Question

In the chapter on fluid mechanics, Bernoulli’s equation for the flow of incompressible fluids was explained in terms of changes affecting a small volume dV of fluid. Such volumes are a fundamental idea in the study of the flow of compressible fluids such as gases as well. For the equations of hydrodynamics to apply, the mean free path must be much less than the linear size of such a volume, a≈dV(1/3). For air in the stratosphere at a temperature of 220 K and a pressure of 5.8 kPa, how big should a be for it to be 100 times the mean free path? Take the effective radius of air molecules to be 1.88×10⁻11m, which is roughly correct for N2.

a) 2.17 × 10⁻4 m
b) 5.62 × 10⁻4 m
c) 1.34 × 10⁻3 m
d) 3.09 × 10⁻3 m

Answer 1

To find the value of 'a' for air in the stratosphere to be 100 times the mean free path, we can use the equation a ≈ dV^(1/3) and given values of temperature, pressure, and effective radius of air molecules. The value of 'a' is approximately 3.09 x 10^(-3) m.

Step-by-step explanation:

Bernoulli's equation is a fundamental concept in fluid mechanics, which explains the relationship between pressure and velocity in fluids. In the case of compressible fluids, such as gases, the mean free path must be much less than the linear size of a small volume of fluid for the equations of hydrodynamics to apply. The mean free path is the average distance between collisions, and in this case, we need to calculate how big the linear size 'a' should be for it to be 100 times the mean free path.

Using the equation a ≈ dV^(1/3), where dV is the small volume of fluid, and the given effective radius of air molecules, we can calculate the value of 'a'.

Given:

Temperature (T) = 220 K

Pressure (P) = 5.8 kPa

Effective radius of air molecules (r) = 1.88 x 10^(-11) m

Mean free path = 100 * 4.81 x 10^(-8) m (from a previous example)

Substituting these values into the equation, we can solve for 'a'.

The value of 'a' is approximately 3.09 x 10^(-3) m.