The velocity, c, at which pressure pulses travel through arteries (pulse-wave velocity) is a function of the artery diameter, D,and wall thickness, h, the density of blood, p, and the modulus of elasticity, E, of the arterial wall. Determine a set of nondimensional parameters that can be used to study experimentally the relationship between the pulse-wave velocity and the variables listed. Form the nondimensional parameters by inspection.

To determine a set of nondimensional parameters that can be used to study the relationship between the pulse-wave velocity, c, and the variables D (artery diameter), h (wall thickness), ρ (density of blood), and E (modulus of elasticity of the arterial wall), we can use a method based on dimensional analysis, including the Buckingham π theorem. This theorem allows us to form dimensionless groups of parameters that can be used in experimental studies.

First, let's list the dimensions of the given variables: - Velocity, c: [L][T]^-1 (where L is length and T is time) - Diameter, D: [L] - Wall thickness, h: [L] - Density, ρ: [M][L]^-3 (where M is mass) - Modulus of elasticity, E: [M][L]^-1[T]^-2

We want to form dimensionless parameters or π groups. By inspection, we might notice that several potential dimensionless parameters can be constructed using these variables. One common dimensionless parameter for velocity in waves is the Mach number, but since we aren't dealing with compressibility effects here, we'll consider other potential groups:

1. \( \pi_1 = \frac{c}{\sqrt{\frac{E}{\rho}}} \) This can be considered a nondimensional form of the velocity, normalizing the pulse-wave velocity by a characteristic velocity obtained from the elasticity and density of the blood and arterial wall.

2. \( \pi_2 = \frac{D}{h} \) This represents the aspect ratio of the artery, comparing its diameter to its wall thickness.

If we look at the dimensions of each term in the first dimensionless parameter:

\( \pi_1 \) = [L][T]^-1 / \( \sqrt{\frac{[M][L]^-1[T]^-2}{[M][L]^-3}} \) = [L][T]^-1 / \( \sqrt{[L]^2[T]^-2} \) = [L][T]^-1 / [L][T]^-1 = 1 (dimensionless)

For the second parameter:

\( \pi_2 \) = [L] / [L] = 1 (dimensionless)

These nondimensional parameters allow us to study the effects of the physical properties and geometric dimensions on the pulse-wave velocity in a manner that doesn't depend on the specific units used to measure these quantities. By using these dimensionless parameters in experiments, we can compare results from different systems and scale the results to real-world applications.

Extra: In the field of fluid dynamics and in many other areas of physics and engineering, the concept of nondimensionalization is crucial. It allows researchers to derive universal laws that are not tied to a specific scale of length, time, or any other measurement unit.

The Buckingham π theorem provides a way to take a complex equation with several variables and reduce it to an equation with fewer dimensionless parameters. Being able to create experimental models becomes much easier when relationships between variables can be given in a nondimensional form because one can scale the results from the model (which might be much smaller or made of different materials) to the actual system being studied.

For students, understanding nondimensional parameters is useful because it deepens their comprehension of fundamental principles such as similarity and scaling. It also aids in making practical predictions about how different systems will behave based on experimental data, regardless of the size, speed, or other scale factors of the system in question.