Differential analysis depends on analyzing an infinitesimal control volume (CV).
General Fluid Motion
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Diffusion
Fluid flow properties can be described by the diffusive process:
\dfrac{\partial \eta}{\partial t}=\nabla\cdot\bm{J}+W
Where \eta is the property in question, \bm{J} is the diffusion flux of the property and provides the direction of diffusion, and W is a source or destruction of the given property. The above equation does not account for convection of properties, that is the effect of the fluid flow moving the fluid property. Therefore:
\dfrac{\partial \eta}{\partial t}+\overrightarrow{\bm{V}}\cdot\nabla\eta=\nabla\cdot\bm{J}+W
The left side of this equation can be described as the material derivative, so to summarize:
\dfrac{D \eta}{D t}=\nabla\cdot\bm{J}+W
Conservation of Mass
The differential equation for conservation of mass can be obtained by setting \eta=m, \bm{J}=\rho\overrightarrow{\bm{V}}, and W=0.
\dfrac{D m}{D t}=\nabla\cdot(\rho\overrightarrow{\bm{V}})
By definition, the mass of a system is steady and does not change, therefore:
0=\nabla\cdot(\rho\overrightarrow{\bm{V}})
If flow is incompressible this equation reduces further:
0=\nabla\cdot\overrightarrow{\bm{V}}
Conservation of Momentum
The differential equation for conservation of momentum can be obtained by setting \eta=\rho\overrightarrow{\bm{V}}, \bm{J}=\bm{\sigma} the stress tensor, and W=\rho\overrightarrow{\bm{f}}.
\dfrac{D (\rho \overrightarrow{\bm{V}})}{D t}=\nabla\cdot\bm{\sigma}+\rho\overrightarrow{\bm{f}}
If flow is incompressible then:
\rho\dfrac{D \overrightarrow{\bm{V}}}{D t}=\nabla\cdot\bm{\sigma}+\rho\overrightarrow{\bm{f}}
Replacing the stress tensor according to Stoke’s hypothesis achieves the famous Navier-Stoke’s equation:
\rho\dfrac{D \overrightarrow{\bm{V}}}{D t}=-\nabla p+\mu\nabla^2\overrightarrow{\bm{V}}+\rho\overrightarrow{\bm{f}}
Change in Temperature
The differential equation for temperature change can be obtained by setting \eta=T, \bm{J}=\alpha\nabla T the stress tensor, and W=S.
\dfrac{D T}{D t}=\alpha\nabla^2 T+S