Fundamentals of Fluid Dynamics - Computational Fluid and Particle Dynamics in the Human Respiratory System - page 104

and normal viscous stress components τ xx as

σ xx =− p + τ xx

The formulation of the appropriate stress-strain relationships for a Newtonian fluid
is based on Newton's law of viscosity and given as:

μ ∂u
∂y

2 μ ∂u
∂x

τ xx = τ yx = (5.8)

where μ is the molecular dynamic viscosity that relates stresses to linear deformation
rate. For the x -velocity component we combine Eqs. (5.6), (5.7), and (5.8), and cancel
out the volume term xyz , which gives:

∂x + μ ∂ 2 u

+ ρ F B

∂ 2 u
∂y 2

ρ Du

∂p

Dt =− ∂x 2 +

(5.9)

The acceleration term is the total derivative of u (the rate of change of velocity of a
moving fluid particle) which can be expressed in 2D as

Du
Dt =

∂u
∂t +

u ∂u v ∂u

∂y ∂x +

(5.10)

Putting Eq. (5.10) into (5.9), and rearranging by dividing through by ρ , the balance
of momentum equation in the x -direction becomes:

ν ∂ 2 u

F B

∂ 2 u
∂y 2

∂u
∂t +

u ∂u v ∂u

1
ρ

∂p
∂x + ∂x + ∂y =− ∂x 2 +

+ (5.11)

and its corresponding y -direction component is:

∂y + ν ∂ 2 v

F B

∂v
∂t +

u ∂v v ∂v

1
ρ

∂p ∂ 2 v

∂y 2

∂x + ∂y =− ∂x 2 +

+ (5.12)

Note that ν is the kinematic viscosity which is related to the dynamic viscosity, μ as
ν

μ
ρ .
Physical Interpretation The negative product of mass and acceleration may be
interpreted as the so-called inertia force. The balance of momentum equation is
then rewritten in a form that represents the balance of four forces (inertial, pressure,
viscous, and body force) acting on a body. Let us consider the y -momentum equation
and its application in the vertical respiratory flows in the tracheal region. Each force is
reflected by different terms in the equation where the inertial force appears through the
local acceleration and convection term; the pressure force via the pressure gradient;
and the viscous force via the diffusion term. These terms are clearly identified in
Eq. (5.13).

=

F B
body force

ν ∂ 2 v ν ∂ 2 v

∂y 2
diffusion

∂v
∂t

u ∂v v ∂v

∂y
convection

∂p
∂ y
pressure gradient

1
ρ

+

∂x +

=− +

∂x 2 +

+ (5.13)

local acceleration