Question

Which of the following is the correct form of the mass divergence form of the continuity equation for a compressible fluid? 
[In the given equations, \(\rho\) is the density and \(\mathbf{V}\) the three-dimensional velocity vector of the fluid.] 
(i) \(\dfrac{\partial \rho}{\partial t} + \nabla \times (\rho \mathbf{V}) = 0\) 
(ii) \(\dfrac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{V}) = 0\) 
(iii) \(\dfrac{\partial \mathbf{v}}{\partial t} + \rho \, \nabla \cdot \mathbf{v} = 0\) 
(iv) \(\dfrac{\partial \rho}{\partial t} + \mathbf{v} \cdot \nabla \rho = 0\) 
 

• A. (i) and (ii)
• B. (ii)
• C. (i) and (iv)
• D. (iii)

Hint:

Always remember: the continuity equation expresses \textbf{conservation of mass}. For compressible fluids, it is written with the divergence of the mass flux \((\rho \mathbf{V})\).

Answer 1

The Correct Option is B

Step 1: General form of continuity equation.
The mass conservation equation for a compressible fluid is: \[ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{V}) = 0 \] This is called the mass divergence form of the continuity equation.

Step 2: Analyze each option.
- (i) Incorrect, because it uses curl (\(\nabla \times\)) instead of divergence (\(\nabla \cdot\)). Curl of mass flux has no physical meaning for conservation of mass.
- (ii) Correct, this is the standard compressible continuity equation.
- (iii) Incorrect, because it applies the derivative to velocity instead of density.
- (iv) Incorrect, because it represents only the material derivative of density without the divergence term.


Step 3: Conclusion.
Only (ii) represents the correct form. Final Answer:
\[ \boxed{\dfrac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{V}) = 0} \]