Wall and interfacial shear stress in pressure driven two-
phase laminar strati®ed pipe ¯ow
D. Biberg*, G. Halvorsen
Institute for Energy Technology, P.O. Box 40, N-2027 Kjeller, Norway
Received 26 November 1997; received in revised form 2 November 1999
Abstract
Two-phase pressure driven laminar strati®ed pipe ¯ow is studied with emphasis on the wall and
interfacial shear stresses. The basic solution of the Navier±Stokes equations is recast into a simpler form,
alleviating physical interpretation and constituting a convenient basis for further developments. Utilizing
two-phase symmetry facilitates writing the velocity ®eld for each phase as one generic expression, which
is then split into a linear combination of two terms. The ®rst term equals the single-phase free surface
¯ow of either phase due to the given driving forces. The second term links the phases together and
represents the shear ¯ow given by the interfacial drag from the opposite phase. The corresponding
expression for the interfacial shear stress now emerges naturally as part of the solution of the boundary
value problem. The wall shear stresses are obtained by formal dierentiation. The limiting behaviour of
the wall and interfacial shear stresses in the triple points, where the ¯uid±¯uid interface meets the pipe
wall, is obtained by application of residue calculus. Surprisingly, it proves possible to integrate out the
Fourier integrals in the expressions for the mean wall and interfacial shear stresses. The expressions for
the mean wall shear stresses are demonstrated to be equivalent to the ¯uid momentum balances, thus
con®rming consistency. The new and remarkably simple closed form expression for the mean interfacial
shear stress, however, represents the local solution of the boundary value problem at the interface, not
obtainable from a regular force balance. It thus complements the momentum balances, facilitating a
simple computation of the exact general solution for the mean wall and interfacial shear stresses for a
given holdup and pressure drop in a given pipe containing a given pair of ¯uids. The equation system
developed here forms the basis for the inversion of the laminar strati®ed pipe ¯ow problem in terms of
¯ow rates. The corresponding, however, considerably more transparent, channel ¯ow solution is utilized
as a guide to the pipe ¯ow problem. 7 2000 Elsevier Science Ltd. All rights reserved.
Keywords: Two-phase; Laminar; Strati®ed; Pipe ¯ow; Shear stress
International Journal of Multiphase Flow 26 (2000) 1645±1673
0301-9322/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0301 - 93 22 (9 9) 00 10 9- 3
www.elsevier.com/locate/ijmulflow
* Corresponding author.