Let's solve the problem step by step: --- ## Problem Restatement **Given:** - Incompressible flow field with velocity components: - \( u 2x \) - \( v = -2y \) - \( w = \) **Question:** Is this a physically possible flow field? Explain. **Hint:** Any physically possible incompressible flow field must satisfy conservation of mass. --- ## Step 1: Conservation of Mass for Incompressible Flow For an incompressible flow, the continuity equation (conservation of mass) is: \[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = \] --- ## Step 2: Calculate the Partial Derivatives Given: - \( u = 2x \) - \( v = -2y \) - \( w = \) Calculate each term: - \(\frac{\partial u}{\partial x} = \frac{\partial}{\partial x}(2x) = 2\) - \(\frac{\partial v}{\partial y} = \frac{\partial}{\partial y}(-2y) = -2\) - \(\frac{\partial w}{\partial z} = \frac{\partial}{\partial z}() = \) Add them up: \[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 2 + (-2) + = \] --- ## Step 3: Conclusion The velocity field **satisfies the continuity equation** for incompressible flow. ### **Final Answer** **Yes, this is a physically possible flow field.** **Reason:** The given velocity components satisfy the incompressible continuity equation, which is the required condition for any physically possible incompressible flow. --- ### **Summary Table** | Component | Value | Derivative | |-----------|---------|--------------------| | \(u\) | \(2x\) | \(\frac{\partial u}{\partial x} = 2\) | | \(v\) | \(-2y\) | \(\frac{\partial v}{\partial y} = -2\) | | \(w\) | \(\) | \(\frac{\partial w}{\partial z} = \) | | **Sum** | | **** | --- **Therefore, the flow field is physically possible.**