Page 223 - Schaum's Outline of Theory and Problems of Applied Physics

208 FLUIDS IN MOTION [CHAP. 17

The rate of flow of a liquid through a pipe obeys Poiseuille’s law
4
πr p
R =
8ηL
where L is the length of the pipe, r is its radius, p is the pressure difference between the ends of the pipe, and
η is the viscosity of the liquid. Evidently the rate of flow depends most strongly on the radius of the pipe.

SOLVED PROBLEM 17.15
Bernoulli’s equation holds for incompressible, nonviscous fluids. How is this relationship changed when
the viscosity of a fluid is not negligible?

The effect of viscosity, which is a kind of friction, is to dissipate mechanical energy into heat. Hence the quantity
2
1
p + dgh + dv decreases along the direction of flow of a viscous fluid.
2
SOLVED PROBLEM 17.16
The grease nipple on a bearing has a hole 1 mm in diameter and 6 mm long. The grease being used has
3
a viscosity of 80 P. How much pressure is needed to force 0.2 cm of grease into the nipple in 5 s?
Here r = 0.5mm = 5 × 10 −4 m, L = 6mm = 6 × 10 −3 m, and
3
(0.2cm ) −8 3
R = 3 = 4 × 10 m /s
(10 cm /m )(5s)
3
6
Hence the required pressure is
3
8ηLR (8)(80 P)(6 × 10 −3 m)(4 × 10 −8 m /s) 5
p = = = 7.8 × 10 Pa
πr 4 π(5 × 10 )
−4 4
This is about 7.7 atm.

REYNOLDS NUMBER
The nature of the fluid flow in a particular situation (that is, whether laminar or turbulent) depends on the density d
and viscosity η of the fluid, its average velocity v, and a characteristic dimension D of the system according to
the Reynolds number N R given by

dv D
N R =
η
The Reynolds number has no units associated with it. For fluid flow in a pipe, D is the pipe diameter. In a pipe,
N R < 2000 corresponds to laminar flow, and N R > 3000 corresponds to turbulent flow. If N R is between 2000
and 3000, the flow may be of either kind and may shift back and forth between them.
The Reynolds number is an important quantity because it provides a basis for experiments that use a small
model system to replace a full-size one. If N R is the same for both, they are said to be dynamically similar, and
the pattern of fluid flow will be the same for both. Wind tunnel tests of airplane models and towing tank tests of
ship models give useful results only when dynamic similarity is obeyed.

SOLVED PROBLEM 17.17
Water at 20 C flows at 1.5 m/s through a tube whose inside diameter is 3 mm. The viscosity of water
◦
at this temperature is 1.0 × 10 −3 P. (a) Determine the nature of the flow in the tube by calculating its
Reynolds number. (b) Find the maximum velocity for laminar flow in the tube.