Pressure-gradient terms¶
The pressure-gradient terms:
−1hξiδξip
contribute to the energy balance as follows:
−N3∑k=1N2∑j=1N1+12∑i=12Ju11hξ1δξ1p=N3∑k=1N2∑j=1N1∑i=1Jp1Jδξ1(Jhξ1u1),−N3∑k=1N2−12∑j=12N1∑i=1Ju21hξ2δξ2p=N3∑k=1N2∑j=1N1∑i=1Jp1Jδξ2(Jhξ2u2),−N3−12∑k=12N2∑j=1N1∑i=1Ju31hξ3δξ3p=N3∑k=1N2∑j=1N1∑i=1Jp1Jδξ3(Jhξ3u3).
The sum of these three relations is
N3∑k=1N2∑j=1N1∑i=1Jp{1Jδξ1(Jhξ1u1)+1Jδξ2(Jhξ2u2)+1Jδξ3(Jhξ3u3)},
which is zero because the component inside the wavy parentheses is the incompressibility constraint.