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Sheet 1
Fluid flow
1. Water is held behind a dam of width, w, and a height, h. Determine the
resultant force exerted by the water dam. Find the average pressure.
2. An iceberg floating in seawater with a hidden volume under sea, is
extremely dangerous for ships. What fraction of the iceberg lies under the
water level? ρsw=1030kg/m3 , ρice=917kg/m3
3. Each second 5300 m3 of water flows down a 670m wide cliff at the top of
a certain falls. The water is approximately 2m deep as it reaches the cliff.
What is the speed of water at that instant?
4. A large storage tank, open at the top and filled with water, develops a
small hole in its side at a point 16m below the water level. If the rate of flow
from the leak is 2.5*10-3 m3/min. Determine the speed at which the water
leaves the hole and the diameter of the hole.
5. An airplane has a mass of 1.6*104 kg. During level flight the velocity of
air on the upper side of the wing is 60m/s while that on the lower is 45m/s.
What maximum load can the airplane carry, assuming that the wind
distribution above and below the wing follows the streamline laminar
distribution and the wing area is 50m2?
6. A Pitot tube is used to determine the velocity of air flow by measuring
the difference between the total pressure and the static pressure. If the fluid in
the tube is mercury (ρ’=13.60g/cm3) and if ∆H=5cm, find the speed the air
flow (ρair=1030kg/m3).
7. A Venturi tube is used to measure the flow rate of gasoline in m3/s, given
that the difference in the mercury arms of the manometer is 3.5cm, the radius
of the outlet tube is 1cm and that of the inlet is 2cm. The density of gasoline
is 700kg/m3 .
8. A hypodermic syringe contains a medicine with density of 1.05g/cm3.
The cylindrical barrel of the syringe has a c.s.a. of 2.5*10 -5m2 and that of the
needle is 10-8 m2. A force of magnitude 2N acts on the plunger, making the
medicine squirt horizontally into the blood stream with a pressure 1.5 Patm
What is the speed of medicine as it leaves the needle’s tip?
9. If the maximum velocity in a large blood vessel section is 5m/s, estimate
the force acting on a spherical erythrocyte falling across from the axial
stream to the vessel inner wall. The radius of the falling erythrocyte is
0.01μm and blood density is 1.5g/cm3.
10. Find the force components in both x and y directions exerted on the inner
wall of the bent pipe shown in terms of angle α , fluid density ρ, cross
sectional areas, pressures and velocities at the entrance and exit of the bent
pipe shown fig.(1).
A1,P1
α
A2,P2
fig.(1)
11. By equating the driving force to the rate of outflow of
momentum deduce a formula that expresses the velocity at the
interface between pipe1 and pipe2 shown in fig.(2).
A2, v2, P2
A1, v1, P1
fig.(2)
Sheet 2
Physics of blood flow
Circuit Topology
1. What is the minimum number of arterioles branching from an artery of
radius 2mm satisfying each of the following conditions:
i. the diameter of a single arteriole is approximately 0.16mm.
ii. it is required that the velocity in the arterioles drops to half its value in
the artery.
2.i. The velocity of the fluid flowing in a pipe can be measured using
Venturi tube. Show how it can be used to measure blood flow rate through
arteries. Illustrate your answer using drawing.
ii. If the radius of the main Venturi tube is 1cm and that of the constriction
is 1mm, derive the constant expressing the relation between the two areas, the
gravity constant, and the density of mercury that should be handed over with
the Venturi tube for velocity and flow rate measurement purposes.
3.i. Explain how erythrocytes are concentrated towards the center of the
blood vessels.
ii. Compare between the velocities of two microorganisms, one of them is
moving at a distance 1.5 mm from the axis and another along the axis of an
artery whose diameter is 4 mm
4. i. Each ventricle of the heart pumps 5.6 liters of blood every minute
for a normal activity person. If the arterial pressure pulsates between (12080)mm Hg and the average pressure is one third the amplitude of the pulse
towards the diastolic pressure value, find an estimated value of the total
peripheral resistance (TPR)encountered through blood circulation.
ii. Knowing that this quantity of blood (flow rate) is distributed between
the legs and other body systems simultaneously. Determine the average
peripheral resistance, Rl, of the legs that enable them to receive about 0.52
liters/min of blood. Find the equivalent peripheral resistance of the body
systems, RS . Find power dissipated in legs.
5. Blood can be considered as a non-Newtonian fluid and thus Poiseuille’s
equation does not apply exactly to blood flow. On the other hand it is still a
reasonable approximation for the study of blood flow in vessels. Discuss the
main drawbacks that restrict our application of such equation.
6. Use Poiseuille’s equation to deduce the percentage change in radius , r,
of an artery that dilates in order to decrease its peripheral resistance, Rp , by
25% of its original value. Deduce the change in the blood flow rate, into this
artery. Consider that the pressure difference across its ends, ΔP, its length, l,
and blood viscosity, η.
7. i. If the blood flow rate fed to a certain body tissue is required to
increase from 0.4 l/min to 0.7 l/min. Calculate the change in its peripheral
resistance such that the average pressure, across its end arterioles, remain
constant.
ii. If this tissue is modeled as a flexible tube of radius 1.5mm, what would
be the change needed in the radius to cause this increase. Hence, determine
the percentage change in the maximum velocity of blood through it.
8. For a running adult the blood flow rate increases from 5l/min to 8l/min.
Assuming that blood pressure is maintained constant, what change must
occur in the overall resistance of the blood?
9. Assume that three body organs are in cascade. The blood quantity fed
into the first organ is 2l/min. If the pressure drop across the second organ is
20mmHg , find its approximate peripheral resistance, R p2 .
10. If the total blood flow from the heart is Q. Assume that the arteries
feed three main parts of the body having peripheral resistances of R p1, Rp2
and Rp3 respectively. What is the share of each organ of blood if they are fed
simultaneously with the same pressure drop across their end arterioles and
venules.
11. Discuss the effect of the dilating of an artery such that its radius is
increased by one third its original radius, on both the flow rate and the flow
velocity.
12. Estimate the change in the pressure gradient and that in the volume
flow rate if: a. the viscosity coefficient decreases by 2%.
b. the radius of the vessel decreases by 2%.
Hence deduce percentage change in the power dissipated per unit length.
13. If the radius of a blood vessel is 0.5cm, what is the pressure gradient
needed to cause the flow of 1.2 liters/min of blood through it? Assume
the viscosity coefficient is 0.02poise.
14.The heart pumps 70 cm3 of blood 72 times per minute at an average
pressure of 90mmHg, into a 2.1 cm radius aortic section. What is its
maximum velocity? Assume that blood density is 1.01g/cm3 and
viscosity 0.04poise. Is the flow turbulent or laminar?
15. Give an R-C circuit section that models the blood flow in the aortic
section given in problem 14. Consider the pressure difference between
systole and diastole is 40mmHg.
16. Crude Petroleum is transported from the Red sea coast to the
Medeteranian coast via 1800 km pipe-lines. The working pressure
difference between the two end points is 107 N/m2 and the viscosity
coefficient of petroleum is 0.25 kg/m.s. If the maximum storage-tank
capacitance is 48*103 m3 and time required for the transportation must
not to exceed 12 hrs., find the radius of the pipes.