19. Fluid Mechanics

(I) Calculate the average speed of blood flow in the major arteries of the body, which have a total cross-sectional area of about 2.0 cm². Use the data of Example 13–14.

A drinking fountain shoots water about 12 cm up in the air from a nozzle of diameter 0.60 cm (Fig. 13–67). The pump at the base of the unit (1.1 m below the nozzle) pushes water into a 1.2-cm-diameter supply pipe that goes up to the nozzle. What gauge pressure does the pump have to provide? Ignore the viscosity; your answer will therefore be an underestimate.

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(II) Water at a gauge pressure of 3.8 atm at street level flows into an office building at a speed of 0.78 m/s through a pipe 5.0 cm in diameter. The pipe tapers down to 2.8 cm in diameter by the top floor, 16 m above (Fig. 13–57), where the faucet has been left open. Calculate the flow velocity and the gauge pressure in the pipe on the top floor. Assume no branch pipes and ignore viscosity.

(II) Water at a gauge pressure of 3.8 atm at street level flows into an office building at a speed of 0.78 m/s through a pipe 5.0 cm in diameter. The pipe tapers down to 2.8 cm in diameter by the top floor, 16 m above (Fig. 13–57), where the faucet has been left open. Calculate the gauge pressure in the pipe on the top floor. Assume no branch pipes and ignore viscosity.

(III) (a) Show that the flow speed measured by a venturi meter (see Fig. 13–32) is given by the relation $v_1=A_2\sqrt{\frac{2(P_1-P_2)}{\rho (A_1^2-A_2^2)}}$. (b) A venturi meter is measuring the flow of water; it has a main diameter of 3.5 cm tapering down to a throat diameter of 1.0 cm. If the pressure difference is measured to be 18 mm-Hg, what is the speed of the water entering the venturi throat?

(II) Suppose the top surface of the vessel in Fig. 13–58 is subjected to an external pressure P₂ in addition to atmospheric pressure P₀. If P₂ = 0.85 atm and y₂ - y₁ = 2.4 m, determine v₁ for water.

(I) A gardener feels it is taking too long to water a garden with a 3/8 -in. - diameter hose. By what factor will the time be cut using a 5/8 - in. - diameter hose instead? Assume nothing else is changed.

You are watering your lawn with a hose when you put your finger over the hose opening to increase the distance the water reaches. If you are holding the hose horizontally, and the distance the water reaches increases by a factor of 4, what fraction of the hose opening did you block?

If cholesterol buildup reduces the diameter of an artery by 25%, by what % will the blood flow rate be reduced, assuming the same pressure difference?

A pump supplies water to a 1.59-cm inner diameter hose that tapers down to a 0.953-cm-diameter nozzle. The nozzle is aimed so water comes out at a 45° angle and lands 3.0 m away. The nozzle is 0.60 m above ground level, and the pump output is essentially at ground level. What pressure is supplied by the pump?

Estimate the air pressure inside a category 5 hurricane, where the wind speed is 300 km/h (Fig. 13–56).

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(II) A viscometer consists of two concentric cylinders, 10.20 cm and 10.60 cm in diameter. A liquid fills the space between them to a depth of 15.0 cm. The outer cylinder is fixed, and a torque of 0.024 m N keeps the inner cylinder turning at a steady rotational speed of 57 rev/min. What is the viscosity of the liquid?

(I) Engine oil (assume SAE 10, Table 13–3) passes through a fine 1.80-mm-diameter tube that is 10.2 cm long. What pressure difference is needed to maintain a flow rate of 5.8 mL/min?

(II) Poiseuille’s equation does not hold if the flow velocity is high enough that turbulence sets in. The onset of turbulence occurs when the Reynolds number, Re, exceeds approximately 2000. Re is defined as $Re=\frac{2\bar{v}r\rho }{\eta }$ where $\bar{v}$ is the average speed of the fluid, ρ is its density, is its viscosity, and r is the radius of the tube in which the fluid is flowing. Determine if blood flow through the aorta is laminar or turbulent when the average speed of blood in the aorta (r = 1.2 cm) during the resting part of the heart’s cycle is about 35 cm/s.

(III) A long vertical hollow tube with an inner diameter of 1.00 cm is filled with SAE 10 motor oil. A 0.900-cm-diameter, 30.0-cm-long 150-g rod is dropped vertically through the oil in the tube. What is the maximum speed attained by the rod as it falls?