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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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