1 Integral continuity and transport

Integral continuity

Let’s look at a specific quantity \(\Psi\) in an arbitrary volume \(\mathcal C\). We can think of it as mass per unit volume, or energy per unit volume, etc. Assuming continuity or local conservation means

  1. \(\Psi\) can increase (decrease) according to influx (outflux) \(\mathbf F\)

  2. \(\Psi\) can increase (decrease) according to production (decay) \(\mathbf S\)

  3. there is no other mechanism

Translated into equations, this yields

\[ \begin{aligned} \underbrace{\frac{d}{dt} \int_{\mathcal C} \Psi d \mathbf x}_{ \text{ rate of change of $\Psi$ in $\mathcal C$}} &= \underbrace{- \int_{\partial \mathcal C} \mathbf F \cdot \mathbf n d \mathbf \sigma}_{ \text{ in-/outflow across the surface (1.)}} + \underbrace{ \int_{\mathcal C} \mathbf S d \mathbf x}_{ \text{ production/decay in (2.)}} \end{aligned} \]

Divergence theorem

Applying the so-called divergence theorem yields an integral formulation with only volume integrals \[ \begin{aligned} \frac{d}{dt} \int_{\mathcal C} \Psi d \mathbf x & = - \int_{\mathcal C} \nabla \cdot \mathbf F d \mathbf x + \int_{\mathcal C} \mathbf S d \mathbf x \end{aligned} \]

and since \(\mathcal C\) is an arbitrary control volume, we yield the strong formulation \[ \begin{aligned} \partial_t \Psi & = - \nabla \cdot \mathbf F + \mathbf S \end{aligned} \]

Note, that the time derivative now comes as a partial derivative, as \(\Psi\) may vary with space, and \(\nabla\) stands for the Hamilton vectorial operator “nabla”, in index notation \(\nabla = \mathbf{e}_i \partial_i\), where \[ \begin{aligned} \nabla f = \text{grad} f , \quad \nabla \cdot f = \text{div} f ,\quad \text{and} \quad \nabla \times f = \text{curl} f . \end{aligned} \]

The strong formulation also assumes differentiability of \(\Psi, \mathbf F\), and \(\mathbf S\).

Transport

The two basic types of transport fluxes are:

  1. Advective transport at velocity \(\mathbf v\), hence \(\mathbf F = \Psi \mathbf v\), and

  2. Diffusive transport with a diffusive flux \(\mathbf F = \mathbf J\), for instance given as gradient-driven transport \(\mathbf J = - D \nabla \Psi\), in which \(D\) stands for the diffusion coefficient (Fourier’s law / Fick’s law)

We now get in operator notation:

\[ \begin{aligned} \partial_t \Psi + \nabla \cdot \left( \Psi \mathbf v \right) & = - \nabla \cdot \mathbf J + S \end{aligned} \]

  • If \(\Psi\) is a scalar then \(\nabla \Psi\) is a vector.
  • The diffusion coefficient can be a scalar \(D\) or a second order tensor \(\mathbf D\) (anisotropic diffusion). Both \(D \nabla \Psi\) and \(\mathbf D \nabla \Psi\) will be vectors.
  • In each case \(\nabla \cdot D \nabla \Psi\) and \(\nabla \cdot \mathbf D \nabla \Psi\) will be a scalar again. It has the same dimension as \(\Psi\).

Notation in components

It is an informative exercise to spell out the generic balance law in components.

For a scalar specific quantity \(\Psi\) and a scalar diffusion coefficient \(D\), the componentwise balance law reads:

\[ \begin{aligned} \partial_t \Psi &+ \partial_x \left( \Psi v_x \right) + \partial_y \left( \Psi v_y \right) + \partial_z \left( \Psi v_z \right) \\[1em] & = \partial_x \left( D \partial_x \Psi \right) + \partial_y \left( D \partial_y \Psi \right) + \partial_z \left( D \partial_z \Psi \right) + S \end{aligned} \]

Notation in components

Diffusion might differs with orientation. The diffusion coefficient is then given as a second order tensor. Hence

\[ \begin{aligned} \mathbf D = \left( \begin{array}{ccc} d_{xx} & d_{xy} & d_{xz} \\ d_{yx} & d_{yy} & d_{yz} \\ d_{zx} & d_{zy} & d_{zz} \end{array} \right) \Rightarrow \mathbf D \nabla \Psi = \left( \begin{array}{ccc} d_{xx} \partial_x \Psi &+ d_{xy} \partial_y \Psi &+ d_{xz} \partial_z \Psi \\ d_{yx} \partial_x \Psi &+ d_{yy} \partial_y \Psi &+ d_{yz} \partial_z \Psi \\ d_{zx} \partial_x \Psi &+ d_{zy} \partial_y \Psi &+ d_{zz} \partial_z \Psi \end{array} \right) \end{aligned} \]

In this case, the balance law written in components reads

\[ \begin{aligned} \partial_t \Psi + \partial_x \left( \Psi v_x \right) + \partial_y \left( \Psi v_y \right) + \partial_z \left( \Psi v_z \right) = \partial_x & \left( d_{xx} \partial_x \Psi + d_{xy} \partial_y \Psi + d_{xz} \partial_z \Psi \right) \\ + \partial_y & \left( d_{yx} \partial_x \Psi + d_{yy} \partial_y \Psi + d_{yz} \partial_z \Psi \right) \\ + \partial_z & \left( d_{zx} \partial_x \Psi + d_{zy} \partial_y \Psi + d_{zz} \partial_z \Psi \right) + S \end{aligned} \]

In index notation, finally, the (scalar) balance law can be written concicely as

\[ \begin{aligned} \partial_t \Psi + \partial_i \Psi v_i & = \partial_i d_{ij} \partial_j \Psi + S \end{aligned} \]

2 Mass and momentum balance

We can now look at special situations that result in the famous mass and momentum balance.

Mass balance

We consider as a specific, conserved quantity mass and accordingly choose

\[\Psi = \rho\]

hence mass per unit volume.

In addition, we choose

\[ \mathbf J = 0 \quad \text{and} \quad \mathbf S = 0. \]

This yields the well known mass balance:

\[ \partial_t \rho + \nabla \cdot \left( \rho \mathbf v \right) = 0 \]

Momentum balance

We consider momentum as the conserved, specific quantity, hence

\[ \Psi = \rho \mathbf v \] as momentum per unit volume.

In addition we set

\[ \mathbf J = \mathbf \sigma \quad \text{and} \quad \mathbf S = \rho \mathbf b, \]

in which \(\mathbf b\) is a body force and \(\mathbf \sigma\) is the Cauchy stress tensor.

In our course the body force \(\mathbf b\) is often given as the force due to gravitational acceleration denoted by \(\mathbf g\).

Momentum balance

All in all, this yields the well known momentum balance:

\[ \partial_t ( \rho \mathbf v ) + \nabla \cdot \left(\rho \mathbf v \otimes \mathbf v \right) = \nabla \cdot \mathbf \sigma + \rho \mathbf b \]

with terms:

  • \(\partial_t ( \rho \mathbf v )\) : local change of momentum per unit volume

  • \(\nabla \cdot \left( \rho \mathbf v \otimes \mathbf v \right)\) :influx/outflux of momentum into control volume due to advective transport

  • \(\nabla \cdot \mathbf \sigma\) : force action of the ambient continuous medium through its boundary (examples: stretched rod, fluid at rest)

  • \(\rho \mathbf b\) : total mass force acting on the medium, e.g. gravitational force

Momentum balance

We can use the mass balance to re-write momentum balance as

\[ \partial_t \mathbf v + \left( \mathbf v \cdot \nabla \right) \mathbf v = \tfrac{1}{\rho} \nabla \cdot \mathbf \sigma + \mathbf b \]

Identifying the total derivative yields

\[ \frac{D}{Dt} \mathbf v = \tfrac{1}{\rho} \nabla \cdot \mathbf \sigma + \mathbf b, \]

which mimicks Newton’s second law.