fluid_mechanics

For Newtonian
fluids, stresses are
linearly related to
the rate of strain.
1for shearing stresses2
t xy t yx m a 0u
0y 0v
0x b (6.125d)
6.8 Viscous Flow 307
t yz t zy m a 0v
0z 0w
0y b
t zx t xz m a 0w
0x 0u
0z b
(6.125e)
(6.125f)
where p is the pressure, the negative of the average of the three normal stresses; that is,
p 1 1 321s xx s yy s zz 2. For viscous fluids in motion the normal stresses are not necessarily the
same in different directions, thus, the need to define the pressure as the average of the three normal
stresses. For fluids at rest, or frictionless fluids, the normal stresses are equal in all directions. 1We
have made use of this fact in the chapter on fluid statics and in developing the equations for inviscid
flow.2 Detailed discussions of the development of these stress–velocity gradient relationships can
be found in Refs. 3, 7, and 8. An important point to note is that whereas for elastic solids the
stresses are linearly related to the deformation 1or strain2, for Newtonian fluids the stresses are
linearly related to the rate of deformation 1or rate of strain2.
In cylindrical polar coordinates the stresses for incompressible Newtonian fluids are expressed
as 1for normal stresses2
1for shearing stresses2
s rr p 2m 0v r
0r
s uu p 2m a 1 r
s zz p 2m 0v z
0z
0v u
0u v r
r b
t ru t ur m c r 0
0r av u
r b 1 0v r
r 0u d
t uz t zu m a 0v u
0z 1 0v z
r 0u b
(6.126a)
(6.126b)
(6.126c)
(6.126d)
(6.126e)
t zr t rz m a 0v r
(6.126f)
0z 0v z
0r b
The double subscript has a meaning similar to that of stresses expressed in Cartesian coordinates—
that is, the first subscript indicates the plane on which the stress acts, and the second subscript the
direction. Thus, for example, s rr refers to a stress acting on a plane perpendicular to the radial direction
and in the radial direction 1thus a normal stress2. Similarly, t ru refers to a stress acting on a plane
perpendicular to the radial direction but in the tangential 1u
direction2 and is therefore a shearing stress.
6.8.2 The Navier–Stokes Equations
The stresses as defined in the preceding section can be substituted into the differential equations
of motion 1Eqs. 6.502 and simplified by using the continuity equation 1Eq. 6.312 to obtain:
1x direction2
z
w
v
u
r a 0u
0t u 0u
0x v 0u
0y w 0u
0z b 0p 0x rg x m a 02 u
0x 02 u
2 0y 02 u
2 0z b 2
1y direction2
(6.127a)
x
y
r a 0v
0t u 0v
0x v 0v
0y w 0v
0z b 0p 0y rg y m a 02 v
0x 2 02 v
0y 2 02 v
0z 2 b
(6.127b)