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). 
 

Answer 2

For an incompressible flow, mass conservation requires the velocity field to satisfy the continuity equation \( \dfrac{\partial V_x}{\partial x} + \dfrac{\partial V_y}{\partial y} + \dfrac{\partial V_z}{\partial z} = 0 \). We are given \( V_x = 2(x+y) \) and \( V_y = 3(y+z) \), so let's evaluate each candidate for \( V_z \) by checking whether it satisfies this equation together with the boundary condition \( V_z = 0 \) at \( z = 0 \).

First, compute the two known partial derivatives: \( \dfrac{\partial V_x}{\partial x} = 2 \) and \( \dfrac{\partial V_y}{\partial y} = 3 \). Substituting into continuity: \( 2 + 3 + \dfrac{\partial V_z}{\partial z} = 0 \), so \( \dfrac{\partial V_z}{\partial z} = -5 \).

1. 5z: Differentiating gives \( \partial V_z/\partial z = 5 \), which would make the continuity sum equal \( 2+3+5=10 \neq 0 \), violating mass conservation.
2. -5z: Differentiating gives \( \partial V_z/\partial z = -5 \), and the continuity sum becomes \( 2+3-5=0 \), satisfying mass conservation; it also satisfies \( V_z = 0 \) at \( z=0 \).
3. 2x + 5z: Differentiating with respect to \( z \) gives \( \partial V_z/\partial z = 5 \), which fails continuity the same way as option 1, and it also incorrectly introduces an \( x \)-dependence that isn't needed to satisfy the boundary condition.
4. 2x - 5z: While differentiating with respect to \( z \) correctly gives \(-5\), evaluating at \( z=0 \) gives \( V_z = 2x \), which is not zero for all \( x \), violating the stated boundary condition.

Only \( V_z = -5z \) satisfies both the continuity equation and the boundary condition.

Therefore, the correct answer is \( -5z \).