The critical velocity v of a body depends on the coefficient of viscosity `eta` the density d and radius of the drop r. If K is a dimensionless constant then v is equal to

Options

A. `(K eta d)/r`
B. `(Kd)/(eta r)`
C. `(K eta)/(dr)`
D. `(Kr)/(eta r)`

Answer

To find the expression for the critical velocity \( v \) of a body in terms of the coefficient of viscosity \( \eta \), density \( d \), and radius of the drop \( r \), we can follow these steps:

### Step 1: Identify the dependencies
The critical velocity \( v \) depends on:
- Coefficient of viscosity \( \eta \)
- Density \( d \)
- Radius of the drop \( r \)

### Step 2: Establish relationships
From fluid dynamics, we can establish the following relationships:
- \( v \) is directly proportional to the coefficient of viscosity \( \eta \).
- \( v \) is inversely proportional to the density \( d \).
- \( v \) is inversely proportional to the radius \( r \).

### Step 3: Write the proportionality equations
Based on the relationships identified:
1. \( v \propto \eta \)
2. \( v \propto \frac{1}{d} \)
3. \( v \propto \frac{1}{r} \)

### Step 4: Combine the relationships
Combining these proportionalities, we can express \( v \) as:
\[ v \propto \frac{\eta}{d \cdot r} \]

### Step 5: Introduce a dimensionless constant
To convert the proportionality into an equation, we introduce a dimensionless constant \( K \):
\[ v = K \cdot \frac{\eta}{d \cdot r} \]

### Step 6: Final expression
Thus, the critical velocity \( v \) can be expressed as:
\[ v = K \cdot \frac{\eta}{d \cdot r} \]

### Conclusion
The correct option for the critical velocity \( v \) is:
\[ v = K \cdot \frac{\eta}{d \cdot r} \]