Question

Which of the following statements are true? (i) Conservation of mass for an unsteady incompressible flow can be represented as $\nabla\! . \vec V = 0$, where $\vec V$ denotes velocity vector.
(ii) Circulation is defined as the line integral of {vorticity} about a closed curve.
(iii) For some fluids, shear stress can be a nonlinear function of the shear strain rate.
(iv) Integration of the Bernoulli’s equation along a streamline under steady-state leads to the Euler’s equation.

• A. (i), (ii) and (iv) only
• B. (i), (ii) and (iii) only
• C. (i) and (iii) only
• D. (ii) and (iv) only

Hint:

Remember: $\Gamma=\oint \vec V\! . d\vec l$; via Stokes, $\Gamma=\iint (\nabla\times \vec V)\! . \hat n\, dS$. Also, Euler $\Rightarrow$ (integrate) Bernoulli, not the other way around.

Answer 1

The Correct Option is C

Step 1: Analyze (i).
For an incompressible flow, $\rho$ is constant (even if the flow is unsteady), and the continuity equation reduces to $\nabla\! . \vec V=0$. Hence (i) is true.
Step 2: Analyze (ii).
Circulation is defined as the line integral of the {velocity} around a closed curve: \[ \Gamma=\oint_C \vec V . d\vec l. \] By Stokes’ theorem, this equals the surface integral of {vorticity} over any surface spanning $C$, not the line integral of vorticity around $C$. So (ii) is false.
Step 3: Analyze (iii).
Non-Newtonian fluids (e.g., shear-thinning or thickening) have shear stress $\tau$ as a nonlinear function of shear rate $\dot\gamma$. Hence (iii) is true.
Step 4: Analyze (iv).
The Euler equation (steady, inviscid) {integrated} along a streamline yields Bernoulli’s equation. The statement reverses the direction; thus (iv) is false. Final Answer: \[ \boxed{\text{(C) }(i)\ \text{and}\ (iii)\ \text{only}} \]