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4 Fluid Mechanics Applied to Biosystems
4.5 Estimate the change of density of water from the surface to one kilometer
deep in the ocean. By what percent would this increase in density increase a fish’s
buoyancy compared to the buoyancy near the surface of the ocean?
4.6 At a depth of 10 m in the ocean, what will be the buoyant force on a person who
weighs 40 kg and has a body volume of 40 l?
4.7 Suppose you need to quickly determine the viscosity of a jar of blood plasma.
You decide to use the drag on a falling bead. You measure that the bead has a
diameter of 2.075 cm and a mass of 4.80 gm. You find that it takes 10.0 s for the
bead to fall 2.00 cm through the plasma. Calculate the plasma’s viscosity.
4.8 If the pressure difference from one end of an artery to the other end were
10 N/m 2 , the artery is 0.5 cm in diameter, with a length of 5 cm, what would be
the flow rate (in cubic centimeters per second) in the artery?
4.9 How much power would the heart expend to push 100 cubic centimeters of
blood per second through an artery of radius 0.5 cm and length 10 cm with a pressure
difference from on end to the other of 30 mmHg?
4.10 A certain artery (A) of diameter 0.30 cm bifurcates into two arteries (B and C)
of diameters 0.24 cm and 0.18 cm respectively. The speed of the blood in A along
its axis was found to be 1.23 cm/s and the speed in B to be 0.86 cm/s. What speed
would be measured in C along its axis? (Include the effects of blood viscosity.)
4.11 Estimate the number of capillaries in your body. Start with a velocity flow in
the aorta of 110 cm/s and an inner diameter of 20 mm. Use a capillary diameter of
6 μm and a blood speed in the capillary of about 0.4 mm/s.
4.12 Suppose an artery of length 5 cm and diameter 2.6 cm expands in volume
under an increase in pressure of 50 mmHg by 10%. Defining the capacitance of
a given length of artery as the extra volume it can contain per unit pressure change,
what is the capacitance per unit length of the given artery? The inductance per unit
length of an artery is the ratio of the extra pressure due to a change in flow per unit
change in flow with time. From Newton’s law of inertia, this will be the density of
the blood per cross-sectional area of the artery. Find a vibrational frequency of the
artery of length 10 cm, using the formula for the natural vibrational frequency of a
circuit with a capacitor and an inductor in series.
4.13 The speed of an eagle with spread wings and drifting horizontally was
measured 34.0 m/s. The shape of the wings made the air on top move at 35.6 m/s.
Estimate the pressure difference from bottom to top of the wing. If the bird weighs
19.2 N, and a third of the wing surface area has this effective pressure difference
acting, what is the area of the bird’s wings?
4.14 Blood speed in humans is generally less than 50 cm/s. Compare the kinetic
energy per unit volume of blood moving with this speed to the energy stored per
volume due to the pressure difference between your head and foot, when standing.
4 Fluid Mechanics Applied to Biosystems
4.5 Estimate the change of density of water from the surface to one kilometer
deep in the ocean. By what percent would this increase in density increase a fish’s
buoyancy compared to the buoyancy near the surface of the ocean?
4.6 At a depth of 10 m in the ocean, what will be the buoyant force on a person who
weighs 40 kg and has a body volume of 40 l?
4.7 Suppose you need to quickly determine the viscosity of a jar of blood plasma.
You decide to use the drag on a falling bead. You measure that the bead has a
diameter of 2.075 cm and a mass of 4.80 gm. You find that it takes 10.0 s for the
bead to fall 2.00 cm through the plasma. Calculate the plasma’s viscosity.
4.8 If the pressure difference from one end of an artery to the other end were
10 N/m 2 , the artery is 0.5 cm in diameter, with a length of 5 cm, what would be
the flow rate (in cubic centimeters per second) in the artery?
4.9 How much power would the heart expend to push 100 cubic centimeters of
blood per second through an artery of radius 0.5 cm and length 10 cm with a pressure
difference from on end to the other of 30 mmHg?
4.10 A certain artery (A) of diameter 0.30 cm bifurcates into two arteries (B and C)
of diameters 0.24 cm and 0.18 cm respectively. The speed of the blood in A along
its axis was found to be 1.23 cm/s and the speed in B to be 0.86 cm/s. What speed
would be measured in C along its axis? (Include the effects of blood viscosity.)
4.11 Estimate the number of capillaries in your body. Start with a velocity flow in
the aorta of 110 cm/s and an inner diameter of 20 mm. Use a capillary diameter of
6 μm and a blood speed in the capillary of about 0.4 mm/s.
4.12 Suppose an artery of length 5 cm and diameter 2.6 cm expands in volume
under an increase in pressure of 50 mmHg by 10%. Defining the capacitance of
a given length of artery as the extra volume it can contain per unit pressure change,
what is the capacitance per unit length of the given artery? The inductance per unit
length of an artery is the ratio of the extra pressure due to a change in flow per unit
change in flow with time. From Newton’s law of inertia, this will be the density of
the blood per cross-sectional area of the artery. Find a vibrational frequency of the
artery of length 10 cm, using the formula for the natural vibrational frequency of a
circuit with a capacitor and an inductor in series.
4.13 The speed of an eagle with spread wings and drifting horizontally was
measured 34.0 m/s. The shape of the wings made the air on top move at 35.6 m/s.
Estimate the pressure difference from bottom to top of the wing. If the bird weighs
19.2 N, and a third of the wing surface area has this effective pressure difference
acting, what is the area of the bird’s wings?
4.14 Blood speed in humans is generally less than 50 cm/s. Compare the kinetic
energy per unit volume of blood moving with this speed to the energy stored per
volume due to the pressure difference between your head and foot, when standing.