Three capillaries of length `L, (L)/(2) and (L)/(3)` are connected in a series. Their radii are `r, (r)/(2) and (r)/(3)` respectively. If a streamlined flow is to be maintained and pressure difference across the first capillary is `rho`, then the pressure difference across the second capillary will be

Options

A. `2 rho`
B. `8 rho`
C. `rho`
D. `(rho)/(2)`

Answer

To solve the problem step by step, we will use the principles of fluid mechanics, particularly Poiseuille's law, which relates the flow rate through a capillary to the pressure difference across it.

### Step 1: Understand the flow in capillaries
We know that the flow rate \( Q \) through a capillary is given by the formula:
\[ Q = \frac{\pi r^4 \Delta P}{8 \eta L} \]
where:
- \( r \) is the radius of the capillary,
- \( \Delta P \) is the pressure difference across the capillary,
- \( \eta \) is the dynamic viscosity of the fluid,
- \( L \) is the length of the capillary.

### Step 2: Define the parameters for each capillary
Let’s denote the three capillaries as follows:
- Capillary 1: Length \( L \), Radius \( r \), Pressure difference \( \Delta P_1 = \rho \)
- Capillary 2: Length \( \frac{L}{2} \), Radius \( \frac{r}{2} \), Pressure difference \( \Delta P_2 \)
- Capillary 3: Length \( \frac{L}{3} \), Radius \( \frac{r}{3} \), Pressure difference \( \Delta P_3 \)

### Step 3: Write the flow equations for each capillary
Using the flow rate equation for each capillary, we have:
1. For Capillary 1:
\[ Q_1 = \frac{\pi r^4 \rho}{8 \eta L} \]
2. For Capillary 2:
\[ Q_2 = \frac{\pi \left(\frac{r}{2}\right)^4 \Delta P_2}{8 \eta \left(\frac{L}{2}\right)} = \frac{\pi \frac{r^4}{16} \Delta P_2}{4 \eta} = \frac{\pi r^4 \Delta P_2}{64 \eta} \]
3. For Capillary 3:
\[ Q_3 = \frac{\pi \left(\frac{r}{3}\right)^4 \Delta P_3}{8 \eta \left(\frac{L}{3}\right)} = \frac{\pi \frac{r^4}{81} \Delta P_3}{\frac{8L}{3}} = \frac{3\pi r^4 \Delta P_3}{648 \eta} = \frac{\pi r^4 \Delta P_3}{216 \eta} \]

### Step 4: Set the flow rates equal
Since the flow rates must be equal in a series connection, we have:
\[ Q_1 = Q_2 = Q_3 \]

### Step 5: Equate \( Q_1 \) and \( Q_2 \)
Setting \( Q_1 = Q_2 \):
\[ \frac{\pi r^4 \rho}{8 \eta L} = \frac{\pi r^4 \Delta P_2}{64 \eta} \]
Canceling \( \pi r^4 \) and \( \eta \) from both sides gives:
\[ \frac{\rho}{8L} = \frac{\Delta P_2}{64L} \]
Multiplying both sides by \( 64L \):
\[ 8\rho = \Delta P_2 \]
Thus, we find:
\[ \Delta P_2 = 8\rho \]

### Conclusion
The pressure difference across the second capillary is \( 8\rho \).