4.1 INTRODUCTION
91
Lominor flow, low Reynolds number
Fig. 4.2. Illustrative of the difference between laminar and turbulent flow in tubes.
physics of this situation is expressed by the Reynolds equation, from which is derived
from a dimensionless coefficient, the Reynolds number, thus:
Udp
R ~ ~ ,
where R is the Reynolds number, U is the velocity of the particle, d is the diameter of
the particle, p is the density of the particle, and/x is the viscosity of the fluid. For a given
situation the Reynolds number can be used to differentiate two different types of fluid
behavior at the solid boundary, be it a sphere or a confining surface such as a tube or
channel wall. For the low Reynolds numbers the fluid flow is laminar, flow lines running
parallel to the boundary surface; for the high Reynolds numbers the flow is turbulent,
generating eddies and vortices (Fig. 4.2). For flow in tubes the critical Reynolds number separating laminar and turbulent flow is about 2000. For a particle in a fluid the critical number is about 1. Stokes' law of settling, discussed in Chapter 3, is derived from
the Reynolds equation. Note that turbulence is proportional to velocity, but inversely
proportional to viscosity.
A second important coefficient of fluid dynamics is the Froude number. This is essentially the ratio between the force required to stop a moving particle and the force
of gravity; that is, the ratio of the force of inertia and the acceleration due to gravity.
Hence:
F
U
VgL'
where U is the velocity of the particle, L is the force of inertia (i.e., the length traveled
by the particle before it comes to rest), and g is the acceleration due to gravity.
91
Lominor flow, low Reynolds number
Fig. 4.2. Illustrative of the difference between laminar and turbulent flow in tubes.
physics of this situation is expressed by the Reynolds equation, from which is derived
from a dimensionless coefficient, the Reynolds number, thus:
Udp
R ~ ~ ,
where R is the Reynolds number, U is the velocity of the particle, d is the diameter of
the particle, p is the density of the particle, and/x is the viscosity of the fluid. For a given
situation the Reynolds number can be used to differentiate two different types of fluid
behavior at the solid boundary, be it a sphere or a confining surface such as a tube or
channel wall. For the low Reynolds numbers the fluid flow is laminar, flow lines running
parallel to the boundary surface; for the high Reynolds numbers the flow is turbulent,
generating eddies and vortices (Fig. 4.2). For flow in tubes the critical Reynolds number separating laminar and turbulent flow is about 2000. For a particle in a fluid the critical number is about 1. Stokes' law of settling, discussed in Chapter 3, is derived from
the Reynolds equation. Note that turbulence is proportional to velocity, but inversely
proportional to viscosity.
A second important coefficient of fluid dynamics is the Froude number. This is essentially the ratio between the force required to stop a moving particle and the force
of gravity; that is, the ratio of the force of inertia and the acceleration due to gravity.
Hence:
F
U
VgL'
where U is the velocity of the particle, L is the force of inertia (i.e., the length traveled
by the particle before it comes to rest), and g is the acceleration due to gravity.