Discussion
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 v the three dimensional velocity vector of the fluid.]
(i) \(\frac{\partial \rho}{\partial t} + \nabla \times (\rho \mathbf{v}) = 0\)
(ii) \(\frac{\partial \rho}{\partial t} + \nabla . (\rho \mathbf{v}) = 0\)
(iii) \(\frac{\partial \mathbf{v}}{\partial t} + \rho . \nabla \mathbf{v} = 0\)
(iv) \(\frac{\partial \rho}{\partial t} + \mathbf{v} . \nabla \rho = 0\)
Answer: B
The mass divergence form is equation (ii) only.
Conservation of mass for a compressible fluid states that the local rate of change of density is balanced by the divergence of the mass flux:
\(\dfrac{\partial \rho}{\partial t} + \nabla \cdot (\rho\mathbf{v}) = 0\)
Equation (i) uses the curl instead of the divergence, and a curl is a vector while \(\partial\rho/\partial t\) is a scalar, so it cannot be a mass balance. Equation (iii) describes the time change of velocity, not of density. Equation (iv) says that density does not change following a fluid parcel, \(D\rho/Dt = 0\), which is the condition for incompressible flow; the full advective form of continuity is \(D\rho/Dt + \rho\nabla \cdot \mathbf{v} = 0\).
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