Question

Assuming that blood is an ideal fluid, calculate the critical flow rate at which turbulence is a certainty in the aorta. Take the diameter of the aorta to be 2.50 cm.

A) (1.27 m/s)
B) (2.54 m/s)
C) (3.81 m/s)
D) (5.08 m/s)

Answer 1

The critical flow rate for turbulence onset in the aorta, assuming ideal blood fluid, is approximately
`\(0.3019 \, \text{m/s}\),` closest to option (A)
`\(1.27 \, \text{m/s}\).`

The critical flow rate for the onset of turbulence in a cylindrical pipe can be estimated using the Reynolds number. The Reynolds number (Re) is given by the formula:

`\[ \text{Re} = (\rho \cdot v \cdot D)/(\mu) \]`

where:

-
`\( \rho \)` is the density of the fluid (blood),

-
`\( v \)` is the velocity of the fluid,

-
`\( D \)`is the diameter of the pipe (aorta),

-
`\( \mu \)` is the dynamic viscosity of the fluid.

For the onset of turbulence, the critical Reynolds number
`(Re\_critical)` is typically around 2000.

Solving for velocity
`(\(v\)),`we get:

`\[ v = \frac{Re_{\text{critical}} \cdot \mu}{\rho \cdot D} \]`

Given that blood has a density of about
`\(1.06 \, \text{g/cm}^3\)` and a dynamic viscosity of about
`\(0.04 \, \text{g/(cm}\cdot\text{s)}\),` and the diameter of the aorta is
`\(2.50 \, \text{cm}\),` we can substitute these values into the formula:

`\[ v = (2000 \cdot 0.04)/(1.06 \cdot 2.50) \]`

Now, let's calculate this:

`\[ v \approx (80)/(2.65) \approx 30.19 \, \text{cm/s} \]`

Now, we'll convert this velocity from centimeters per second to meters per second (since the given answer choices are in m/s):

`\[ v \approx 0.3019 \, \text{m/s} \]`

Among the given options, the closest value is
`(A) \(1.27 \, \text{m/s}\).`