28.1. Hyperbolic problems and conservation laws#

todo

  • In fluid dynamics: no viscosity, no heat conduction

  • In this limit, the equation may produce non-physical equations, i.e. solutions that are not the limit of the solution of the full problem in the limit of \(\mathbf{F}^d(\nabla \mathbf{u}) \rightarrow \mathbf{0}\)

  • How many equations in how many variables?

  • Linear diffusion flux \(\mathbf{F}^d = \dots\)

  • Entropy condition, to select the physical solution from all the possible solutions

28.1.1. Integral equations#

Many laws can be written as integral balance equations for a material volume \(V_t\),

\[\begin{split}\begin{aligned} & \dfrac{d}{dt} \int_{V_t} \mathbf{u} + \oint_{\partial V_t} \hat{\mathbf{n}} \cdot \mathbf{F}(\mathbf{u}, \nabla \mathbf{u}) = \int_{V_t} \mathbf{s} \\ & \dfrac{d}{dt} \int_{V_t} u_i + \oint_{\partial V_t} n_j \, F_{ji} = \int_{V_t} s_i \\ \end{aligned}\end{split}\]

or for generic volume \(v_t\), using Reynolds’ transport theorem,

\[\dfrac{d}{dt} \int_{v_t} \mathbf{u} + \oint_{\partial v_t} \mathbf{u} \left( \mathbf{v} - \mathbf{v}_b \right) \cdot \hat{\mathbf{n}} + \oint_{\partial v_t} \hat{\mathbf{n}} \cdot \mathbf{F}(\mathbf{u}, \nabla \mathbf{u}) = \int_{v_t} \mathbf{s} \ ,\]

being \(\mathbf{v}\) the velocity of the continuous medium and \(\mathbf{v}_b\) the velocity of the boundary \(\partial v_t\).

The flux can be often written as a sum of a conservative part and a diffusion part as

\[\begin{aligned} \mathbf{F}(\mathbf{u}, \nabla \mathbf{u}) & = \mathbf{F}(\mathbf{u}) + \mathbf{F}^d(\nabla \mathbf{u}) \ , \end{aligned}\]

with slight abuse of notation in writing the conservative flux with the same letter as the full flux.

Under some conditions, the diffusion part goes to zero.

28.1.2. Jump relations#

Collapsing the volume \(v_t\) on a surface, volume terms vanish faster than the surface terms. Jump conditions follow from the arbitrariety of this surface

\[\begin{split}\begin{aligned} \mathbf{0} & = \hat{\mathbf{n}}_1 \cdot \left[ (\mathbf{v}_1-\mathbf{v}_b) \mathbf{u} + \mathbf{F}_1 \right] + \hat{\mathbf{n}}_2 \cdot \left[ (\mathbf{v}_2-\mathbf{v}_b) \mathbf{u} + \mathbf{F}_2 \right] = \\ & = \hat{\mathbf{n}} \cdot \left[ (\mathbf{v}-\mathbf{v}_b) \mathbf{u} + \mathbf{F} \right] = \\ & = \left[ v_n^{rel} \mathbf{u} + \mathbf{F}_n \right] \ . \end{aligned}\end{split}\]

28.1.3. Differential equations#

If the functions are differentiable, with divergence theorem and exploiting the arbitrariety of the volume \(v_t\)

\[\begin{split}\begin{aligned} \partial_t u_i + \partial_j F_{ji} & = s_i \\ \partial_t \mathbf{u} + \nabla \cdot \mathbf{F} & = \mathbf{s} \ . \end{aligned}\end{split}\]