H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 97
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He showed that the cylindricalinterface is unstable, and tends to pinch off,forming bubbles. The most rapidly amplifieddisturbance has an axial I\-avelength ofpghr,lu~l-r,/r,X= 6.48DO(b)(11.'isti)from which we estimate the pinched-off bubblediameter asD = 1.7Do(I 1.i8b)Hole configuration models2rFIGURE11.30.-Stabilityof some capillary-supportedsystems.found that the maximum values of p~hrlaranged from 1.5 to 3.5. Highest values wereobtained with a water-Teflon system, which isprobably most closely modeled by figure11.30(e). These experiments prove that simpleanalyses of this type can yield stability criteriasufficiently accurate for many engineering design purposes.Instabilities in Other Capillary SystemsLamb (ref. 11.34) presents analysis whichshows that a cylinder of liquid is unstable underthe action of surface tension in zero g.
Thecolumn of liquid tends to pinch together andform drops of liquid. If Do is the cylinderdiameter, the most amplified disturbance isfound to have an axial wavelength of\Xz4.5Do(11.77a)T h e implication of the linearized theory is thatthe drops formed on breakup of the cylinder willhave a diameter of approximatelyD = 1.5D0(11.77b)These predictions have been confirmed experimentally and are beauti:ully demonstrnted inthe motion picture of Trefethen (ref.
11.12).Anliker and Beam (ref. 11.36) studied thestability of uniform liquid layers spread overcylinders and spheres and found that the filmon a fully covered sphere or cylinder is al~w-aysunstable. This contradicts the simplifiedanalysis of Lee (ref. 11.37) wvhich dealt primarily with longitudinal ware instabilities.With water, Lee always observed instt~bility,but with oil he did not. I t might be thoughtthat this was a viscous effect, but the analysisof Bellman and Pennington (ref.
11.38) indicates that viscosity does riot alter the criticalBond number. The surface tension value foroil used b y Lee seems high, and hence his oilexperiments are questionable.Otto (ref. 11.39) reports experiments on thestability of a meniscus in a horizontal tube.The critical Bond numbers, based on tuberadius, ranged from about 1 to about 2.5;this \-as not attributed by Otto to contactangle variations, but may well have been dueto differences in contact angle or to contactangle hysteresis, wvhich Satterlee and Reynolds(ref.
11.24) found to be important in theirsloshing studies.11.4LOW-s SLOSHING AND SOME RELATEDPROBLEMSIntroductionThe sloshing analyses presented in previouschapters apply whenever the Bond number issubstantially greater than unity. At low gLIQUID PROPELLANT BEHAVIOR AT L O W AND ZERO GDisturbed interface 7Undisturbed interfaceSubscript notation is used to denote partialdifferentiation with respect to the subscriptedvariable. The boundary condition appropriatefor the inviscid problem a t the solid walls isFIGURE11.31.-Definitionsfor axisymmetric meniscussloshing analysis.these analyses must be modified to account forthe distortion of the undisturbed interfaceshape and the added interface pressure tlifference caused by capillarity.
IThile these newfactors make the mathematics considernblymore complicated, the general approach remains unchanged. In this section, me u-illformulate the sloshing analysis for 2isisymmeti-icmenisci and present the results for t ! p:irti(w~larly useful case of a cylindrical tank with anaxial body force.
For more details of thetreatment, see reference 11.24.General Formulation for Low-g SloshingWe presume that the essential features ofsloshing motions will be revealed by an inviscid,incompressible flow theory. Consider the liquidin the axisyrnmetric tank of figure 11.31. Thebounding walls are denoted by W, the undisturbed interface by S, the disturbed interfaceby S'. The domain occupied by the liquidin the undisturbed and disturbed c o d ,vurationsare D and D', respectively. We denote thevelocity components in the r, 6,and z directionsby u, v, and w, respectively.
The fluid meetsLL uuw W Z L at~ c o n t c t angie, el measured in aplane perpendicular to the contact line, C.Assuming the motion to be inviscid, incompressible, and irrotational, the equation ofcontinuity may be expressed in terms of thevelocity potential, 4, asSince the motion is irrotational, Bernoulli'sequn tion is satisfied throughout the liquiddomain, and in particular at the free surface.The unsteady form of Bernoulli's equationtipplied a t the perturbed surface iswhere $ ( t ) is an arbitrary function of time.Here P is the pressure just inside of the interfaceand V is the total fluid velocity.
I t should bepointed out that the only point a t whichdynamics enters the analysis is through equation(11.82).P is related to the pressure just outside ofthe liquid, P,, byP,-P=uK(11.83)where the total interface curvature, K , isgiven by (ref. 11.24)-4 kinematic relation between the velocityfield and the interface motion is essential tothis analysis. This kinemat,ic relation is (ref.11.24)Here the velocity potential is defined by1s ~ = - + ~ + + ~ s ~ + &sa~(11.55)414THE DYNAMIC BEHAVIOR OF LIQUIDSIf the meniscus is flat, or nearly so, only thefirst term on the right in equation (11.85)need be retained, as was the case in earlierchapters.
However, with the strongly curvedmeniscus, the second term will also contributeto a linearized analysis, and all three termswould have t o be used in a proper nonlineartreatment.For convenience, the arbitrary function oftime in equation (11.52) may be set equal tothe equilibrium liquid pressure a t the vertexof the equilibrium free surface shape, dividedby the liquid density. Then, using the kinematic condition, the free surface conditionreduces toEquation (11.79) is the equation which must,be solved, and the boundary conditions whichmust be imposed are equations (11.81) rtnd(11.86), plus a contact angle condition. Thckinematic connection between the shape of thefree surface and the velocity potential isequntion (11.85).
Obviously, the nonlinearproblem is quite difficult, and in the interestsof obtaining a useful approximate method, wewill nnw linearize the problem by idealizingthat the perturbations are small.The Linearized ProblemSurface boundary condition:lVall boundary condition:Kinematic condition at the surface:h, = - &+A$, on S(11.88d)I n addition, we require a contact pointcondition. We assume that the perturbationin the slope h, a t the contact point is given,to order a, byhr=7h on C(11.88e)The parameter 7 provides a macroscopic wayfor including the effects of contact anglehysteresis (ref.
11.24). I n walls with curvature in 1111 axial plane, 7 also includes thecurvature effects.Now that the problem has been linearized,we can reduce it to the familiar sort of eigenvalue problem for the natural sloshing modes.We put+= J ~ @ ( R9,, 2 ) sin w t(ii.ag)We assume that 4 is of the order e, where a isa small parameter, and express the interface asThe values of 4 and its derivatives on the disas a Taylor,sturbed surface areseries ex~ansion about their values on theundisturbed meniscus. This in effect allowsthe single boundary condition to be "transferred" from S' to S. The linearized problemthen becomes (to order c) :!a y e r e n t i d equation:v24=0 in Dand obtain the dimensionless eigenvalue problem for @, H, and Q:DiJerential eqrlatim:v%=O in D(1 1.90a)Wall bounday condition:(11.88a)@,=OonW(11.gob)IiiI>_.,---.....--..-.---,.
.- .---.-. --.-..LLIQUID PROPELLANT BEHAVIOR AT LOW AND ZERO GSurjace boundary condition:Kinematic condition:z.Contact angle condition:I n spite of the linearization, we still are facedwith a formidable problem; except for verysimple cases, the domain D is not well suitedfor solution of equation (11.90a), for themeniscus shape F(R) is known only numerically.We will now obtain a solution for the specialcase of a cylindrical tank with a flat bottomand a flat interface, and subsequent,lyformulateand use a variational scheme for analysis of morerealistic situations.Solution for a Special CaseFor t,he special case indicated by figure11.32, solutions sat,isfying equation (11.90a)in D are@=J,,(k,, .R) cos m6 cosh [k,,,(L+Z)](11.91)where J,(x) is the Bessel function of order m.Note that m=O corresponds to the first symmetric perturbation, while m = l gives thefirst antisymmetric (lateral sloshing) mode.The eigenvalues k,,, are determined fromequation (11.gob), which givesNote that equation (11.90d) is satisfied byequation (11.91).
Using the kinematic culldition, the surface boundary condition becomesT h i s will be satisfied by solutions of the formof equation (11.91) only for particular valuesof Q, namelyFIGURE11.32.-Geometry for a special case.Note that we \\-ere unable to enforce any conbnctpoint condition. However, i t may easily beshown that t.he contact angle is 'unchanged forthis solution, which therefore corresponds tor=0.The lowest sloshing frequency (thefundamental) is obtained for m = n = l , and isNote that for very large Bond numbers,equation (11.95) gives-Q2/Bo=b=w2ro/g=l.841 tanh (1.S41L)(11.96)which is the result obtained \\-hen surfacetension is neglected.As !oiig as _Re>-1.39,D2 \\-ill be positiveand hence the motion oscillatory.
But forBo< -3.39 SZZ is imaginary, corresponding tos growing disturbance. The neutral stabilit,ypoint is therefore Bo=-3.39, which agrees exactly with the value obtained from the simplerpotential energy considerations. (Note thatthe sign difference is due to the different direction for positive g. Note also that the criticalvalue is.
independent of depth, which is also inagreement with our earlier discussions.)416THE DYNAMIC BEHAVIOR OF LIQUIDSAVariational FormulationIn general, exact solution of the eigenvalueproblem, equation (11.90), is not possible, andapproximate methods are preferable. I n particular, a variational formulntion can be made(ref. 11.24). Following the usual formalisms ofthe calculus of rnrint'ions, one can show that,1°\\-hen and it,s H are eigenfunctions satisfyingequation (11.90), associated with eigenvalue Qz,the functionDet (A,,+Q2Bij) =O(11 .loo)The roots of this expression provide t,he approximation to the eigenvalues.Some considerable reduction in the amountof numerical work can be obtained if the wallgeometry is such that each member of the approximating functions \I.(" can be chosen so asto satisfy v2@=0in D.
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