Water flows in a stream line manner through a capillary tube of radius a. the pressure difference being P and the rate of the flows is Q. If the radius is reduced to `(a)/(4)` and the pressure is increased to 4P, then the rate of flow becomes

Options

A. `Q_(2)=64Q`
B. `Q_(2)=Q//64`
C. `Q_(2)=Q//32`
D. `Q_(2)=Q//16`

Answer

To solve the problem, we will use the Hagen-Poiseuille equation, which describes the flow of a viscous fluid through a cylindrical pipe. The equation is given by:

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

Where:
- \( Q \) = rate of flow
- \( \Delta P \) = pressure difference
- \( r \) = radius of the tube
- \( \eta \) = coefficient of viscosity
- \( L \) = length of the tube

### Step 1: Write the initial condition
Given:
- Initial radius \( r = a \)
- Initial pressure difference \( \Delta P = P \)
- Initial rate of flow \( Q = Q \)

Using the Hagen-Poiseuille equation, we can write:

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

### Step 2: Write the new condition after changes
Now, the radius is reduced to \( r' = \frac{a}{4} \) and the pressure is increased to \( \Delta P' = 4P \).

### Step 3: Substitute the new values into the equation
Using the Hagen-Poiseuille equation for the new conditions:

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

Substituting the new values:

\[ Q' = \frac{\pi (4P) \left(\frac{a}{4}\right)^4}{8 \eta L} \]

### Step 4: Simplify the expression
Now, calculate \( \left(\frac{a}{4}\right)^4 \):

\[ \left(\frac{a}{4}\right)^4 = \frac{a^4}{256} \]

Substituting this back into the equation for \( Q' \):

\[ Q' = \frac{\pi (4P) \frac{a^4}{256}}{8 \eta L} \]

### Step 5: Factor out common terms
Now, simplify the expression:

\[ Q' = \frac{4 \pi P a^4}{256 \cdot 8 \eta L} \]

### Step 6: Further simplify
Calculating the constants:

\[ Q' = \frac{4 \pi P a^4}{2048 \eta L} \]

### Step 7: Relate \( Q' \) to \( Q \)
Now, we can relate \( Q' \) to \( Q \):

\[ Q' = \frac{4}{2048} Q = \frac{1}{512} Q \]

### Final Result
Thus, the new rate of flow \( Q' \) is:

\[ Q' = \frac{Q}{512} \]