Vorticity Equation - Tensor Notation

Tensor Notation

The vorticity equation can be expressed in tensor notation using Einstein's summation convention and the Levi-Civita symbol :

\begin{align} \frac{d\omega_i}{dt} &= \frac{\partial \omega_i}{\partial t} + v_j \frac{\partial \omega_i}{\partial x_j} \\ &= \omega_j \frac{\partial v_i}{\partial x_j} - \omega_i \frac{\partial v_j}{\partial x_j} + e_{ijk}\frac{1}{\rho^2}\frac{\partial \rho}{\partial x_j}\frac{\partial p}{\partial x_k} + e_{ijk}\frac{\partial}{\partial x_j}\left(\frac{1}{\rho}\frac{\partial \tau_{km}}{\partial x_m}\right) + e_{ijk}\frac{\partial B_k }{\partial x_j} \end{align}

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