ISSN 0021-8944, Journal of Applied Mechanics and Technical Physics, 2018, Vol. 59, No. 1, pp. 52–60.
Pleiades Publishing, Ltd., 2018.
Original Russian Text
CALCULATION OF LINEAR STABILITY OF A STRATIFIED GAS–LIQUID FLOW
IN AN INCLINED PLANE CHANNEL
Yu. Ya. Trifonov
Abstract: Linear stability of liquid and gas counterﬂows in an inclined channel is considered. The
full Navier–Stokes equations for both phases are linearized, and the dynamics of periodic disturbances
is determined by means of solving a spectral problem in wide ranges of Reynolds numbers for the
liquid and vapor velocity. Two unstable modes are found in the examined ranges: surface mode
(corresponding to the Kapitsa waves at small velocities of the gas) and shear mode in the gas phase.
The wave length and the phase velocity of neutral disturbances of both modes are calculated as
functions of the Reynolds number for the liquid. It is shown that these dependences for the surface
mode are signiﬁcantly aﬀected by the gas velocity.
Keywords: viscous ﬁlm ﬂow, nonlinear waves, stability.
1. INTRODUCTION AND FORMULATION OF THE PROBLEM
Theoretical investigations of ﬁlm ﬂows were started by Nusselt  who obtained an exact solution of the
Navier–Stokes equations in the case of a free downﬂow of a thin layer of a viscous ﬂuid over a smooth vertical
wall. Various wave regimes of down-ﬂowing ﬁlms were considered theoretically and experimentally in [2, 3]. Linear
stability of the Nusselt solution was studied in [4, 5] by an asymptotic approach; it was shown that this solution is
unstable at Reynolds numbers Re > Re
=5cot(β)/6(β is the angle of inclination of the surface on which the ﬂow
occurs). The unstable disturbances (in what follows, the surface mode) found in [4, 5] are long-wave disturbances,
which correspond to the Kapitsa waves at the nonlinear stage. One more mode of unstable disturbances for a
gravity-induced ﬁlm ﬂow down an inclined plane in the absence of surface tension was discovered in [6, 7]. It was
shown that unstable disturbances of this mode at very small angles of surface inclination β are more dangerous than
disturbances of the surface mode. Based on the full linearized Navier–Stokes equations, Floryan et al.  calculated
the neutral stability curves for both unstable modes of the gravity-induced descending ﬁlm ﬂow with allowance for
surface tension. Yih  performed a pioneering study of the linear stability of a stratiﬁed Poiseuille ﬂow in a plane
channel with prescribed volume fractions of both phases and total mean mass velocities of the ﬂuids in the channel.
It was shown that the emergence of unstable disturbances can be initiated by the diﬀerence in ﬂuid viscosities.
Stability of a stratiﬁed ﬂow of two ﬂuids in an inclined channel with diﬀerent values of ﬂuid densities and volume
fractions was considered in [10–12] in a similar formulation.
The goal of the present work is to study various unstable modes in liquid and gas counterﬂows between
two inclined planes by means of solving the full linearized Navier–Stokes equations. The mean mass velocities for
both phases are used as independent parameters, which makes the present study principally diﬀerent from those in
[9–12]. Similar parameters are used in experiments for studying an important phenomenon of ﬂooding .
Kutateladze Institute of Thermophysics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090
Russia; firstname.lastname@example.org. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 59, No. 1,
pp. 61–70, January–February, 2018. Original article submitted December 6, 2016; revision submitted January 16,
2018 by Pleiades Publishing, Ltd.