Question:
The velocity field on an incompressible flow is given by $\overline{v} = (a_1x + a_2y + a_3z)i + (b_1x + b_2y + b_3z)j + (c_1x + c_2y + c_3z)k$ where $a_1 = 2, c_3 = -4$ then the value of $b_2 = _$
The velocity field on an incompressible flow is given by $\overline{v} = (a_1x + a_2y + a_3z)i + (b_1x + b_2y + b_3z)j + (c_1x + c_2y + c_3z)k$ where $a_1 = 2, c_3 = -4$ then the value of $b_2 = _$
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"Incompressible" is a keyword in physics and math that always implies $\text{div } \overline{v} = 0$.
Updated On: Jul 3, 2026
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The Correct Option is C
Solution and Explanation
Step 1: Concept
For an incompressible flow, the divergence of the velocity field must be zero: $\nabla \cdot \overline{v} = 0$.
Step 2: Meaning
Calculate divergence: $\frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z} = a_1 + b_2 + c_3 = 0$.
Step 3: Analysis
Substitute the given values: $2 + b_2 + (-4) = 0$.
Step 4: Conclusion
Simplify: $b_2 - 2 = 0 \rightarrow b_2 = 2$. Final Answer: (C)
For an incompressible flow, the divergence of the velocity field must be zero: $\nabla \cdot \overline{v} = 0$.
Step 2: Meaning
Calculate divergence: $\frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z} = a_1 + b_2 + c_3 = 0$.
Step 3: Analysis
Substitute the given values: $2 + b_2 + (-4) = 0$.
Step 4: Conclusion
Simplify: $b_2 - 2 = 0 \rightarrow b_2 = 2$. Final Answer: (C)
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