Fundamentals of Fluid Dynamics - Computational Fluid and Particle Dynamics in the Human Respiratory System - page 107

Fig. 5.7 Velocity profiles at different locations of diameter distances from the inlet for u in = 0.01 m/s
and dynamic viscosity of a μ 1 =4 10 − 5 s, and b μ 2 =10 − 5 × kg/m · kg/m · s

these cases, the inlet profile can be prescribed with a uniform velocity inlet, but this
neglects the upstream flow conditions developed by the upper respiratory airway. To
account for some flow development, studies in the literature have used a parabolic
profile at the tracheobronchial inlet for a laminar steady flow (Liu et al. 2003) while
more recent studies also include the influence of the laryngeal jet effect (Luo and
Liu 2009; Zhang et al. 2005a). The development of the velocity flow profile through
internal flows is primarily due to the diffusion term in the momentum equation. To il-
lustrate this, let us consider an internal pipe flow, which again may represent a highly
simplified trachea that has a uniform velocity profile applied to the inlet. Here, the
dimensions of the pipe are given as height L = 15 cm and diameter D = 2 cm. Using
CFD, the development of the velocity profiles between the inlet and the outlet can
be visualized with air (density ρ = 1.2 kg/m 3 ) as the working fluid, for a fixed inlet
velocity u in = 0.05 m/s, and with dynamic viscosities μ 1 =4 10 − 5 kg/m × · s (Case 1)

or μ 2 =10 − 5

s (Case 2).
Although the dynamic viscosity μ is used in the cases above, it is related to the
kinematic viscosity which is presented in Eq. (5.13) by ν

kg/m ·

μ/ρ . Therefore the influ-
ence of the diffusion term is represented by the variation in the dynamic viscosities in
Case 1 and Case 2. Based on a CFD simulation, Fig. 5.7 shows the velocity profiles
at different downstream locations measured by the diameter distances from the inlet

=