Question

The velocity field in an incompressible flow is \( \mathbf{v} = axy \, \mathbf{i} + v_y \, \mathbf{j} + \beta \, \mathbf{k} \), where \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are unit-vectors in the \( (x, y, z) \) Cartesian coordinate system. Given that \( a \) and \( \beta \) are constants, and \( v_y = 0 \) at \( y = 0 \), the correct expression for \( v_y \) is:

• A. \( -\frac{axy}{2} \)
• B. \( -\frac{ay^2}{2} \)
• C. \( \frac{ay^2}{2} \)
• D. \( \frac{axy}{2} \)

Hint:

For incompressible flows, use the condition \( \nabla \cdot \mathbf{v} = 0 \) and apply boundary conditions to determine constants of integration.

Answer 1

The Correct Option is B

Step 1: Apply the incompressibility condition. For an incompressible flow, the divergence of the velocity field must be zero: \[ \nabla \cdot \mathbf{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z} = 0. \] Substitute the given velocity components: \[ v_x = axy, \quad v_y = v_y(y), \quad v_z = \beta. \] Step 2: Compute the divergence. The partial derivatives are: \[ \frac{\partial v_x}{\partial x} = \frac{\partial (axy)}{\partial x} = ay, \quad \frac{\partial v_y}{\partial y} = \frac{\partial v_y}{\partial y}, \quad \frac{\partial v_z}{\partial z} = \frac{\partial \beta}{\partial z} = 0. \] Substitute into the incompressibility condition: \[ ay + \frac{\partial v_y}{\partial y} = 0. \] Step 3: Solve for \( v_y \). Integrate \( \frac{\partial v_y}{\partial y} = -ay \): \[ v_y = -\frac{ay^2}{2} + C, \] where \( C \) is the constant of integration. Step 4: Apply the boundary condition. Given \( v_y = 0 \) at \( y = 0 \): \[ 0 = -\frac{a(0)^2}{2} + C \quad \Rightarrow \quad C = 0. \] Thus, \[ v_y = -\frac{ay^2}{2}. \] Step 5: Conclusion. The correct expression for \( v_y \) is \( -\frac{ay^2}{2} \).