= Dyadic transport of material line elements
{title2=$D_tA=LA+AL^T$}
For a <material line> tangent $\boldsymbol\ell$, $D_t\boldsymbol\ell=L\boldsymbol\ell$ with <velocity gradient> $L_{ij}=\partial_j u_i$. The <product rule> gives $D_t(\boldsymbol\ell\boldsymbol\ell^T)=L\boldsymbol\ell\boldsymbol\ell^T+\boldsymbol\ell\boldsymbol\ell^TL^T$. Consequently the <upper-convected derivative> vanishes for any transported sum of these <outer products>. Under the derivative-index-first convention $G=L^T$ the same formula is $D_tA=G^TA+AG$. The distinction prevents transposition errors in <constitutive equations>.
Back to article page
