Question

For an incompressible flow, the x and y components of the velocity vector are \( V_x = 2(x+y) \), \( V_y = 3(y+z) \), where x, y, and z are in meters and velocities are in m/s. Then the z component of the velocity vector (V\(_z\)) for the boundary condition \( V_z = 0 \) at \( z = 0 \) is

• A. 5z
• B. -5z
• C. 2x + 5z
• D. 2x - 5z

Hint:

In incompressible flow, the continuity equation ensures that the sum of the velocity components’ derivatives is zero. This helps in calculating the missing velocity components.

Answer 1

The Correct Option is B

Step 1: Understanding the boundary condition. 
In incompressible flow, the velocity components must satisfy the incompressibility condition, which is the continuity equation: \[ \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} + \frac{\partial V_z}{\partial z} = 0 \] Given \( V_x = 2(x + y) \) and \( V_y = 3(y + z) \), we can differentiate these components with respect to their respective variables. The z component \( V_z \) can be found by applying the incompressibility condition at \( z = 0 \). 
Step 2: Calculation. 
The partial derivatives are: \[ \frac{\partial V_x}{\partial x} = 2, \quad \frac{\partial V_y}{\partial y} = 3 \] Substituting into the continuity equation: \[ 2 + 3 + \frac{\partial V_z}{\partial z} = 0 \quad \Rightarrow \quad \frac{\partial V_z}{\partial z} = -5 \] Therefore, \( V_z = -5z \). Step 3: Conclusion. 
The z component of the velocity vector is \( V_z = -5z \), and the correct answer is (2).