg z m a 02 w
0x 2 02 w
0y 2 02 w
0z 2 b
p2,
v 0v r
r
0r v u 0v r
r 0u v2 u
r v 0v r
z
0z b
0p
0r rg r m c 1 0
r 0r ar 0v r
0r b v r
r 1 0 2 v r
2 r 2 0u 2 0v u
2 r 2 0u 02 v r
0z d 2
(6.128a)
x
θ
r
v z
vθ
v r
y
r a 0v u
0t
v 0v u
r
0r v u 0v u
r 0u v rv u
v 0v u
r z
0z b
1 0p
r 0u rg u m c 1 0
r 0r ar 0v u
0r b v u
r 1 0 2 v u
2 r 2 0u 2 0v r
2 r 2 0u 02 v u
0z d 2
(6.128b)
1z direction2
r a 0v z
0t
v 0v z
r
0r v u 0v z
r 0u v 0v z
z
0z b
0p
0z rg z m c 1 0
r 0r ar 0v z
0r b 1 0 2 v z
r 2 0u 2
02 v z
0z 2 d
(6.128c)
To provide a brief introduction to the use of the Navier–Stokes equations, a few of the
simplest exact solutions are developed in the next section. Although these solutions will prove to
be relatively simple, this is not the case in general. In fact, only a few other exact solutions have
been obtained.
6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Fluids
A principal difficulty in solving the Navier–Stokes equations is because of their nonlinearity arising
from the convective acceleration terms 1i.e., u 0u0x, w 0v0z, etc.2. There are no general analytical
schemes for solving nonlinear partial differential equations 1e.g., superposition of solutions cannot
be used2, and each problem must be considered individually. For most practical flow problems, fluid
particles do have accelerated motion as they move from one location to another in the flow field.
Thus, the convective acceleration terms are usually important. However, there are a few special cases
for which the convective acceleration vanishes because of the nature of the geometry of the flow