Question

Which of the equations suit this model?

(a) \(\frac{\partial}{\partial t}\iiint_V\rho dV + \iint_s \rho \vec{V}.\vec{dS} = 0\)

(b) \(\frac{D}{Dt}\iiint_V\rho dV = 0\)

(c) \(\frac{D\rho}{Dt}+\rho \nabla.\vec{V}=0\)

(d) \(\frac{\partial\rho}{\partial t}+\nabla.(\rho \vec{V})=0\)

The question was posed to me in an international level competition.

I'm obligated to ask this question of Continuity Equation in portion Governing Equations of Fluid Dynamics of Computational Fluid Dynamics

Answer 1

The correct option is (c) \(\frac{D\rho}{Dt}+\rho \nabla.\vec{V}=0\)

To elaborate: \(\frac{\partial\rho}{\partial t}+\nabla.(\rho \vec{V})=0\) is the non-conservative differential equation. Non-conservative differential equation is given by an infinitesimally small fluid element moving along with the flow.