3 Balance laws
3.1 Integral continuity and Transport
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
\(\Psi\) can increase (decrease) according to influx (outflux) \(\mathbf F\)
\(\Psi\) can increase (decrease) according to production (decay) \(\mathbf S\)
there is no other mechanism
Translated into equations, this yields
\[ \begin{aligned} \underbrace{\frac{d}{dt} \int_{\mathcal C} \Psi d \mathbf x}_{ \text{ rate of change of $\Psi$ in $\mathcal C$}} &= \underbrace{- \int_{\partial \mathcal C} \mathbf F \cdot \mathbf n d \mathbf \sigma}_{ \text{ in-/outflow across the surface (1.)}} + \underbrace{ \int_{\mathcal C} \mathbf S d \mathbf x}_{ \text{ production/decay in (2.)}} \end{aligned} \tag{3.1}\]
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 \[ \begin{aligned} \partial_t \Psi & = - \nabla \cdot \mathbf F + \mathbf S \end{aligned} \tag{3.2}\]
Note that,
the time derivative now comes as a partial derivative, as \(\Psi\) may vary with space
\(\nabla\) is the Hamilton vectorial operator “nabla”, in index notation \(\nabla = \mathbf{e}_i \partial_i\), where \[ \begin{aligned} \nabla f = \text{grad} f , \qquad \nabla \cdot f = \text{div} f ,\qquad \nabla \times f = \text{curl} f . \end{aligned} \]
The latter step also assumes some regularity of \(\Psi, \mathbf F\), and \(\mathbf S\). Basically we have to demand that they are continuous and differentiable. Otherwise the strong formulation doesn’t make any sense.
The two basic types of transport are:
Advective transport at velocity \(\mathbf v\), hence \(\Psi \mathbf v\), and
Diffusive transport with a diffusive flux \(\mathbf J\), for instance given as gradient-driven transport \(\mathbf J = - D \nabla \Psi\), in which \(D\) stands for the diffusion coefficient. Fourier’s law (heat conduction) and Fick’s law (mass diffusion) are examples for gradient-driven diffusion.
We now get in operator notation:
\[ \begin{aligned} \partial_t \Psi + \nabla \cdot \left( \Psi \mathbf v \right) & = - \nabla \cdot \mathbf J + S \end{aligned} \tag{3.3}\]
It is a good exercise to write down the generic balance law Equation 3.3 in components and in index notation. For a scalar diffusion coefficient, the componentwise scalar balance law reads:
\[ \begin{aligned} \partial_t \Psi &+ \partial_x \left( \Psi v_x \right) + \partial_y \left( \Psi v_y \right) + \partial_z \left( \Psi v_z \right) \\[1em] & = \partial_x \left( D \partial_x \Psi \right) + \partial_y \left( D \partial_y \Psi \right) + \partial_z \left( D \partial_z \Psi \right) + S \end{aligned} \tag{3.4}\]
Whenever diffusion differs with the orientation (we will for instance see in a later chapter on Chapter 8 Darcy’s law that such a situation might arise when considering flow in a layered porous medium), the diffusion coefficient is given as a second order tensor. Hence
\[ \begin{aligned} \mathbf D = \left( \begin{array}{ccc} d_{xx} & d_{xy} & d_{xz} \\ d_{yx} & d_{yy} & d_{yz} \\ d_{zx} & d_{zy} & d_{zz} \end{array} \right) \Rightarrow \mathbf D \nabla \Psi = \left( \begin{array}{ccc} d_{xx} \partial_x \Psi &+ d_{xy} \partial_y \Psi &+ d_{xz} \partial_z \Psi \\ d_{yx} \partial_x \Psi &+ d_{yy} \partial_y \Psi &+ d_{yz} \partial_z \Psi \\ d_{zx} \partial_x \Psi &+ d_{zy} \partial_y \Psi &+ d_{zz} \partial_z \Psi \end{array} \right) \end{aligned} \tag{3.5}\]
In this case, the generic 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
\[ \begin{aligned} \partial_t \Psi + \partial_i \Psi v_i & = \partial_i d_{ij} \partial_j \Psi + S \end{aligned} \tag{3.6}\]
We can now look at special situations that will result in the balance equations for mass and momentum.
3.2 Mass balance
At first, we consider mass as a conserved quantity and accordingly choose \(\Psi = \rho\) being mass per unit volume. In addition, we set
\[ \mathbf J = 0 \quad \text{and} \quad \mathbf S = 0. \]
This yields the well known mass balance:
\[ \partial_t \rho + \nabla \cdot \left( \rho \mathbf v \right) = 0 \tag{3.7}\]
3.3 Momentum balance
Next, we consider momentum and choose \(\Psi = \rho \mathbf v\), hence being 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 class the body force \(\mathbf b\) is often given as the force due to gravitational acceleration \(\mathbf g\).
All in all, this yields the well known momentum balance:
\[ \partial_t ( \rho \mathbf v ) + \nabla \cdot \left(\rho \mathbf v \otimes \mathbf v \right) = \nabla \cdot \mathbf \sigma + \rho \mathbf b \tag{3.8}\]
The terms corresponds from left to right:
\(\partial_t ( \rho \mathbf v )\) : 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