2. Derivation of the Continuity Equation¶
2.1. Mass is not gained or lost - so the rate of change of mass is zero¶
\[{D M \over Dt} = 0\]
2.1.1. For a system (where ‘system’ = identifiable group of matter):¶
\[{D \over D t} \int_{SYS} dM = {D \over D t} \int_{SYS} \rho dV = 0\]
2.1.2. For a control volume (via Reynolds Transport Theorem):¶
\[\underbrace{{\partial \over \partial t} \int_{CV} \rho dV}_{\text{(1) Rate of change of mass in CV}} + \underbrace{\int_{CS} \rho \vec V \cdot \hat n dA}_{\text{(2) Net rate of flow of mass across CS}} = 0\]
2.2. For (1) Rate of change of mass in CV¶
2.2.1. Consider a small element \(\delta x \delta y \delta z\):¶
\[{\partial \over \partial t} \int_{CV} \rho dV = {\partial \rho \over \partial t} \delta x \delta y \delta z\]
2.2.2. For incompressible flow:¶
\[{\partial \rho \over \partial t} = 0\]
2.3. For (2) Net rate of flow of mass across CS¶
2.3.1. Rate of mass flow in x-direction, \(\rho\) is the same throughout. \(u\) is the x-component of mass flow rate.¶
2.3.2. Taylor series expansion leaving CV:¶
\[\dot m_{out} = \rho u (x + {\delta x \over 2}) \delta y \delta z \simeq \rho \left[u + {\partial u \over \partial x} {\delta x \over 2} \right]\delta y \delta z\]
2.3.3. Taylor series expansion entering CV:¶
\[\dot m_{in} = \rho u (x - {\delta x \over 2}) \delta y \delta z \simeq \rho \left[u - {\partial u \over \partial x} {\delta x \over 2} \right]\delta y \delta z\]
2.3.4. Net rate of mass outflow in x:¶
\[\dot m_{x} = \dot m_{out} - \dot m_{in} = \rho {\partial u \over \partial x} \delta x \delta y \delta z\]
2.3.5. Similarly net rate of mass outflow in y:¶
\[\dot m_{y} = \rho {\partial v \over \partial y} \delta x \delta y \delta z\]
2.3.6. Similarly net rate of mass outflow in z:¶
\[\dot m_{z} = \rho {\partial w \over \partial z} \delta x \delta y \delta z\]