Conservation Equations

What is the general conservation principle, in words?

Rate of change of substance in system = rate of substance entering $-$ rate of substance leaving $+$ rate of internal growth of substance.

What is a flux?

The total amount of substance passing through a cross-sectional area per unit time. E.g. a river's discharge in $\mathrm{m^{3}\,s^{-1}}$.

What is a flux density?

The amount of substance passing through a surface per unit cross-sectional area per unit time (the flux through a surface of unit area).

What is a constitutive relation? Give examples.

An assumed rule, often found experimentally, relating physical quantities; it often gives the form of a flux or flux density.

Examples: Ohm's law, Hooke's law, Fourier's law.

What is the compartmental form of the conservation equation?

$\dfrac{dM}{dt}=Q+R$, where $M$ is the stored quantity, $Q$ the net flux through the boundaries (flux in $-$ flux out) and $R$ the net rate of internal growth/decay.

What is the integral form of the conservation equation?

$\dfrac{d}{dt}\displaystyle\int_{\omega}\phi\,dV=-\int_{\partial\omega}\mathbf{f}\cdot\mathbf{n}\,dS+\int_{\omega}r\,dV$, where $\phi$ is the density, $\mathbf{f}$ the flux density, $r$ the source per unit volume, $\mathbf{n}$ the outward unit normal, $\omega$ the region and $\partial\omega$ its boundary.

Why is there a minus sign on the flux integral in the integral conservation law?

$\mathbf{n}$ points outwards, so $\mathbf{f}\cdot\mathbf{n}>0$ means substance leaving. The amount entering is therefore $-\displaystyle\int_{\partial\omega}\mathbf{f}\cdot\mathbf{n}\,dS$.

What does the divergence theorem state (vector and scalar forms)?

For a vector field: $\displaystyle\int_{V}\nabla\cdot\mathbf{f}\,dV=\int_{\partial V}\mathbf{f}\cdot\mathbf{n}\,dS$.

For a scalar field: $\displaystyle\int_{V}\nabla f\,dV=\int_{\partial V}f\,\mathbf{n}\,dS$.

What is the differential (local) form of the conservation equation, and when is it valid?

$\dfrac{\partial\phi}{\partial t}+\nabla\cdot\mathbf{f}=r$.

Valid at every point, provided the arbitrary region $\omega$ is fixed in space.

How do you derive the differential form of conservation from the integral form?

(1) Use the divergence theorem on the flux term: $\dfrac{d}{dt}\displaystyle\int_\omega\phi\,dV+\int_\omega(\nabla\cdot\mathbf{f}-r)\,dV=0$.

(2) $\omega$ fixed, so move $\dfrac{d}{dt}$ inside as $\dfrac{\partial}{\partial t}$: $\displaystyle\int_\omega\left(\dfrac{\partial\phi}{\partial t}+\nabla\cdot\mathbf{f}-r\right)dV=0$.

(3) This holds for every $\omega$, so the integrand is zero.

Why must the integrand vanish everywhere in the conservation derivation (localisation argument)?

If the integrand were positive (or negative) at some point, by continuity we could choose a small region $\omega$ around that point where it is positive throughout, making the integral non-zero. Contradiction, since the integral is zero for every $\omega$. So the integrand is $0$ everywhere.

What is the one-dimensional conservation law?

$\dfrac{\partial\phi}{\partial t}+\dfrac{\partial f}{\partial x}=r$, where $f$ is now a scalar flux in the $x$-direction.

What is the time derivative of an integral over a region moving with velocity $\mathbf{u}$?

If $V$ moves with velocity $\mathbf{u}$: $\dfrac{d}{dt}\displaystyle\int_{V}f\,dV=\int_{V}\dfrac{\partial f}{\partial t}\,dV+\int_{\partial V}f\,\mathbf{u}\cdot\mathbf{n}\,dS=\int_{V}\left(\dfrac{\partial f}{\partial t}+f(\nabla\cdot\mathbf{u})+\mathbf{u}\cdot\nabla f\right)dV$.

What vector identity expands $\nabla\cdot(f\mathbf{u})$?

$\nabla\cdot(f\mathbf{u})=f(\nabla\cdot\mathbf{u})+\mathbf{u}\cdot\nabla f$.

What is the conservation equation when the contents move with velocity $\mathbf{u}$?

$\dfrac{\partial\phi}{\partial t}+\phi(\nabla\cdot\mathbf{u})+(\mathbf{u}\cdot\nabla)\phi+\nabla\cdot\mathbf{f}=r$, equivalently $\dfrac{\partial\phi}{\partial t}+\nabla\cdot(\phi\mathbf{u})+\nabla\cdot\mathbf{f}=r$.

Which form of the conservation law should you use in which situation?

Clearly defined, fixed boundaries: $\dfrac{dM}{dt}=Q+R$.

Boundaries whose shape/size changes: the integral form.

Arbitrary small fixed regions (fields varying in space): $\dfrac{\partial\phi}{\partial t}+\nabla\cdot\mathbf{f}=r$.

Moving contents: the transport form.

How do you find the volume of a compartment whose cross-sectional area varies with height?

If $A(z)$ is the horizontal area at height $z$ and $h(t)$ the surface height, then $W=\displaystyle\int_{0}^{h(t)}A(z)\,dz$ and $\dfrac{dW}{dh}=A(h)$ (fundamental theorem of calculus).

How fast does the surface height of a reservoir change?

By the chain rule $\dfrac{dW}{dt}=\dfrac{dW}{dh}\dfrac{dh}{dt}=A(h)\dfrac{dh}{dt}$, so $\dfrac{dW}{dt}=Q_{\text{in}}-Q_{\text{out}}$ gives $\dfrac{dh}{dt}=\dfrac{Q_{\text{in}}-Q_{\text{out}}}{A(h)}$.

A larger surface area means the level changes more slowly.

How do you derive the one-dimensional river (area) conservation equation?

For a slice $[x,x+dx]$: $\dfrac{\partial}{\partial t}(A\,dx)=\bar uA\big|_{x}-\bar uA\big|_{x+dx}$.

Taylor expand: $\bar uA\big|_{x+dx}=\bar uA\big|_{x}+dx\,\dfrac{\partial(\bar uA)}{\partial x}+O(dx^{2})$.

Divide by $dx$ and let $dx\to0$.

What is the river conservation (St Venant) equation?

$\dfrac{\partial A}{\partial t}+\dfrac{\partial(A\bar u)}{\partial x}=0$.

This is the 1D conservation law with $\phi=A$ and flux $f=A\bar u$

What Taylor expansion is used in conservation derivations, and what does it give?

$F(x+dx)=F(x)+dx\,\dfrac{\partial F}{\partial x}(x)+O(dx^{2})$. So $F(x)-F(x+dx)\approx-dx\,\dfrac{\partial F}{\partial x}$: net inflow into a slice equals minus the flux gradient times its width.