Page 98 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

PHYSICAL AND FLOW PROPERTIES OF BLOOD 75

FIGURE 3.4 Cross section of artery showing the change of volume and
volume flow.

∂A ∂
+ ( Au) = 0 (3.12)
∂t ∂z
∂u ∂u 1 ∂P
+ u + = 0 (3.13)
∂t ∂z ρ ∂z
where A(z) = tube cross-sectional area
u(z) = uniform axial velocity of blood
P(z) = pressure in the tube
r = fluid viscosity
z = axial coordinate
t = time
The elasticity of the tube wall can be prescribed by relationship between the local tube pressure and
the cross-sectional area, P(A).
We further assume that the perturbation is small, while the wave length is very large compared
with the tube radius. Thus, the nonlinear inertia variables are negligible and the linearized conserva-
tion equation of mass and momentum become, respectively,
∂A ∂u
=−A (3.14)
∂t ∂z
∂u 1 ∂P
=− (3.15)
∂t ρ ∂z

Next, we differentiate Eq. (3.14) with respect to t and Eq. (3.15) with respect to z, and upon adding
the results we obtain the following wave equation:

2
2
2
∂ P ρ ∂ ∂ P −2 ∂ P
A
2 = 2 = c 2 (3.16)
∂z A ∂P ∂t ∂t