Question

The velocity field in a fluid is given to be \[ \vec{V} = (4xy)\,\hat{i} + 2(x^{2} - y^{2})\,\hat{j}. \] Which of the following statement(s) is/are correct?

• A. The velocity field is one-dimensional.
• B. The flow is incompressible.
• C. The flow is irrotational.
• D. The acceleration experienced by a fluid particle is zero at $(x=0,\,y=0)$.

Hint:

To test incompressibility, compute divergence. To test irrotationality, compute vorticity. Zero velocity at a point immediately gives zero convective acceleration at that point.

Answer 1

The Correct Option is B, C, D

Step 1: Check if the velocity field is one-dimensional.
The velocity components are \[ u = 4xy,\qquad v = 2(x^2 - y^2). \] Both depend on both variables \(x\) and \(y\). A one-dimensional field would depend on only one spatial variable. Hence, the velocity field is not one-dimensional. Statement (A) is incorrect.
Step 2: Check incompressibility using \[ \nabla \cdot \vec{V} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}. \] Compute each term: \[ \frac{\partial u}{\partial x} = \frac{\partial (4xy)}{\partial x} = 4y, \] \[ \frac{\partial v}{\partial y} = \frac{\partial (2(x^2 - y^2))}{\partial y} = -4y. \] Thus, \[ \nabla \cdot \vec{V} = 4y - 4y = 0. \] Zero divergence implies an incompressible flow. Statement (B) is correct.
Step 3: Check irrotationality using vorticity \[ \omega_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}. \] Compute each derivative: \[ \frac{\partial v}{\partial x} = \frac{\partial (2(x^2 - y^2))}{\partial x} = 4x, \] \[ \frac{\partial u}{\partial y} = \frac{\partial (4xy)}{\partial y} = 4x. \] So, \[ \omega_z = 4x - 4x = 0. \] Since the vorticity is zero everywhere, the flow is irrotational. Statement (C) is correct.
Step 4: Check acceleration at $(x=0, y=0)$.
Fluid acceleration is \[ \vec{a} = (u \frac{\partial}{\partial x} + v \frac{\partial}{\partial y})\vec{V}. \] At the point \((0,0)\): \[ u(0,0) = 4(0)(0) = 0,\qquad v(0,0) = 2(0 - 0) = 0. \] If both velocity components are zero at that point, then \[ \vec{a}(0,0) = 0. \] Thus, acceleration is zero at the origin. Statement (D) is correct.
Final Answer: (B), (C), (D)