H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 95
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In fact, it is possible to perform thesame experiment with a tube 0.2 centimeter indiameter. However, if one att,empts to inverta column of similar length in rt 3.0-centimeterdiameter tube, t,he mercury invariably fallsdown and out of t,he tube. We therefore expec,tthat there is a critical radius below which the404THE DYNAMIC B E ~ V I O ROF LIQUIDSinterface is stable, and above which it is unstable.
The determination of this radius andthe associated critical Bond number will beconsidered momentarily.TABLE11 .l.-Method of Determining the MeniscusConfiguration as a Function of Liquid Volumefor a Spherical Tank@@@v",la"1v;v, - va-E2 -g+eRIR;Sin BL,IR;(lo-CosB)haw7(e) -el1L 'a5Liquid volume 1container volume1.0L(3R SR,Calculations below for 8 = 20'. pg~:lo * 20IFIGURE11.18.-Menisciin a epherical tank.FIGURE11.19.- menisci in inverted tubes.IA first thought might be to attribute thesupport of the heavy liquid at the top of thetube to capillary forces. However, one findsthat wetting fluids can also be suspended atthe top of a glass tube, as in the alcohol-in-glassthermometer.
In fact, by being careful in theinversion process, water can be suspended in a0.6-centimeter-diameter tube. For wettingliquids, the surface tension forces tend to assistgravity in pulling the liquid down. In nonwetting liquids, some slight assisting supportis given by surface tension, but the liquid isbasically supported by pressure. Atmosphericpressure acts on the interface, and a reducedpressure acts at the top end of the tube (fig.ll.l9(a)).Since the ratio of weight to pressure force isindependent of diameter, it is possible to supportfluid in any diameter tube, provided, of course,that the length is small enough so that a positiveabsolute pressure results at the top of the tube.But we observe instability of the column inlarge tubes, while in the smaller tubes the interface is stabilized by surface tension.What would happen if we opened the top endof the tube, equalizing the pressures? Ofcourse the liquid would fall down.
This wouldnot be considered an instability because thesuspended configuration is impossible in itself,and one can only sensibly discuss stabilitywhen the static configuration could in principleoccur. But suppose we restricted the bottomend of the tube, causing an interface of different curvature to be formed, as shown infigure 11.19(c). Suppose the interface curvature is such as to produce a pressure in theliquid at the hole which is higher than atmos-,LIQUID PROPELLANT BEHAVIOR AT LOW AND ZERO Gpheric, perhaps even high enough to supportthe liquid column.
The tallest column will bepossible when the radius of curvature of theprotruding interface is least, for this yields thegreatest "capillary pressure." Here me see anexample of a system in which capillary forcessimultaneously provide both support andstability.The stability of a meniscus may be analyzedin a number of ways. One can do a dynamicanalysis of the liquid motion, and then look fornormal modes of vibration which grow inamplitude with increasing time. This is adifficult computational t,ask, for although theproblem may be linearized in a study of smalldisturbances, the domain in which the solutionmust be obtained is not simple. I t is possiblcto show that the eigenvalue equation giving thestability limit obtained from an inviscid theoryof this type is exactly as would be obtainedsimply b y invoking the thermodynamic criterion of marginal stability; that is, that variations in the meniscus shape produce no variationsin the total potential energy (refs.
11.21 and11.22). This condition marks the boundarybetween states of minimum free energy (stablestates) and maximum free energy (unstablestates). We will use the latter method, and insection 11.4 will demonstrate for a simple casethat the dynamic stability analysis indeed yieldsthe same critical Bond number.According to our thermodynamic criteria,for a meniscus shap'e to be stable there must beno small perturbation of its shape which canlead to a reduction in the total potential energy.For example, consider the meniscus formed bythe liquid a t the top of a spherical container, asshown in figure 11.20(a). Suppose the meniscussomehow finds itself with exactly the sameshape, but with its posicion rotated sEght!yabout the center of the sphere.
The capillarypotential energy would be unchanged, but thegravitational potential energy would be decreased, and hence the total potential energywould be reduced. Consequently, the meniscus in an "inverted" spherical tank is alwaysunstable. This fact has important applicationin propellant tank design.All "inverted" menisci become unstable a tsufficiently high g's. The limit of stability isThe inverted$ * mAn unstableperturbation--9//'(a)< Ilcrit' gcrit'gcrit*StableNeutrally stableUnstable1(b)FIGURE11.20.-Stabilityof menisci.-DisturbedinterfaceUndisturbed interface-a+aFIGURE11.21.-Liquidin an inverted channel.given by the point a t which the total potentialenergy of the capillary system ceases to be uminimum, and starts becoming a maximum.A simple analogy helpful in grasping this concept is given in figure 11.20(b).T o illustrate the methods, we will now carryout the stability analysis for a simple case.Consider the liquid having 0=90° in an invertedtwo-dimensional channel, as shown in figure11.21.
The capillary potential energy per unitof channel length is, for ti two-dimensionaldisturbance,THE DYNAMIC BEHAVIOR OF LIQUIDSpEc=S-TrJ1+(gydxThen, substituting back into equation (11.68),we findBoC,lt=r2/4=2.46(11.70)number marks the onset of this instability.Now-, the stability criteria obtained from thepotential energy considerations admit anyY(,y) which satisfies the proper contact conditions. Hence, if instability occurs for unormal-mode perturbation, both methods willyield identical mlues for the critical Bondnumber.Concus (ref. 11.31) has used this approach totreat the instability in the ti\-o-dimensionalchannel for contact angles other than 90'.He found that the 90' case was the moststable, and that the critical Bond number forfully wetting or nontvetting liquids was onlyabout 0.72 (compared to 2.46 for 0=90°).This same trend is f o ~ ~ nind asisymmetricmenisci, which are of more practical interest.However, an important result of a later studyby Concus (ref.
11.22) indicated that an_vmeniscus for which the curvature changes signwas unstable. While this \\-as shown onlyfor u two-dimensional meniscus, it might bespeculated that it would be true for itnvmeniscus, when applied to the totnl meuiscrlscurvature. These criteria might provide M goodway of estimating critical Bond numbers formenisci of irregular shape.The extension of the stability analysis t ogeneralized axisyrnmetric menisci is considerably more complicated ; the details differ somewhat from the previous analysis, but they arethe same in concept. A summary of theanalysis is given in reference 11.1, and weshall simply present the results here.The critical Bond number may be relatedto the radius of curvature (r,) of the bounding~vallsa t the contact point in ct plane containingthe axis, to the contact angle, and to the anglea! which the meniscus makes with the axis.This relationship may be expressed nondimensionally in the formThis is identical with the result obtained t ~ sa byproduct of the sloshing analysis.
I n thesloshing analysis, one considers the normalmode oscillations, each of which yields t iperiodic interface perturbation Y,,(,Y). If theBond number is such that one of these normalmodes grows in time, rather than oscillating,the system is unstable. The critical BondThis relat,ionship was determined by numericalsolution and is shown graphically in figure11.22. The meanings of the symbols aregiven on that fi,oure.The curves of figure 11.22 can be used t oexamine the stttbility of practically any axisym-(11'65a)Hence, for small perturbations y(x), the variation in capillary energy is, to first orderS+"-a(-1 dydX2 dx6p,73e~g --(u.6jb)The varitttion in gravitational potential energyper unit of length from the equilibrium (flatin terfnce) value isDenoting Y=y/a, X=x/a, B ~ = ~ g a ~ /thea,totnl varitttion in potential energy map beespressed nondimensionully ttsFor stability, 6 P E = 0 for any small virtualvariation Y ( X ) .
The neutrally stable caseoccurs when 6PE vanishes. Hence, in a stablestateTo find the critical Bond number, we shouldseek the shape perturbation P(X) which minimizes the ratio 11/12above. This leads us toan isoperimetric problem in the calculus ofvariations (ref. 1123) ; the solution whichmaintains the contact angle fixed isLIQUID PROPELLANT BEHAVIOR AT LOW mDR, sinZERO G8FIGURE11.22.-Neutral stability curves for axisyrnrnetric menisci.metric meniscus. Several important generalconclusions may be drawn from them directly:(1) An "inverted meniscus" is unstable a t allBond numbers for positive wall cur\-ature(concave with respect to the meniscus) such thatthe parameter r / ( r , sin 8 ) is greater than1.This includes the spherical tank case previouslydiscussed.+229-64s 0-65-27(2) The critical Bond number for zero wallcurvature (r,= co ) depends only upon the anglewhich the meniscus makes with respect to thessis, and is otherwise independent of the contactangle. This allo\vs the stability criteria forconical capillaries to be represented in themanner of figure 11.23.(3) The critical Bond number for \vull-bound408T H E DYNAMIC BEHAVIOR OFT~~~~~~---V-Drop or bubble volumeI90'60"30'O0-30°-60'-90'aFIGURE11.24.-Stabilityof wall-bound drops and bubbles.curvature occurs at the bottom edge of uvertical tube, or in a horizontal orifice; if themeniscus is flat, the critical Bond number is14.6.
Experiments of Duprez (ref. 11.27)support this re diction. The case a = O O givesthe m a s i m o ~ ~sizei of t i 1011:: bubble which C L I I ~persist in H straight vertical tube, arid is intigeemen t ~ v iht the theory 21 n d experimentsreported by Bretherton (ref. 1 1 2 8 ) . Twoexamples illustrating the use of these curves inFIGURE11.23.-Critical Bond number for axisyn~metrictisisymmetric meniscus stnbility tinalysis tiremenisci in a conical tank.given in figure 11.25.There are a number of important tipp1ic:~tionsof these results in the design of liquid stortigedrops tind bubbles mtiy be determined fromsystems for rlse in space. If it is desirable thtitfigure 11.22, and expressed in terms of a criticti1the liquid not remain in the inverted position,volume.
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