fluid_mechanics

8.2 Fully Developed Laminar Flow 395
The governing
differential equations
can be simplified
by
appropriate assumptions.
We will not repeat the detailed steps used to obtain the laminar pipe flow from the Navier –
Stokes equations 1see Section 6.9.32 but will indicate how the various assumptions used and steps applied
in the derivation correlate with the analysis used in the previous section.
General motion of an incompressible Newtonian fluid is governed by the continuity equation
1conservation of mass, Eq. 6.312 and the momentum equation 1Eq. 6.1272, which are rewritten
here for convenience:
0V
0t
V 0
V V p
r g n2 V
(8.13)
(8.14)
For steady, fully developed flow in a pipe, the velocity contains only an axial component, which
is a function of only the radial coordinate 3V u1r2î 4. For such conditions, the left-hand side of
the Eq. 8.14 is zero. This is equivalent to saying that the fluid experiences no acceleration as it
flows along. The same constraint was used in the previous section when considering F ma for
the fluid cylinder. Thus, with g gkˆ the Navier –Stokes equations become
V 0
p rgkˆ m 2 V
(8.15)
The flow is governed by a balance of pressure, weight, and viscous forces in the flow direction,
similar to that shown in Fig. 8.10 and Eq. 8.10. If the flow were not fully developed 1as in an entrance
region, for example2, it would not be possible to simplify the Navier –Stokes equations to that
form given in Eq. 8.15 1the nonlinear term V V would not be zero2, and the solution would be
very difficult to obtain.
Because of the assumption that V u1r2î, the continuity equation, Eq. 8.13, is automatically
satisfied. This conservation of mass condition was also automatically satisfied by the
incompressible flow assumption in the derivation in the previous section. The fluid flows across
one section of the pipe at the same rate that it flows across any other section 1see Fig. 8.82.
When it is written in terms of polar coordinates 1as was done in Section 6.9.32, the component
of Eq. 8.15 along the pipe becomes
0p
0x rg sin u m 1 r
0 0u
ar
0r 0r b
(8.16)
Since the flow is fully developed, u u1r2 and the right-hand side is a function of, at most, only
r. The left-hand side is a function of, at most, only x. It was shown that this leads to the condition
that the pressure gradient in the x direction is a constant— 0p0x ¢p/. The same condition
was used in the derivation of the previous section 1Eq. 8.32.
It is seen from Eq. 8.16 that the effect of a nonhorizontal pipe enters into the Navier–Stokes
equations in the same manner as was discussed in the previous section. The pressure gradient in
the flow direction is coupled with the effect of the weight in that direction to produce an effective
pressure gradient of ¢p/ rg sin u.
The velocity profile is obtained by integration of Eq. 8.16. Since it is a second-order equation,
two boundary conditions are needed—112 the fluid sticks to the pipe wall 1as was also done
in Eq. 8.72 and 122 either of the equivalent forms that the velocity remains finite throughout the
flow 1in particular u 6 q at r 02 or, because of symmetry, that 0u0r 0 at r 0. In the derivation
of the previous section, only one boundary condition 1the no-slip condition at the wall2 was
needed because the equation integrated was a first-order equation. The other condition
10u0r 0 at r 02 was automatically built into the analysis because of the fact that t m dudr
and t 2t w rD 0 at r 0.
The results obtained by either applying F ma to a fluid cylinder 1Section 8.2.12 or solving
the Navier –Stokes equations 1Section 6.9.32 are exactly the same. Similarly, the basic assumptions
regarding the flow structure are the same. This should not be surprising because the two methods
are based on the same principle—Newton’s second law. One is restricted to fully developed laminar
pipe flow from the beginning 1the drawing of the free-body diagram2, and the other starts with
the general governing equations 1the Navier –Stokes equations2 with the appropriate restrictions
concerning fully developed laminar flow applied as the solution process progresses.