Question

For a steady and incompressible flow, the velocity field $\vec{V}$ in Cartesian $(x,y,z)$ coordinate system is given as: \[ \vec{V} = 5x \, \mathbf{i} - P y \, \mathbf{j} + 3 \mathbf{k}. \] Here, $\mathbf{i}, \mathbf{j}, \mathbf{k}$ are unit vectors along $x, y, z$ directions, respectively, and $P$ is a constant. Which one of the following options is the correct value of $P$ that satisfies the conservation of mass for the given velocity field?

• A. 5
• B. -5
• C. 8
• D. 2

Hint:

Always check $\nabla \cdot \vec{V}$ for incompressible flows. If divergence $\neq 0$, the velocity field is not physically possible for an incompressible fluid.

Answer 1

The Correct Option is A

Step 1: Continuity equation for incompressible flow.
For a fluid that is incompressible and steady, the mass conservation law reduces to: \[ \nabla \cdot \vec{V} = 0. \] This means the net divergence of the velocity field at every point in space must vanish.

Step 2: Compute divergence of given velocity field.
Given: \[ \vec{V} = 5x \, \mathbf{i} - P y \, \mathbf{j} + 3 \mathbf{k}. \] So, \[ \nabla \cdot \vec{V} = \frac{\partial}{\partial x}(5x) + \frac{\partial}{\partial y}(-Py) + \frac{\partial}{\partial z}(3). \] Compute each term: - $\frac{\partial}{\partial x}(5x) = 5$, - $\frac{\partial}{\partial y}(-Py) = -P$, - $\frac{\partial}{\partial z}(3) = 0$ (since 3 is constant w.r.t $z$). So, \[ \nabla \cdot \vec{V} = 5 - P. \]

Step 3: Apply incompressibility condition.
For incompressible flow: \[ \nabla \cdot \vec{V} = 0 \;\;\Rightarrow\;\; 5 - P = 0. \] Thus, \[ P = 5. \]

Step 4: Verify with options.
Option (A) = 5 matches our result. All other options are incorrect. Final Answer:
\[ \boxed{5} \]