A liquid solution of coefficient of viscosity `2 xx 10^(-3) Nsm^(-2)` made to drive with velocity of `10^(-4)" m/s"` along the Xylem vessels of radius 3/4 m and length 7cm. Calculate the pressure difference across the length of Xylem vessel.

Answer

To calculate the pressure difference across the length of the xylem vessel, we can use Poiseuille's Law, which relates the flow of a viscous fluid through a cylindrical pipe. The formula is given by:

\[ Q = \frac{\pi P r^4}{8 \eta L} \]

Where:
- \( Q \) is the volumetric flow rate,
- \( P \) is the pressure difference,
- \( r \) is the radius of the pipe,
- \( \eta \) is the coefficient of viscosity,
- \( L \) is the length of the pipe.

### Step 1: Convert all units to SI units
- Coefficient of viscosity, \( \eta = 2 \times 10^{-3} \, \text{N s/m}^2 \)
- Velocity, \( V = 10^{-4} \, \text{m/s} \)
- Radius, \( r = \frac{3}{4} \, \text{m} = 0.75 \, \text{m} \)
- Length, \( L = 7 \, \text{cm} = 0.07 \, \text{m} \)

### Step 2: Calculate the cross-sectional area \( A \)
The cross-sectional area \( A \) of the xylem vessel can be calculated using the formula for the area of a circle:

\[ A = \pi r^2 \]

Substituting the radius:

\[ A = \pi (0.75)^2 = \pi \times 0.5625 \approx 1.7671 \, \text{m}^2 \]

### Step 3: Relate volumetric flow rate \( Q \) to velocity and area
The volumetric flow rate \( Q \) can also be expressed as:

\[ Q = A \cdot V \]

Substituting the values:

\[ Q = 1.7671 \, \text{m}^2 \times 10^{-4} \, \text{m/s} \approx 1.7671 \times 10^{-4} \, \text{m}^3/s \]

### Step 4: Rearranging Poiseuille's Law to find pressure difference \( P \)
We can rearrange Poiseuille's Law to solve for \( P \):

\[ P = \frac{8 \eta L Q}{\pi r^4} \]

### Step 5: Calculate \( r^4 \)
First, calculate \( r^4 \):

\[ r^4 = (0.75)^4 = 0.3164 \, \text{m}^4 \]

### Step 6: Substitute all values into the equation for \( P \)
Now substitute the values into the rearranged equation:

\[ P = \frac{8 \times (2 \times 10^{-3}) \times (0.07) \times (1.7671 \times 10^{-4})}{\pi \times 0.3164} \]

Calculating the numerator:

\[ \text{Numerator} = 8 \times 2 \times 10^{-3} \times 0.07 \times 1.7671 \times 10^{-4} \approx 1.0007 \times 10^{-7} \]

Calculating the denominator:

\[ \text{Denominator} = \pi \times 0.3164 \approx 0.9932 \]

Now calculate \( P \):

\[ P \approx \frac{1.0007 \times 10^{-7}}{0.9932} \approx 1.0075 \times 10^{-7} \, \text{Pascals} \]

### Final Result
The pressure difference across the length of the xylem vessel is approximately:

\[ P \approx 1.0075 \times 10^{-7} \, \text{Pascals} \]