Answer Happy

The term (w : Duappears in the vorticity transport equation. This term acts as a source/sink of vorticity by stretching/compressing or rotating vortex lines. In index notation this term is written as w;dju iwhere Oju is the velocity gradient tensor. The velocity gradient tensor can be decomposed into two parts dju;= {(Oju; + 0,4;) + 2(0,4; – 0,4;) The first term is the strain rate tensor €; = (0,0; + 0,4;) = ди; The second term is the rotation tensor Dj = {(@ju; – 0,4;) Show that the strain rate tensor, éjj , is a symmetric tensor and the rotation tensor, Sij , is an anti- symmetric tensor. Prove the identities wk Ekijijand 12 EkijWij where ū = V xūis the vorticity. Show that the vortex stretching/tilting term only depends on the symmetric part of the velocity gradient tensor. In other words, show W;dju; = Wiejor show W; 12;; = 0. == == 1 2