If two capillary tubes of radii `r_1 and r_2` and having length `l_1 and l_2` respectively are connected in series across a heaed of pressure p, find the rate of flow of the liqid through the tubes, if `eta` is the coefficient of viscosity of the liquid.

Explanation

To find the rate of flow of a liquid through two capillary tubes connected in series, we can follow these steps:

### Step 1: Understand the setup
We have two capillary tubes with radii \( r_1 \) and \( r_2 \), and lengths \( l_1 \) and \( l_2 \) respectively. They are connected in series across a pressure head \( P \).

### Step 2: Define the pressure differences
Let \( P_1 \) be the pressure difference across the first capillary tube and \( P_2 \) be the pressure difference across the second capillary tube. The total pressure difference across both tubes is given by:
\[ P = P_1 + P_2 \]

### Step 3: Calculate the resistance of each tube
The resistance \( R \) to flow through a capillary tube is given by the formula:
\[ R = \frac{8 \eta l}{\pi r^4} \]
where \( \eta \) is the viscosity of the liquid, \( l \) is the length of the tube, and \( r \) is the radius of the tube.

For the first tube:
\[ R_1 = \frac{8 \eta l_1}{\pi r_1^4} \]

For the second tube:
\[ R_2 = \frac{8 \eta l_2}{\pi r_2^4} \]

### Step 4: Calculate total resistance
Since the tubes are connected in series, the total resistance \( R_{total} \) is the sum of the individual resistances:
\[ R_{total} = R_1 + R_2 = \frac{8 \eta l_1}{\pi r_1^4} + \frac{8 \eta l_2}{\pi r_2^4} \]

### Step 5: Substitute into the flow rate equation
The rate of flow \( Q \) through the tubes can be expressed using the formula:
\[ Q = \frac{P}{R_{total}} \]
Substituting for \( R_{total} \):
\[ Q = \frac{P}{\frac{8 \eta l_1}{\pi r_1^4} + \frac{8 \eta l_2}{\pi r_2^4}} \]

### Step 6: Simplify the expression
Factoring out \( \frac{8 \eta}{\pi} \):
\[ Q = \frac{P \cdot \pi}{8 \eta} \cdot \frac{1}{\frac{l_1}{r_1^4} + \frac{l_2}{r_2^4}} \]

### Final Result
Thus, the rate of flow of the liquid through the two capillary tubes is given by:
\[ Q = \frac{\pi P}{8 \eta} \left( \frac{1}{\frac{l_1}{r_1^4} + \frac{l_2}{r_2^4}} \right) \]