Question

In 2D viscous incompressible flow, the continuity equation is:

• A. \( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \)
• B. \( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial x} = 0 \)
• C. \( \frac{\partial u}{\partial y} = \frac{\partial v}{\partial x} \)
• D. \( \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 v}{\partial y^2} \)

Hint:

In incompressible flow, the continuity equation ensures that the volume flow rate is constant across any cross-section of the fluid. It is essential for analyzing fluid dynamics in 2D flow systems.

Answer 1

The Correct Option is A

In fluid mechanics, the continuity equation expresses the principle of conservation of mass for a flowing fluid. For incompressible flow, the density of the fluid remains constant. In 2D flow, the continuity equation can be written as:
\[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \] where:
- \( u \) is the velocity component in the \( x \)-direction,
- \( v \) is the velocity component in the \( y \)-direction,
- \( \frac{\partial u}{\partial x} \) and \( \frac{\partial v}{\partial y} \) represent the rate of change of the velocity components with respect to their respective coordinates.
This equation essentially states that the net flow of mass into any point in the fluid must be balanced by the net flow out of that point, ensuring conservation of mass in an incompressible fluid.
Thus, the correct form of the continuity equation for 2D incompressible flow is \( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \).