Conservation Laws - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 54

9 1 mmHg

133

879 kg
m 3

81 m
s 2

1m
100 cm

p 2 5 : : ð Þ :

751 17 mmHg 1 45 cm 5 780 28 mmHg

: 32 Pa

p blood 5 p 2 2 ρ blood g ð z p 2 z 2 Þ

9

1050 kg
m 3

81 m
s 2

1m
100 cm

1 mmHg
133

p blood 5 780 : 28 mmHg : ð 25 cm Þ 760 : 96 mmHg

2 5

:

32 Pa

The following example illustrates a few principles that should be remembered when
working on fluid statics problems. The first is that the pressure at an interface between
two different fluids is always the same. This is how we can equate the pressure at location 1
and 2, 2 and P . If the fluid is continuous (same density), any location at the same height
has the same pressure. Therefore, you can move around the bends without calculating
each pressure change around those bends. Also, the dashed line through fluid one has the
same pressure as location 2. Finally, pressure should increase as the elevation decreases
and pressure should decrease as the elevation increases.
There are many biofluid problems in which density will vary. These types of fluids are
compressible fluids and the density function would need to be stated within the problem.
The density function would need to be given as a function of pressure and/or height.
Once this function is known, then Equation 3.10 can be used to solve for the pressure dis-
tribution throughout the fluid. As an example, the density of most gases depends on the
pressure and the temperature of the system. The ideal gas law represents this relationship
and should be familiar to most students. The ideal gas law states that

p

5 ρ

RT ð 3 : 12 Þ

where R is the universal gas constant (8.314 J/(g mol K)) and T is the absolute temperature
(in Kelvin). The problem with using this relationship is that it introduces a new variable,
T , into the equation, which may vary with height as well. We will typically make the
assumption in this textbook that temperature fluctuations within the body can be
neglected. This means that for humans, the temperature will be assumed to be 310.15 K
(37 C), unless stated otherwise. Using the ideal gas law, the pressure variation in a com-
pressible fluid, with a constant temperature is

dp
dz 52 ρ

p
RT g z

g z

5

dp
p 52

g z
RT dz

RT z

p 1
p 0 52

g z

z 1

ln ð p Þ

z 0

52

p 1
p 0

g z
RT ð

ln ð p 1 Þ 2 ln ð p 0 Þ 5 ln z 1 z 0 Þ

2