Question

Which one of the following statements is TRUE about the continuity equation \( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \) (where \( u \), \( v \), \( w \) are the velocity components along the \( x \), \( y \), and \( z \) coordinates respectively):

• A. The equation is valid only for steady incompressible flows.
• B. The equation is valid for both steady and unsteady incompressible flows.
• C. The equation is valid only for steady compressible flows.
• D. The equation is valid only for unsteady compressible flows.

Hint:

The continuity equation is a fundamental principle in fluid dynamics, applicable to both steady and unsteady incompressible flows.

Answer 1

The Correct Option is B

The continuity equation represents the conservation of mass in a fluid flow. For an incompressible flow, the density remains constant, and the equation simplifies to: \[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \] This equation holds for both steady and unsteady incompressible flows because it expresses the condition that the net mass flux into any differential volume must be zero. In other words, the rate at which mass enters any region of the flow must be equal to the rate at which mass leaves that region. For steady flows, this implies no change in the velocity field with time, while for unsteady flows, it allows time-dependent variations in velocity. The equation does not apply to compressible flows because in those cases, the density is not constant, and the equation must be modified to account for changes in density. Thus, the correct answer is (B).
Final Answer: (B) The equation is valid for both steady and unsteady incompressible flows.