Conservation Of Mass

• Fluid Mechanics Fundamentals and Applications, Fourth Edition, Yunus Cengel, John Cimbala

pp-445

Derivation Using the Divergence Theorem

Conservation of mass for a CV:

\[\cfrac{\text{d}m_{sys}}{\text{d}t}=\cfrac{\text{d}}{\text{d}t}\int_{\text{CV}}\rho \text{d}V +\int_{\text{CS}}\rho (\mathbf{v}-\mathbf{v}_{\text{CS}})\cdot\mathbf{n}\text{d}A\]

\[\int_{\text{CV}}\cfrac{\partial \rho}{\partial t}dV+\int_{\text{CS}}\rho \vec{V}\cdot \vec{n}dA=0\]

\[\cfrac{\text{d}m_{sys}}{\text{d}t}=\cfrac{\text{d}}{\text{d}t}\int_{\text{CV}}\rho \text{d}V +\int_{\text{CS}}\rho (\mathbf{v}-\mathbf{v}_{\text{CS}})\cdot\mathbf{n}\text{d}A= \int_{\text{CV}}\cfrac{\partial \rho}{\partial t}dV+\int_{\text{CS}}\rho \vec{V}\cdot \vec{n}dA=0\]

Divergence theorem:

\[\int_{V}\nabla\cdot\vec{G}=\oint_{A}\vec{G}\cdot\vec{n}dA\]

\[\int_{\text{CV}}\cfrac{\partial \rho}{\partial t}dV+\int_{\text{CV}}\nabla\cdot(\rho \vec{V})dV=0\]

We now combine the two volume integrals into one,

\[\int_{\text{CV}}\left[\cfrac{\partial \rho}{\partial t}+\nabla\cdot(\rho \vec{V})\right]dV=0\]

Continuity equation:

\[\cfrac{\partial \rho}{\partial t}+\nabla\cdot(\rho \vec{V})=0\]

Alternative Form of the Continuity Equation

\[\cfrac{\partial \rho}{\partial t}+\nabla\cdot(\rho \vec{V})=\underbrace{\cfrac{\partial \rho}{\partial t} +\vec{V}\cdot\nabla\rho}_{\text{Material derivative of } \rho}+\rho\nabla\cdot\vec{V}=0\]

\[\cfrac{\text{d}\rho}{\text{d} t}+\rho\nabla\cdot\vec{V}=0\]