# Problem Statement Given velocity components for a flow field: - \( u = 2xy \) - v = -x^2y) - \( w = \) We are asked: **Is this a physically possible incompressible flow field?** *Hint: Any physically possible incompressible flow field must satisfy conservation of mass.* --- # Step 1: Conservation of Mass for Incompressible Flow The conservation of mass (continuity equation) for an incompressible fluid is: \[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = \] --- # Step 2: Calculate the Partial Derivatives Given: - \( u = 2xy \) - \( v = -x^2y \) - \( w = \) Let's compute the required partial derivatives: ## \( \frac{\partial u}{\partial x} \) \[ \frac{\partial u}{\partial x} = \frac{\partial (2xy)}{\partial x} = 2y \] ## \( \frac{\partial v}{\partial y} \) \[ \frac{\partial v}{\partial y} = \frac{\partial (-x^2y)}{\partial y} = -x^2 \] ## \( \frac{\partial w}{\partial z} \) \[ \frac{\partial w}{\partial z} = \frac{\partial ()}{\partial z} = \] --- # Step 3: Substitute into the Continuity Equation \[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 2y + (-x^2) + = 2y - x^2 \] For incompressible flow, this sum **must be zero everywhere**: \[ 2y - x^2 = \implies 2y = x^2 \] --- # Step 4: Interpretation - The equation \( 2y - x^2 = \) is **not true for all (x, y, z)** in the flow field. - It is only satisfied on the curve \( y = \frac{x^2}{2} \), not everywhere. --- # Step 5: Conclusion ## Is this a physically possible incompressible flow field? **No**, this velocity field does **not** represent a physically possible incompressible flow field everywhere, because the continuity equation is not satisfied for arbitrary \((x, y, z)\). --- ## Summary Table | Term | Value | |-------------|------------------| | \( u \) | \( 2xy \) | | \( v \) | \( -x^2y \) | | \( w \) | \( \) | | Continuity | \( 2y - x^2 \) | | Incompressible? | **NO** | --- ## Final Statement **The flow field is not physically possible for an incompressible fluid because it does not satisfy the continuity equation everywhere in the domain.**