Page 352 - Modelling in Transport Phenomena A Conceptual Approach

332 CHAPTER 9. STEADY MICROSCOPIC BALANCES WITH GEN.

At the liquid-air interface, the jump momentum balance’ indicates that the
normal and tangential components of the total stress tensor are equal to each
other, i.e.,
at x = o P’ = pA for all z (9.1-47)
at x = o T,”, = 7tz for all z (9.1-48)

Since both PL and PA depend only on z, then
dPL dPA
-- -- (9.1-49)
dz dz
From Eqs. (9.1-46) and (9.1-49) one can conclude that
dPL
-
dz = PAS (9.1-50)
Substitution of Eq. (9.1-50) into Eq. (9.1-43) gives

-p s- (9.1-51)
L @v,t
- (PL -PA) 9
Since pL >> pA, then pL - pA M pL and l3q. (9.1-51) takes the form

(9.1-52)

This analysis shows the reason why the pressure term does not appear in the
equation of motion when a fluid flows under the action of gravity. This point
is usually overlooked in the literature by simply stating that “free surface + no
pressure gradient.”
For simplicity, superscripts in Eq. (9.1-52) will be dropped for the rest of the
analysis with the understanding that properties are those of the liquid. Therefore,
the governing equation takes the form

(9.1-53)

Integration of Eq. (9.1-53) twice leads to

(9.1-54)

The boundary conditions are
dvz
at x=O -=O (9.1-55)
dx
at x=6 vz=O (9.1-56)

‘For a thorough discussion on jump balances, see Slattery (1999).