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:

$$\frac{\text{d}{m}_{sys}}{\text{d}t}=\frac{\text{d}}{\text{d}t}{\int }_{\text{CV}}\rho \text{d}V+{\int }_{\text{CS}}\rho (\mathbf{v}-{\mathbf{v}}_{\text{CS}})\cdot \mathbf{n}\text{d}A$$

$${\int }_{\text{CV}}\frac{\partial \rho }{\partial t}dV+{\int }_{\text{CS}}\rho \overrightarrow{V}\cdot \overrightarrow{n}dA=0$$

$$\frac{\text{d}{m}_{sys}}{\text{d}t}=\frac{\text{d}}{\text{d}t}{\int }_{\text{CV}}\rho \text{d}V+{\int }_{\text{CS}}\rho (\mathbf{v}-{\mathbf{v}}_{\text{CS}})\cdot \mathbf{n}\text{d}A={\int }_{\text{CV}}\frac{\partial \rho }{\partial t}dV+{\int }_{\text{CS}}\rho \overrightarrow{V}\cdot \overrightarrow{n}dA=0$$

Divergence theorem:

$${\int }_V\mathrm{\nabla }\cdot \overrightarrow{G}={\oint }_A\overrightarrow{G}\cdot \overrightarrow{n}dA$$

$${\int }_{\text{CV}}\frac{\partial \rho }{\partial t}dV+{\int }_{\text{CV}}\mathrm{\nabla }\cdot (\rho \overrightarrow{V})dV=0$$

We now combine the two volume integrals into one,

$${\int }_{\text{CV}}[\frac{\partial \rho }{\partial t}+\mathrm{\nabla }\cdot (\rho \overrightarrow{V})]dV=0$$

Continuity equation:

$$\frac{\partial \rho }{\partial t}+\mathrm{\nabla }\cdot (\rho \overrightarrow{V})=0$$

Alternative Form of the Continuity Equation

$$\frac{\partial \rho }{\partial t}+\mathrm{\nabla }\cdot (\rho \overrightarrow{V})=\underset{\text{Material derivative of }\rho }{\underbrace{\frac{\partial \rho }{\partial t}+\overrightarrow{V}\cdot \mathrm{\nabla }\rho }}+\rho \mathrm{\nabla }\cdot \overrightarrow{V}=0$$

$$\frac{\text{d}\rho }{\text{d}t}+\rho \mathrm{\nabla }\cdot \overrightarrow{V}=0$$