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4 Fluid Mechanics Applied to Biosystems
To integrate over height in the atmosphere, we need to know how air density depends
on air pressure. For air modeled as an ideal gas at constant temperature, this would
be ρ a = (m/k B T )p, where m is the average mass of a molecule in the air. To
account for the decrease of the acceleration of gravity with height, we use Newton’s
Law of Gravity to express
g(z) = g 0
1
(1 + z/R)
2
.
(4.4)
If the temperature were constant, then integration would give
p = p o exp
−
mg 0
k B T
z
1 + z/R
,
(4.5)
where R is the radius of the Earth, z is the height above the Earth’s surface, and p o
is the atmospheric pressure at sea level. A similar expression applies for the density
of air. But the temperature is not constant. A plot of the temperature variation with
height is shown in Fig. 16.1. As T (z) is rather complicated, numerical integration
of the relation dp/dz to find the pressure with height is used. Direct measure of
pressure above the Earth can be performed by weather balloons up to 40 km (in the
stratosphere). The Earth’s atmosphere makes a transition to the solar environment at
about 100 km. Above about 120 km, solar wind dominates the temperature, which
can reach greater than 1000 ◦ C above 150 km. However, the atmosphere is so thin
there that little energy is available to transfer as heat to astronauts who leave their
spaceship (as ‘space walkers’).
Our inner ear is sensitive to air pressure when the Eustachian tube is not open.
Airplane travelers are quite aware of this sensitivity. It is caused by the fact that
the atmospheric pressure diminishes with height. The air behind the ear drums are
connected to the pharynx (upper throat) by the Eustachian tubes. When the tube is
open, the pressure of the air in the inner ear is the same as atmospheric. However, the
tube is normally closed until you strongly swallow. If you have not, then going up in
the airplane makes a difference in pressure between the middle ear and the inner ear,
causing the ear drum to distort outward. As a protective mechanism for the drum,
nerves associated with the drum send a pain signals to your brain. Swallowing can
open the Eustachian tubes from the nasal cavities to the middle ears, letting air move
to equalize pressure on the eardrum.
For water in the oceans, the density ρ is nearly constant. (The increase of water
density from the surface to a depth of 1000 m is only about 0.3%.) If taken to have
a constant density, the Eq. (4.3) for the pressure a distance |z| below the surface of
a fluid integrates to
p = p o + ρg |z| .
(4.6)
Divers must take caution to let air out of their lungs as they rise from the depths, so
that alveoli in the lungs are not ruptured.
4 Fluid Mechanics Applied to Biosystems
To integrate over height in the atmosphere, we need to know how air density depends
on air pressure. For air modeled as an ideal gas at constant temperature, this would
be ρ a = (m/k B T )p, where m is the average mass of a molecule in the air. To
account for the decrease of the acceleration of gravity with height, we use Newton’s
Law of Gravity to express
g(z) = g 0
1
(1 + z/R)
2
.
(4.4)
If the temperature were constant, then integration would give
p = p o exp
−
mg 0
k B T
z
1 + z/R
,
(4.5)
where R is the radius of the Earth, z is the height above the Earth’s surface, and p o
is the atmospheric pressure at sea level. A similar expression applies for the density
of air. But the temperature is not constant. A plot of the temperature variation with
height is shown in Fig. 16.1. As T (z) is rather complicated, numerical integration
of the relation dp/dz to find the pressure with height is used. Direct measure of
pressure above the Earth can be performed by weather balloons up to 40 km (in the
stratosphere). The Earth’s atmosphere makes a transition to the solar environment at
about 100 km. Above about 120 km, solar wind dominates the temperature, which
can reach greater than 1000 ◦ C above 150 km. However, the atmosphere is so thin
there that little energy is available to transfer as heat to astronauts who leave their
spaceship (as ‘space walkers’).
Our inner ear is sensitive to air pressure when the Eustachian tube is not open.
Airplane travelers are quite aware of this sensitivity. It is caused by the fact that
the atmospheric pressure diminishes with height. The air behind the ear drums are
connected to the pharynx (upper throat) by the Eustachian tubes. When the tube is
open, the pressure of the air in the inner ear is the same as atmospheric. However, the
tube is normally closed until you strongly swallow. If you have not, then going up in
the airplane makes a difference in pressure between the middle ear and the inner ear,
causing the ear drum to distort outward. As a protective mechanism for the drum,
nerves associated with the drum send a pain signals to your brain. Swallowing can
open the Eustachian tubes from the nasal cavities to the middle ears, letting air move
to equalize pressure on the eardrum.
For water in the oceans, the density ρ is nearly constant. (The increase of water
density from the surface to a depth of 1000 m is only about 0.3%.) If taken to have
a constant density, the Eq. (4.3) for the pressure a distance |z| below the surface of
a fluid integrates to
p = p o + ρg |z| .
(4.6)
Divers must take caution to let air out of their lungs as they rise from the depths, so
that alveoli in the lungs are not ruptured.