Question

The general relationship between shear stress, \( \tau \), and the velocity gradient \[ \tau = k \left( \frac{du}{dy} \right)^n, \] where \( k \) is a constant with appropriate units. The fluid is Newtonian if

• A. \( n > 1 \)
• B. \( n < 1 \)
• C. \( n = 1 \)
• D. \( n = 0 \)

Hint:

For a Newtonian fluid, the shear stress is directly proportional to the velocity gradient, meaning \( n = 1 \) in the power-law relationship.

Answer 1

The Correct Option is C

For a fluid to be Newtonian, the shear stress should be directly proportional to the velocity gradient (i.e., the rate of change of velocity with respect to distance). The equation provided, \( \tau = k \left( \frac{du}{dy} \right)^n \), defines a power-law relationship.

Step 1: Understanding the behavior of Newtonian fluids.
In a Newtonian fluid, the shear stress \( \tau \) is linearly proportional to the velocity gradient \( \frac{du}{dy} \). This means that for a Newtonian fluid, \( n = 1 \), and the equation simplifies to \( \tau = k \frac{du}{dy} \).

Step 2: Conclusion.
Thus, the fluid is Newtonian if \( n = 1 \), making option (C) the correct answer.

Final Answer: \text{(C) \( n = 1 \)}