Question

A flow velocity field \( \vec{V} = \vec{V}(x, y) \) for a fluid is represented by \( \vec{V} = 3 \hat{i} + (5x) \hat{j \). In the context of the fluid and the flow, which one of the following statements is CORRECT?}

• A. The fluid is incompressible and the flow is rotational.
• B. The fluid is incompressible and the flow is irrotational.
• C. The fluid is compressible and the flow is rotational.
• D. The fluid is compressible and the flow is irrotational.

Hint:

Remember, incompressibility is indicated by a zero divergence of the velocity field, and a non-zero curl indicates rotational flow. These calculations are foundational in fluid dynamics to determine the basic characteristics of flow behavior.

Answer 1

The Correct Option is A

Step 1: Check for compressibility by evaluating the divergence of the velocity field. The divergence of \( \vec{V} \) is given by: \[ \nabla \cdot \vec{V} = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y}. \] Substituting \( V_x = 3 \) and \( V_y = 5x \): \[ \nabla \cdot \vec{V} = 0 + 0 = 0. \] Since the divergence is zero, the fluid is incompressible. Step 2: Check for rotational or irrotational flow by evaluating the curl of the velocity field. The curl of \( \vec{V} \) in two dimensions is given by: \[ \nabla \times \vec{V} = \left( \frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y} \right) \hat{k}. \] Substituting the components: \[ \nabla \times \vec{V} = (5 - 0) \hat{k} = 5 \hat{k}. \] Since the curl is not zero, the flow is rotational.