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Capillary/C-InstabSh03-27.tex
1
Viscous Potential Flow Analysis
of Capillary Instability
T. Funada and D.D. Joseph
University of Minnesota
Aug 2001
Draft printed March 28, 2002
This paper is dedicated to Klaus Kirchg¨assner on the occasion of his 70th birthday.
Contents
1 Introduction
1
2 Governing equations and dimensionless parameters
2.1 Linearized disturbance equations . . . . . . . . . .
2.2 Dispersion relation for fully viscous flow (FVF) . .
2.3 More viscous fluid outside . . . . . .

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Capillary/C-InstabSh03-27.tex
1
Viscous Potential Flow Analysisof Capillary Instability
T. Funada and D.D. JosephUniversity of MinnesotaAug 2001
Draft printed March 28, 2002
This paper is dedicated to Klaus Kirchg¨assner on the occasion of his 70th birthday.
Contents
1 Introduction 12 Governing equations and dimensionless parameters 3
2.1 Linearized disturbance equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Dispersion relation for fully viscous ﬂow (FVF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 More viscous ﬂuid outside . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Dispersion relation for viscous potential ﬂow (VPF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Growth rate curves,
vs.
64 Maximum growth rates and wavenumbers,
and
vs.
115
vs.
for IPF 166 Conclusions and discussion 16
Abstract
Capillary instability of a viscous ﬂuid cylinder of diam-eter
surrounded by another ﬂuid is determined by aReynolds number
, a viscosity ratio
and a density ratio
. Here
is the capillary collapse velocity based on the more vis-cous liquid which may be inside or outside the ﬂuid cylin-der. Results of linearized analysis based on potential ﬂowof a viscous and inviscid ﬂuid are compared with theunapproximated normal mode analysis of the linearizedNavier-Stokes equations. The growth rates for the inviscidﬂuid are largest,the growth rates of the fully viscous prob-lemaresmallestandthoseofviscouspotentialﬂowarebe-tween. We ﬁnd that the results from all three theories con-verge when
is large with reasonable agreement betweenviscous potential and fully viscous ﬂow with
.The convergence results apply to two liquids as well as toliquid and gas.
1 Introduction
Capillary instability of a liquid cylinder of mean radius
leading to capillary collapse can be described as a neck-down due to surface tension
in which ﬂuid is ejectedfrom the throat of the neck, leading to a smaller neck andgreater neckdown capillary force as seen in the diagram inﬁgure 1.1.The dynamical theory of instability of a long cylindri-cal column of liquid of radius
under the action of capil-lary force was given by Rayleigh (1879) following earlierwork by Plateau (1873) who showed that a long cylin-der of liquid is unstable to disturbances with wavelengthsgreater than
. Rayleigh showed that the effect of in-ertia is such that the wavelength
corresponding to themode of maximum instability is
exceed-ing very considerably the circumference of the cylinder.Theideathatthewavelengthassociatedwithfastestgrow-ing growth rate would become dominant and be observedin practice was ﬁrst put forward by Rayleigh (1879). Theanalysis of Rayleigh is based on potential ﬂow of an invis-cid liquid neglecting the effect of the outside ﬂuid. (Look-
Capillary/C-InstabSh03-27.tex
2ing forward, we here note that it is possible and useful todo an analysis of this problem based on the potential ﬂowof a viscous ﬂuid).An attempt to account for viscous effects was madeby Rayleigh (1892) again neglecting the effect of the sur-rounding ﬂuid. One of the effects considered is meant toaccount for the forward motion of an inviscid ﬂuid with aresistance proportional to velocity. The effect of viscos-ity is treated in the special case in which the viscosityis so great that inertia may be neglected. He shows thatthe wavelength for maximum growth is very large, strictlyinﬁnite. He says, “... long threads do not tend to dividethemselves into drops at mutual distances comparable towith the diameter of the cylinder, but rather to give wayby attenuation at few and distant places.”Weber (1931) extended Rayleigh’s theory by consider-ing an effect of viscosity and that of surrounding air on thestability of a columnar jet. He showed that viscosity doesnot alter the value of the cut-off wavenumber predicted bytheinviscidtheoryandthattheinﬂuenceoftheambientairis not signiﬁcant if the forward speed of the jet is small.Indeed the effects of the ambient ﬂuid, which can be liq-uid or gas, might be signiﬁcant in various circumstances.The problem, yet to be considered for liquid jets, is the su-perposition of Kelvin-Helmholtz and capillary instability.Tomotika(1935)consideredthestabilitytoaxisymmet-ric disturbances of a long cylindrical column of viscousliquid in another viscous ﬂuid under the supposition thatthe ﬂuids are not driven to move relative to one another.He derived the dispersion relation for the fully viscouscase (his (33), our (2.17); he solved it only under the as-sumptionthat thetimederivativeintheequationof motioncan be neglected but the time derivative in the kinematiccondition is taken into account (his (34)). These approxi-mations lead herein to the asymptotic solution in the limitof
, in which the wavenumber giving maximum in-stability, say
in our notation, depends only upon theviscosity ratio
, where
is the viscosity of liquid in another ﬂuid of viscosity
;
takes a max-imum as
at
(which givesthe critical
), while
is reduced to zero as
(single ﬂuid column of high viscosity studied byRayleigh (1982)) and
(single hollow in a ﬂuid of high viscosity), as shown in his ﬁgure 2. The parameter
is important for
as is shown ﬁgure 5.1 but not forsmall
(Stokes ﬂow); inertia is not important as
.The effect of viscosity on the stability of a liquid cylin-der when the surrounding ﬂuid is neglected and on a hol-low (dynamically passive) cylinder in a viscous liquid wastreated brieﬂy by Chandraseckhar (1961). The parameter
which can be identiﬁed as a Reynolds numberbased on a velocity
appears in the dispersion rela-tion derived there.Tomotika’s problem was studied by Lee and Flumer-felt (1981) without making the approximations used byTomotika, focusing on the elucidation of various limitingcases deﬁned in terms of three dimensionless parameters,a density ratio, a viscosity ratio and the Ohnesorge num-ber
. They showed for variousvalues of
and a ﬁxed value of the density ratio that
is bounded below by Tomotika’s limiting case (
)and above by the inviscid case (
) that is inde-pendent of
; refer to their ﬁgure 4.
R u
Capillary Force
/
r r = R+
Figure 1.1:
Capillary instability. The force
forces ﬂuid from the throat, decreasing
leading to collapse.
In this paper we treat the general fully viscous problemconsidered by Tomotika. This problem is resolved com-pletelywithout approximation andis appliedto 14 pairsof viscous ﬂuids. Theories based on viscous and inviscid po-tential ﬂows are constructed and compared with the fullyviscous analysis and with each other.It is perhaps necessary to call attention to the fact itis neither necessary or desirable to put the viscosities to
Capillary/C-InstabSh03-27.tex
3zero when considering potential ﬂows. The Navier-Stokesequations are satisﬁed by potential ﬂow; the viscous termis identically zero when the vorticity is zero but the vis-cous stresses are not zero (Joseph and Liao 1994). It is notpossibletosatisfytheno-slipconditionatasolidboundaryor the continuity of the tangential component of velocityand shear stress at a ﬂuid-ﬂuid boundary when the veloc-ity is given by a potential. The viscous stresses enter intothe viscous potential ﬂow analysis of free surface prob-lems through the normal stress balance (2.10) at the inter-face. Viscous potential ﬂow analysis gives good approx-imations to fully viscous ﬂows in cases where the shearsfromthegasﬂowarenegligible;theRayleigh-Plessetbub-ble is a potential ﬂow which satisﬁes the Navier-Stokesequations and all the interface conditions. Joseph, Be-langer and Beavers (1999) constructed a viscous poten-tial ﬂow analysis of the Rayleigh-Taylor instability whichcan scarcely be distinguished from the exact fully viscousanalysis. Similar agreements were demonstrated for vis-coelastic ﬂuids by Joseph, Beavers & Funada (2002). In arecent paper, Funada and Joseph (2001) analyzed Kelvin-Helmholtz instability of a plane gas-liquid layer using vis-cous potential ﬂow. This problem is not amenable to anal-ysisforthefullyviscouscaseforseveralreasonsidentiﬁedin their paper. The study leads to unexpectedresults whichappear to agree with experiments.Thepresentproblemofcapillaryinstabilitycanbefullyresolvedinthefully viscousandpotentialﬂowcasesanditallows us to precisely identify the limits in which differentapproximations work well.
2 Governing equations and dimen-sionless parameters
The problem formulation for the capillary instability of a viscous cylinder in another viscous ﬂuid was formu-lated by Tomotika (1935). It is based on a normal modeanalysis of the linearized Navier Stokes equations. To-motika’s problem was resolved for many limiting cases byLee and Flumerfelt (1981); they also recognized that thesolution was controlled by three dimensionless parame-ters,
,
and a Reynolds number
where
. A brief review of thegoverning equations in dimensionless form is given be-low to facilitate comparison with viscous and inviscid po-tential ﬂow. Consider the stability of a liquid cylinder of radius
(
) with viscosity
and density
sur-rounded by another ﬂuid with viscosity
and density
under capillary forces generated by interfacial tension
.Our convention is that
. In the inverse problemthe viscous liquid is outside. The analysis is done in cylin-drical coordinates
and only axisymmetric distur-bances independent of
are considered.The governing Navier-Stokes equations and interfaceconditions for disturbance of the cylinder at rest are madedimensionless with the following scales
length, velocity, time, pressure
where
(2.1)
2.1 Linearized disturbance equations
The system of equations for small disturbances are givenby
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)with
(2.8)The kinematic condition at the interface
is given by
(2.9)The normal stress balance at the interface is given by
(2.10)The velocity normal to the interface and the velocity tan-gential to the interface are continuous as
(2.11)The tangential stress balance at the interface is given by
(2.12)
Capillary/C-InstabSh03-27.tex
4
2.2 Dispersion relation for fully viscous ﬂow (FVF)
Following Tomotika, the velocities are expressed with a stream function
:
(2.13)and the basic variables are expressed in normal modes:
(2.14)
(2.15)
(2.16)where the modiﬁed Bessel functions of the ﬁrst order are denoted by
for the ﬁrst kind and
for the second kind. Sub-stitution of (2.14)-(2.16) into (2.11), (2.12) and (2.10) leads to the solvability condition, which is given as the dispersionrelation:
(2.17)where
(2.18)
(2.19)
(2.20)with
(2.21)For small
,
and
may be expanded around
up to the ﬁrst order terms, which yields the expansion of (2.17)-(2.20)and the resultant dispersion relation is Equation(34) in Tomotika’s paper; that is,
(a function of
and
)
.
2.3 More viscous ﬂuid outside
The equations are the same except that subscripts
and
are interchanged,
are replaced with
(2.22)The capillary collapse is still controlled by the more viscous ﬂuid
where
is now the viscosity of thesurrounding ﬂuid. We shall index all our results with
,
,
.

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