Page 103 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

80 BIOMECHANICS OF THE HUMAN BODY

and for the y direction one obtains
∂ ( P + vρ ′ + gzρ ) = 0
2
∂y (3.34)

Integration yields
2
P + ρ gz + ρ v′ = constant
(3.35)
As v′ must vanish near the wall, the values of P + rgz will be larger near the wall. That implies that,
in a turbulent boundary layer, the pressure does not change hydrostatically (is not height or depth
dependent), as is the case of laminar flow. Equation (3.35) implies that the pressure is not a function
of y, and thus,
∂P ∂ ⎛ dU ⎞
= ⎜ μ − uv ⎟ (3.36)
ρ
′′
y
∂x ∂ ⎝ dy ⎠
Since P is independent of y, integration in y yields
dU ∂ P
μ − ρuv ′′ = y + C 1 (3.37)
dy x ∂
where C = t is the shear stress near the wall.
1 0
We see that in addition to the convective, pressure, and viscous terms, we have an additional term,
which is the gradient of the nonlinear term ru′v′, which represents the average transverse transport
of longitudinal momentum due to the turbulent fluctuations. It appears as a pseudo-stress along with
the viscous stress m∂U/∂y, and is called the Reynolds stress. This term is usually large in most
turbulent shear flows (Lieber and Giddens, 1988).

3.3.6 Flow in Curved Tubes
The arteries and veins are generally not straight uniform tubes but have some curved structure, espe-
cially the aorta, which has a complex three-dimensional curved geometry with multiplanar curvature.
To understand the effect of curvature on blood flow, we will discuss the simple case of steady laminar
flow in an in-plane curved tube (Fig. 3.8). When a steady fluid flow enters a curved pipe in the
horizontal plane, all of its elements are subjected to a centripetal acceleration normal to their
original directions and directed toward the bend center. This force is supplied by a pressure gradient
in the plane of the bend, which is more or less uniform across the cross section. Hence, all the fluid
elements experience approximately the same sideways acceleration, and the faster-moving elements
with the greater inertia will thus change their direction less rapidly than the slower-moving ones. The
net result is that the faster-moving elements that originally occupy the core fluid near the center of
the tube are swept toward the outside of the bend along the diametrical plane, and their place is taken
by an inward circumferential motion of the slower moving fluid located near the walls. Consequently,
the overall flow field is composed of an outward-skewed axial component on which is superimposed
a secondary flow circulation of two counterrotating vortices.
The analytical solution for a fully developed, steady viscous flow in a curved tube of circular
cross section was developed by Dean in 1927, who expressed the ratio of centrifugal inertial forces
to the viscous forces (analogous to the definition of Reynolds number Re) by the dimensionless
Dean number,
r
De = Re (3.38)
R curve