Blood Flow in Arteries and Veins - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 161

x

@ v 1

μ

@ p

y 2 ρ

g y c 1

x 5

1

@ @ μ

x 2
2 1

1
μ

@ p

y 2 ρ

v ð x Þ 5 g y c 1 x 1 c 2

@ μ

Using the boundary conditions of

v ð 0 Þ 5 0 and v ð 2 r 3 Þ 5 0

the solution of the velocity profile becomes

x 2
2 1

x

1
μ

@ p

y 2 ρ

r 3 ρ
μ

1
μ

@ p

v ð x Þ 5 g y g y 2

@ μ @ y

The solution for section 2 becomes more difficult because you would need to solve 2 coupled
partial differential equations ( u is a function of x/y and v is a function of x/y ).

2 u 2 u 2 u

@ u

u @

u

v @

u

w @

u

g x 2 @

p @

x 2 1 @ y 2 1 @

ρ

5 ρ

x 1 μ t 1 x 1 y 1 @ @ @ @ z @ @ @ @ z 2

2 v 2 v 2 v

@ v

@

u @

v

v @

v

w @

v

g y 2 @

p @

x 2 1 @ y 2 1 @

ρ

5 ρ

y 1 μ t 1 x 1 y 1 @ @ @ z @ @ @ @ z 2

u @

u

v @

u

52 @

p @

2 u

x 2 1 @

2 u

ρ

x 1 μ

x 1 y 2

@ @ y @ @ @

u @

v

v @

v

g y 2 @

p @

2 v

x 2 1 @

2 v

ρ

x 1

5 ρ

y 1 μ

@ @ y @ @ @ y 2

The easiest solution to the preceding two equations is through the use of numerical methods,
which will not be addressed here. Because we did not provide the exact details about the flow
properties, we will leave the discussion at this point. However, if you can make the assumption
that the x-direction velocity component does not vary much with x (i.e.,

@ u

0) and that the

@ x

@ v

y-direction velocity component does not vary much with y (i.e., 0), then this problem will

@ y

simplify significantly.

5.10 PULSATILE FLOWAND TURBULENCE

In most of the examples that we have used so far, we have always made the assumption
that blood flow is steady. Due to the cardiac cycle (Section 4.3), the blood flow rate that is
being ejected from the heart is not constant. In fact, the systemic blood flow mimics the
left ventricular pressure waveform in time ( Figure 5.22 ). Does this affect the calculations
that we have been carrying out?
One can argue that the steady flow assumption does or does not affect the calculations
that we have done so far. Because the flow is pulsatile, then the change in velocity with
regard to time cannot be zero (see any Navier-Stokes example where we ignored @ v
@ t ). In