u = c[tau]W/[Omega]p,
and
H = u²/2g = c²[tau]²W²/2g[Omega]²p²,
.: dH/H + (2g[Omega]²p/c[tau]W²) dp + [zeta] dl/m = 0. (4a)
For tubes of uniform section m is constant; for steady motion W is
constant; and for isothermal expansion [tau] is constant. Integrating,
log H + g[Omega]²p²/W²c[tau] + [zeta]l/m = constant; (5)
for
l = 0, let H = H0, and p = p0;
and for
l = l, let H = H1, and p = p1.
log (H1/H0) + (g[Omega]²}/W²c[tau]) (p1² - p0²) + [zeta]l/m = 0. (5a)
where p0 is the greater pressure and p1 the less, and the flow is from
A0 towards A1.
By replacing W and H,
log (p0/p1) + (gc[tau]/u0²p0²)(p1² - p0² + [zeta]l/m = 0 (6)