Fundamentals of Fluid Dynamics - Computational Fluid and Particle Dynamics in the Human Respiratory System - page 102

be equal to the inlet velocity u 1 . However if we consider the realistic model, the
two cross-sections shown have different cross-sectional areas, where A 1 = 1.5 cm 2
and A 2 = 3.0 cm 2 . This means that A 1 =½ A 2 and u 1 =2 u 2 (from mass conservation)
which means that the flow decelerates from the larynx down through the trachea dur-
ing steady inhalation. Conversely during steady exhalation, the flow moves from a
larger to a smaller cross-section, from A 2 to A 1 , which means that the flow accelerates
as it approaches the larynx.

5.2.2 Momentum Balance

The balance of momentum is a fundamental law of physics based on Newton ' s second
law of motion, which states that the sum of forces that act on a control volume equals
the product of its mass, m, and acceleration, a , of the control volume, which is
precisely the time rate of change of its momentum:
F =

ma
F is the sum of all forces, namely body forces and surface forces, acting on a
control volume. We can also rewrite the mass m as the product of the density ρ
and volume x y z , and rewrite the acceleration as

D U

Dt , which then gives us the

following:

F body + F surface =

( ρxyz ) D U
Dt

(5.6)

Body forces act over the entire volume and as such are sometimes called volume
forces. Typically these forces include gravity, centrifugal, Coriolis and electromag-
netic forces, which act at a distance to the control volume. These effects are usually
incorporated by introducing them into the momentum equations as additional source
terms that add to the contribution of the surface forces. If we denote the body force
per unit mass acting on the fluid element in the x -direction as F B , then the total body
force acting over the entire fluid element is

body force = F B ρ ( xyz )

Surface forces are those forces that act on the surface of the fluid element causing it
to deform (Fig. 5.3 ).
These surface forces include the normal stress σ xx, a combination of pressure p
exerted by the surrounding fluid and normal viscous stress components τ xx that act
perpendicular to fluid element surface, and also the tangential stresses τ yx and τ zx that
act along the surfaces of the fluid element. The surface forces only for the velocity
component u (i.e. one-dimensional) are shown in Fig. 5.4 .