Question

Which one of the following is the mass conservation equation?

• A. \( \frac{D}{Dt} \iiint_V \rho \, \vec{v} \cdot \hat{n} \, dV = 0 \)
• B. \( \frac{\partial}{\partial t} \iiint_V \rho \, dV = 0 \)
• C. \( - \frac{\partial}{\partial t} \iiint_V \rho \, dV = \iint_S \rho \vec{v} \cdot \hat{n} \, ds \)
• D. \( - \frac{D}{Dt} \iiint_V \rho \, dV = \iint_S \rho \vec{v} \cdot \hat{n} \, ds \)

Hint:

To identify the mass conservation equation, ensure it balances the temporal rate of change of mass within a volume with the net mass flux across its surface.

Answer 1

The Correct Option is C

Step 1: Recall the mass conservation equation. 
The principle of mass conservation states that the rate of change of mass within a control volume must be equal to the net flux of mass through the control surface. Mathematically, this is expressed as: \[ \frac{\partial}{\partial t} \iiint_V \rho \, dV + \iint_S \rho \vec{v} \cdot \vec{n} \, ds = 0, \] where: - \( \rho \) is the density, - \( \vec{v} \) is the velocity vector, - \( \vec{n} \) is the outward unit normal vector on the surface \( S \), - \( V \) is the control volume. 

Step 2: Simplify the equation. 
Rewriting the mass conservation equation: \[ -\frac{\partial}{\partial t} \iiint_V \rho \, dV = \iint_S \rho \vec{v} \cdot \vec{n} \, ds. \] This matches Option (C). 

Step 3: Analyze other options. 
Option (A): Incorrect, as it describes a volumetric flux without accounting for mass conservation.
Option (B): Incorrect, as it only considers the temporal change of mass in the volume and neglects surface flux.
Option (D): Incorrect, as it misrepresents the derivative form of the conservation equation.

Conclusion: The mass conservation equation is \( -\frac{\partial}{\partial t} \iiint_V \rho \, dV = \iint_S \rho \vec{v} \cdot \vec{n} \, ds \).