Question

The Poiseuille's law applicable to saturated flow in soils states that the flow of water in capillary is

• A. Directly proportional to square of radius of capillary
• B. Inversely proportional to square of radius of capillary
• C. Directly proportional to fourth power of radius of capillary
• D. Inversely proportional to fourth power of radius of capillary

Hint:

Poiseuille’s law: \[ \boxed{ Q = \frac{\pi r^4 \Delta P}{8\mu L} } \] Very important relation: \[ \boxed{ Q \propto r^4 } \] Small increase in pore radius causes a very large increase in flow.

Answer 1

The Correct Option is C

Concept: Poiseuille’s law describes laminar flow through a circular capillary tube. For saturated flow in soils, soil pores are assumed to behave like very small capillary tubes. According to Poiseuille’s law: \[ Q = \frac{\pi r^4 \Delta P}{8\mu L} \] where:
• \(Q\) = discharge
• \(r\) = radius of capillary
• \(\Delta P\) = pressure difference
• \(\mu\) = dynamic viscosity
• \(L\) = length of capillary The important observation is: \[ Q \propto r^4 \]

Step 1: Writing Poiseuille’s law. The mathematical expression is: \[ Q = \frac{\pi r^4 \Delta P}{8\mu L} \] This equation governs viscous laminar flow through capillary tubes.

Step 2: Understanding the proportionality. From the formula: \[ Q \propto r^4 \] This means:
• If radius doubles, discharge increases \(16\) times
• Flow is extremely sensitive to capillary radius

Step 3: Checking all options carefully. Option (A): \[ Q \propto r^2 \] Incorrect. \[ \boxed{ \text{Option (A) is incorrect} } \] Option (B): \[ Q \propto \frac{1}{r^2} \] Incorrect. \[ \boxed{ \text{Option (B) is incorrect} } \] Option (C): \[ Q \propto r^4 \] Correct according to Poiseuille’s law. \[ \boxed{ \text{Option (C) is correct} } \] Option (D): \[ Q \propto \frac{1}{r^4} \] Incorrect. \[ \boxed{ \text{Option (D) is incorrect} } \] Final Conclusion: According to Poiseuille’s law: \[ \boxed{ Q \propto r^4 } \] Thus the flow is directly proportional to the fourth power of capillary radius. Hence the correct answer is: \[ \boxed{ (C) } \]