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 \( \nabla \) the three-dimensional velocity vector of the fluid.] 
[(i)] $\displaystyle \frac{\partial \rho}{\partial t} + \nabla \times (\rho \mathbf{v}) = 0$
[(ii)] $\displaystyle \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0$ 
[(iii)] $\displaystyle \frac{\partial \mathbf{v}}{\partial t} + \rho \cdot \nabla \mathbf{v} = 0$ 
[(iv)] $\displaystyle \frac{\partial \rho}{\partial t} + \mathbf{v} \cdot \nabla \rho = 0$

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

Hint:

For a compressible fluid, the correct form of the mass continuity equation is \( \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0 \), representing mass conservation.

Answer 1

The Correct Option is B

The mass continuity equation for a compressible fluid is based on the principle of conservation of mass. The correct form of this equation in differential form is: \[ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0 \] This equation reflects the fact that the rate of change of mass within a control volume is equal to the net mass flux through the control surface.
Let's review each equation:
1. Equation (i):
\[ \frac{\partial \rho}{\partial t} + \nabla \times (\rho \vec{v}) = 0 \] This equation is incorrect because the term \( \nabla \times (\rho \vec{v}) \) represents the vorticity and is not relevant to the mass continuity equation. This term should be a divergence, not a curl.
2. Equation (ii):
\[ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0 \] This is the correct form of the continuity equation for a compressible fluid. The term \( \nabla \cdot (\rho \vec{v}) \) represents the divergence of the mass flux, which is the proper term for mass conservation.
3. Equation (iii):
\[ \frac{\partial \vec{v}}{\partial t} + \rho \nabla \cdot \vec{v} = 0 \] This equation represents the change in velocity, which is not related to the mass continuity equation. This is a form of the momentum equation.
4. Equation (iv):
\[ \frac{\partial \rho}{\partial t} + \vec{v} \cdot \nabla \rho = 0 \] This equation is a simplification of the continuity equation, but it is valid only for cases where the fluid is incompressible, which is not applicable in this case for compressible fluids.
Thus, the correct answer is (B) (ii).