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HW1



= Title 1 =

[[media:egm6341.s10.p1-1.png|Lecture p.1-1]]

Title 1.1
$$\alpha = \beta$$

The subscript $$t$$ in $$\Omega_t$$ is used to indicate the collection of material points $$\{ X \}$$ that passes through the fixed spatial domain $$\Omega_t$$ at time $$t$$, i.e., $$X = \phi_t^{-1} (x)$$, where $$\phi_t $$ designates the flow mapping at time $$t$$, such that $$x = \phi_t (X) = \phi (X,t) $$.


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$$  \displaystyle \rho \frac {D E} {Dt} = {\mathbf U} \cdot \frac {\rho D \mathbf U} {Dt} = {\mathbf U \cdot {\rm div} \boldsymbol \tau} = \frac{\partial \tau_{ij}} {\partial x_i} U_j = \frac{\partial} {\partial x_i} \left( \tau_{ij} U_j \right ) - \tau_{ij} \frac{\partial U_j} {\partial x_i} = {\rm div} (\mathbf U \cdot \boldsymbol \tau) - \frac{1}{2} \left( \tau_{ij} \frac{\partial U_j} {\partial x_i} + \tau_{ji} \frac{\partial U_i} {\partial x_j} \right ) = {\rm div} (\mathbf U \cdot \boldsymbol \tau) - \tau_{ij} \frac{1}{2} \left( \frac{\partial U_j} {\partial x_i} + \frac{\partial U_i} {\partial x_j} \right ) = {\rm div} (\mathbf U \cdot \boldsymbol \tau) - \tau_{ij} S_{ij} = {\rm div} (\mathbf U \cdot \boldsymbol \tau) - \boldsymbol \tau {\boldsymbol :} \mathbf S $$ (N)
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Reynolds transport theorem
Taking the total time derivative of the kinetic energy $$\mathcal K$$ of the control volume  $$\Omega_t$$ as expressed in Eq.(6), keeping the material points  $$X$$ inside    $$\Omega_t$$ fixed, yields


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$$  \displaystyle \left. \frac{d \mathcal K}{d t} \right|_{X fixed} = \left. \frac{d}{dt} \left( \int\limits_{\Omega_t} \rho (x,t) E (x,t) d \Omega_t \right ) \right|_{X fixed} = \int\limits_{\Omega_t} \rho \frac{D E}{Dt} d \Omega_t $$     (A1)
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