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?}
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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.
The fluid is incompressible and the flow is rotational.
The fluid is incompressible and the flow is irrotational.
The fluid is compressible and the flow is rotational.
The fluid is compressible and the flow is irrotational.
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The Correct Option isA
Solution and Explanation
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.