General Balance Equations and Continuity Equation
Based on the derivation in (Bird, et. al. 2007).
The general balance equations (GBE) are the main tennets of continuum
mechanics. Proper derivation of these equations is absolutely essential
to theral-hydraulics. They are also incredibly useful for other physical
systems, and when coupled with the electronic charge conservation
equation, could (ostensibly) solve any non-quantum mechanical problem.
General Balance Equation
The general balance equation for any property is given by:
\[\begin{split}\underbrace{\frac{D}{Dt}\intop_{V_{m}}\psi dV}_{\substack{\text{change of }\psi\\
\text{per unit time}\\
\text{in volume }V_{m}
}
}=-\underbrace{\oint_{S_{m}}\mathbb{J}\cdot\vec{n}dS}_{\substack{\text{influx of }\psi\text{ across}\\
\text{surface }S_{m}
}
}+\underbrace{\int_{V_{m}}\dot{\psi_{g}}dV}_{\substack{\text{generation of }\psi\\
\text{in volume }V_{m}
}
}\end{split}\]
We can use Reynold’s Transport theorem, which states that
\(\frac{D}{Dt}\int_{V_{m}}\psi dV=\int_{V_{m}}\left[\frac{\partial\psi}{\partial t}+\nabla\cdot\left(\psi\vec{v}\right)\right]dV\)
and Greene Theorem, which thats
\(\oint_{S_{m}}\mathbb{J}\cdot\vec{n}dS=\int_{V_{m}}\nabla\cdot\mathbb{J}dV\).
Inputting these two expressions to first convert the material derivative
to time and spatial derivative and to convert the surface integral to a
volume integral, respectively. With these two changes and combining
volume integrals, we then get
\[\int_{V_{m}}\left[\frac{\partial\psi}{\partial t}+\nabla\cdot\left(\psi\vec{v}\right)=-\nabla\cdot\mathbb{J}+\dot{\psi_{g}}\right]dV\]
and after differentiating:
\[\begin{split}\underbrace{\frac{\partial\psi}{\partial t}}_{\substack{\text{change of }\psi\\
\text{per unit time}\\
\text{per unit volume}
}
}\underbrace{\nabla\cdot\left(\psi\vec{v}\right)}_{\substack{\text{convection by}\\
\text{material motion}\\
\text{per unit volume}
}
}=-\underbrace{\nabla\cdot\mathbb{J}}_{\substack{\text{influx of }\psi\text{ per}\\
\text{surface }S_{m}\text{ per}\\
\text{unit volume}
}
}+\underbrace{\dot{\psi_{g}}}_{\substack{\text{generation of }\psi\\
\text{per unit volume}
}
}\end{split}\]
Continuity Equation
Apply the GBE, using \(\psi = \rho\), \(\mathbb{J} = 0\), and
\(\dot{\psi_{g}} = 0\), we get
\[\frac{\partial\rho}{\partial t}+\nabla\cdot\left(\rho\vec{v}\right)=0\]
which is equivalent to (by FOIL):
\(\frac{\partial\rho}{\partial t}+\nabla\cdot\left(\rho\vec{v}\right)=\frac{\partial\rho}{\partial t}+\vec{v}\cdot\nabla\rho+\rho\nabla\cdot\vec{v}\)
and using the material derivative’s definition:
\(\frac{D\rho}{Dt}\equiv\frac{\partial\rho}{\partial t}+\vec{v}\cdot\nabla\rho\)
in the above equation, we get:
\[\frac{1}{\rho}\frac{D\rho}{Dt}=-\nabla\cdot\vec{v}\]
and assuming incompressible: \(\frac{D\rho}{Dt}=0\), it is apparent
that:
\[\nabla\cdot\vec{v}=0\]
which is the continuity equation.
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