and putting
dw dv du dw dv du
-- - -- = 2[xi], -- - -- = 2[eta], -- - -- = 2[zeta], (2)
dy dz dz dx dx dy
d[xi] d[eta] d[zeta]
----- + ------ + ------- = 0, (3)
dx dy dz
the equations of motion may be written
du dH
-- - 2v[zeta] + 2w[eta] + -- = 0, (4)
dt dx
dv dH
-- - 2w[xi] + 2u[zeta] + -- = 0, (5)
dt dy
dw dH
-- - 2u[eta] + 2w[xi] + -- = 0, (6)
dt dz
where
_
/
H = | dp/[rho] + V + ½q², (7)
_/
q² = u² + v² + w², (8)
and the three terms in H may be called the pressure head, potential
head, and head of velocity, when the gravitation unit is employed and
½q² is replaced by ½q²/g.