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Ch 12: Fluid Mechanics
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 12: Fluid MechanicsProblem 42
Chapter 12, Problem 42

BIO. Artery Blockage. A medical technician is trying to determine what percentage of a patient's artery is blocked by plaque. To do this, she measures the blood pressure just before the region of blockage and finds that it is 1.20×104 Pa, while in the region of blockage it is 1.15×104 Pa. Furthermore, she knows that blood flowing through the normal artery just before the point of blockage is traveling at 30.0 cm/s, and the specific gravity of this patient's blood is 1.06. What percentage of the cross-sectional area of the patient's artery is blocked by the plaque?

Verified step by step guidance
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First, understand that the problem involves fluid dynamics, specifically the application of Bernoulli's equation and the continuity equation to determine the blockage in an artery.
Convert the velocity of blood flow from cm/s to m/s for consistency in units. Since 1 cm = 0.01 m, the velocity is 0.30 m/s.
Use Bernoulli's equation, which states that the sum of the pressure energy, kinetic energy per unit volume, and potential energy per unit volume is constant along a streamline. The equation is: P+12ρv2+ρgh=constant, where P is the pressure, ρ is the density, v is the velocity, g is the acceleration due to gravity, and h is the height. Assume height changes are negligible.
Calculate the density of blood using its specific gravity. Specific gravity is the ratio of the density of a substance to the density of water. Since the specific gravity is 1.06, the density of blood is ρ=1.06×1000=1060 kg/m³.
Apply the continuity equation, which states that the product of cross-sectional area and velocity is constant for incompressible flow: A1v1=A2v2. Use Bernoulli's equation to find the velocity in the blocked region and then solve for the change in cross-sectional area to find the percentage of blockage.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Bernoulli's Principle

Bernoulli's Principle states that in a fluid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or potential energy. This principle is crucial for understanding how changes in blood pressure and velocity relate to the narrowing of an artery due to plaque. It helps explain the pressure difference observed before and within the blockage.
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Continuity Equation

The Continuity Equation in fluid dynamics asserts that the mass flow rate must remain constant from one cross-section of a pipe to another. For incompressible fluids like blood, this means that the product of cross-sectional area and velocity is constant. This concept is essential for determining how the velocity of blood changes as the artery narrows due to plaque buildup.
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Specific Gravity

Specific gravity is the ratio of the density of a substance to the density of a reference substance, typically water for liquids. In this context, the specific gravity of blood (1.06) is used to calculate its density, which is necessary for applying Bernoulli's equation and understanding the dynamics of blood flow through the artery. It provides a basis for comparing the blood's properties to those of water.
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Related Practice
Textbook Question

A small circular hole 6.00 mm in diameter is cut in the side of a large water tank, 14.0 m below the water level in the tank. The top of the tank is open to the air. Find (a) the speed of efflux of the water and (b) the volume discharged per second.

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Textbook Question

Home Repair. You need to extend a 2.50-inch-diameter pipe, but you have only a 1.00-inch-diameter pipe on hand. You make a fitting to connect these pipes end to end. If the water is flowing at 6.00 cm/s in the wide pipe, how fast will it be flowing through the narrow one?

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Textbook Question

A shower head has 20 circular openings, each with radius 1.0 mm. The shower head is connected to a pipe with radius 0.80 cm. If the speed of water in the pipe is 3.0 m/s, what is its speed as it exits the shower-head openings?

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Textbook Question

A soft drink (mostly water) flows in a pipe at a beverage plant with a mass flow rate that would fill 220 0.355-L cans per minute. At point 2 in the pipe, the gauge pressure is 152 kPa and the cross-sectional area is 8.00 cm2. At point 1, 1.35 m above point 2, the cross-sectional area is 2.00 cm2. Find the (b) volume flow rate. (c) flow speeds at points 1 and 2.

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Textbook Question

Water is flowing in a pipe with a varying cross-sectional area, and at all points the water completely fills the pipe. At point 1 the cross-sectional area of the pipe is 0.070 m2, and the magnitude of the fluid velocity is 3.50 m/s. (c) Calculate the volume of water discharged from the open end of the pipe in 1.00 hour.

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Textbook Question

At one point in a pipeline the water's speed is 3.00 m/s and the gauge pressure is 5.00×104 Pa. Find the gauge pressure at a second point in the line, 11.0 m lower than the first, if the pipe diameter at the second point is twice that at the first.

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