If Q is the rate of flow of liquid through a capillary tube of length l and radius r at constant pressure P, then the rate of flow of liquid through a capillary tube when radius is reduced to one third and length of tube is doubled

Options

A. `(Q)/(162)`
B. `(Q)/(32)`
C. `(Q)/(64)`
D. `(Q)/(81)`

Answer

To solve the problem, we need to determine the new rate of flow \( q' \) through a capillary tube when the radius is reduced to one-third and the length is doubled. We will use the formula for the rate of flow through a capillary tube, which is given by Poiseuille's law.

### Step-by-Step Solution:

1. **Understanding the Initial Condition**:
The initial rate of flow \( q \) through the capillary tube is given by:
\[ q = \frac{P \pi r^4}{8 \eta l} \]
where:
- \( P \) is the pressure,
- \( r \) is the radius of the tube,
- \( \eta \) is the viscosity of the liquid,
- \( l \) is the length of the tube.

2. **Modifying the Dimensions**:
According to the problem, the radius is reduced to one-third:
\[ r' = \frac{r}{3} \]
and the length of the tube is doubled:
\[ l' = 2l \]

3. **Calculating the New Rate of Flow**:
The new rate of flow \( q' \) can be expressed as:
\[ q' = \frac{P \pi (r')^4}{8 \eta (l')} \]
Substituting \( r' \) and \( l' \):
\[ q' = \frac{P \pi \left(\frac{r}{3}\right)^4}{8 \eta (2l)} \]

4. **Simplifying the Expression**:
Now, calculate \( (r')^4 \):
\[ (r')^4 = \left(\frac{r}{3}\right)^4 = \frac{r^4}{81} \]
Thus, substituting this into the equation for \( q' \):
\[ q' = \frac{P \pi \frac{r^4}{81}}{8 \eta (2l)} = \frac{P \pi r^4}{162 \eta l} \]

5. **Finding the Ratio of the Flow Rates**:
Now, we can find the ratio of the initial flow rate \( q \) to the new flow rate \( q' \):
\[ \frac{q}{q'} = \frac{\frac{P \pi r^4}{8 \eta l}}{\frac{P \pi r^4}{162 \eta l}} = \frac{162}{8} = 20.25 \]

6. **Finding \( q' \) in terms of \( q \)**:
Rearranging gives:
\[ q' = \frac{q}{20.25} \]

### Final Result:
Thus, the new rate of flow \( q' \) when the radius is reduced to one-third and the length is doubled is:
\[ q' = \frac{q}{162} \]