What is the pressure drop (in mm Hg) in the blood as it passes through a capillary `1mm` long and `2mu m` in radius if the speed of the blood through the centre of the capillary is `0.66mm/s`? (The viscosity of whole blood is `4xx10^(-3) PI)`

Explanation

To solve the problem of finding the pressure drop in the blood as it passes through a capillary, we can use the formula derived from Poiseuille's law for viscous flow in a cylindrical pipe. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the given values
- Length of the capillary (L) = 1 mm = \(1 \times 10^{-3}\) m
- Radius of the capillary (R) = 2 µm = \(2 \times 10^{-6}\) m
- Speed of blood (V_max) = 0.66 mm/s = \(0.66 \times 10^{-3}\) m/s
- Viscosity of blood (η) = \(4 \times 10^{-3} \, \text{Pa.s}\)

### Step 2: Use the formula for pressure drop
The pressure drop (ΔP) across a length of capillary can be calculated using the formula:
\[ \Delta P = \frac{8 \eta L V_{max}}{R^2} \]
This formula is derived from Poiseuille's law for laminar flow in a cylindrical tube.

### Step 3: Substitute the values into the formula
Substituting the known values into the formula:
\[ \Delta P = \frac{8 \times (4 \times 10^{-3}) \times (1 \times 10^{-3}) \times (0.66 \times 10^{-3})}{(2 \times 10^{-6})^2} \]

### Step 4: Calculate the denominator
Calculating the denominator:
\[ (2 \times 10^{-6})^2 = 4 \times 10^{-12} \]

### Step 5: Calculate the numerator
Calculating the numerator:
\[ 8 \times (4 \times 10^{-3}) \times (1 \times 10^{-3}) \times (0.66 \times 10^{-3}) = 8 \times 4 \times 0.66 \times 10^{-9} = 21.12 \times 10^{-9} \]

### Step 6: Calculate the pressure drop
Now substituting the values into the equation:
\[ \Delta P = \frac{21.12 \times 10^{-9}}{4 \times 10^{-12}} = 5.28 \times 10^{3} \, \text{Pa} \]

### Step 7: Convert pressure drop to mm Hg
To convert the pressure from Pascals to mm Hg, we use the conversion factor:
\[ 1 \, \text{mm Hg} = 133.322 \, \text{Pa} \]
Thus,
\[ \Delta P \text{ (in mm Hg)} = \frac{5.28 \times 10^{3}}{133.322} \approx 39.6 \, \text{mm Hg} \]

### Step 8: Final Answer
The pressure drop in the blood as it passes through the capillary is approximately **39.6 mm Hg**.

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