9.1. Conduction#

\[\begin{split}\begin{aligned} \rho c \frac{\partial T}{\partial t} & = - \nabla \cdot \vec{q} + \rho r \\ \end{aligned}\end{split}\]

Fourier model of heat condcution flux wants the heat flux to be proportional to the temperature gradient, \(\nabla T\). The most general expression for linear non-isotropic media reads

\[\vec{q} = - \mathbb{K} \cdot \nabla T \ ,\]

with \(\mathbb{K}\) the conductivity tensor (a second-oreder (semi)definite (symmetric? todo check, Onsager reciprocal relations) positive to comply with the second principle of thermodynamics: “heat transfers energy from hot to cold regions”). The behavior of isotorpic media is described by isotropic tensors; thus1 the conductivity tensor becomes \(\mathbb{K}^{iso} = k \mathbb{I}\), and the heat flux reads

\[\vec{q} = - k \nabla T \ .\]

Differential problem. It follows that conduction in a … medium in region \(\Omega\) for time \(t \in [t_0, t_1]\) is governed by the differential problem represented by the PDE

\[\begin{aligned} \rho c \frac{\partial T}{\partial t} & = \nabla \cdot \left( k \, \nabla T \right) + \rho r \qquad \vec{r} \in \Omega \ , \end{aligned}\]

supplied with proper initial conditions in every point \(\vec{r} \in \Omega\)

\[T(\vec{r}, t_0) = T_0(\vec{r}) \qquad \vec{r} \in \Omega\]

and boundary conditions for every point on the boudary of the domain, \(\vec{r} \in \partial \Omega\) for every time \(t \in [t_0, t_1]\). Different boundary conditions may represent different physical processes:

  • known temperature on \(S_D\) (Dirichlet boundary conditions: the value of the unknown function is prescribed),

    \[T(\vec{r}, t) = T_D (\vec{r}, t) \qquad \vec{r} \in S_D\]

  • known heat flux on \(S_N\) (Neumann boundary conditions: the value of the directional derivative of the unknown function is prescribed, in the normal direction w.r.t. the boundary),

    \[\hat{n}(\vec{r},t) \cdot \nabla T(\vec{r}, t) = \phi_N (\vec{r}, t) \qquad \vec{r} \in S_N\]

  • known value of a linear combination of the value of the unknown function and its directional derivative (Robin boundary conditions),

    \[\alpha T(\vec{r},t) + \beta \hat{n}(\vec{r},t) \cdot \nabla T(\vec{r}, t) = h(\vec{r},t) \qquad \vec{r} \in S_R\]

todo

  • derivation of this equation from governing equations of continuum mechanics: list assumptions, thermodynamic conditions (to transform internal energy \(e\) as a function of \(T\))

  • link to Mathematics:Numerical methods for PDEs:Elliptic (steady) and Parabolic (unsteady) problems

1

Isotrpic 2-nd order tensors are the identity tensor and its multiples, \(\mathbb{A}^{iso} = a \mathbb{I}\). The dot product of 2-nd order identity tensor and a vector (field) gives the vector itself, \(\mathbb{I} \cdot \vec{v} = \vec{v}\).