2.2. Differential Balance Equations of aaa physical quantities

2.2.1. Balance equation in physical space

In this section, differential form of balance equations is derived using time \(t\) and physical coordinate \(\vec{r}\) as independent variables of fields representing physical quantities \(f(\vec{r},t)\).

Conservative form - Eulerian description in physical space.

(2.1)\[\begin{split}\begin{aligned} & \dfrac{\partial \rho }{\partial t} + \nabla \cdot \left( \rho \vec{v} \right) = 0 \\ & \dfrac{\partial }{\partial t} \left( \rho \vec{v} \right) + \nabla \cdot \left( \rho \vec{v} \otimes \vec{v} \right) = \rho \vec{g} + \nabla \cdot \mathbb{T} \\ & \dfrac{\partial}{\partial t} \left( \rho e^t \right) + \nabla \cdot \left( \rho e^t \vec{v} \right) = \rho \vec{g} \cdot \vec{v} + \nabla \cdot \left( \mathbb{T} \cdot \vec{v} \right) - \nabla \cdot \vec{q} + \rho r \end{aligned}\end{split}\]

Convective form - Lagrangian description in physical space. Using vector calculus identities to evaluate partial derivatives of products, mass equation and relation (1.3) to write partial derivative w.r.t. material derivative,

\[\begin{split}\begin{aligned} & \dfrac{D \rho }{D t} = - \rho \nabla \cdot \vec{v} \\ & \rho \dfrac{D \vec{v}}{D t} = \rho \vec{g} + \nabla \cdot \mathbb{T} \\ & \rho \dfrac{D e^t }{D t} = \rho \vec{g} \cdot \vec{v} + \nabla \cdot \left( \mathbb{T} \cdot \vec{v} \right) - \nabla \cdot \vec{q} + \rho r \end{aligned}\end{split}\]

Proof

todo

Arbitrary description in physical space. Using relation (1.3) to write material derivatives w.r.t. time derivative at constant \(\vec{r}_b\)

\[\begin{split}\begin{aligned} & \left.\dfrac{\partial \rho }{\partial t}\right|_{\vec{r}_b} + \left( \vec{v} - \vec{v}_b \right) \cdot \nabla \rho = - \rho \nabla \cdot \vec{v} \\ & \rho \left.\dfrac{\partial \vec{v}}{\partial t}\right|_{\vec{r}_b} + \rho \left( \vec{v} - \vec{v}_b \right) \cdot \nabla \vec{v} = \rho \vec{g} + \nabla \cdot \mathbb{T} \\ & \rho \left.\dfrac{\partial e^t }{\partial t}\right|_{\vec{r}_b} + \rho \left( \vec{v} - \vec{v}_b \right) \cdot \nabla e^t = \rho \vec{g} \cdot \vec{v} + \nabla \cdot \left( \mathbb{T} \cdot \vec{v} \right) - \nabla \cdot \vec{q} + \rho r \end{aligned}\end{split}\]

2.2.2. Balance equations in reference space

In this section, differential form of balance equations is derived using time \(t\) and material coordinate \(\vec{r}_0\) as independent variables of fields representing physical quantities \(f_0(\vec{r}_0,t) = f(\vec{r}(\vec{r}_0, t), t)\), exploiting the change of variables \(\vec{r}(\vec{r}_0, t)\) and its inverse transformation - assumed to exist (with the same consideration done in the kinematics sections: while it’s likely that a global invertible transformation w.r.t. the original reference configuration doesn’t exist, limiting the time interval and space domain a “piecewise” invertible transformation w.r.t. intermetdiate states exists).