Breakup Criterion Research Articles

Classroom Note: The Lagrange--Charpit Method

Previous article Next article Classroom Note: The Lagrange--Charpit MethodManuel DelgadoManuel Delgadohttps://doi.org/10.1137/S0036144595293534PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractWe give a rigorous description of the Lagrange--Charpit method used to find a complete integral of a nonlinear p.d.e. adapted for a university course in differential equations.[1] Google Scholar[2] S. S. Demidov, The study of partial differential equations of the first order in the 18th and 19th centuries, Archive for History of Exact Sciences, 26 (1982), pp. 325–350. a32 AHESAN 0003-9519 Arch. Hist. Exact Sci. CrossrefISIGoogle Scholar[3] Google Scholar[4] Google Scholar[5] Google Scholar[6] R. Hermann, Geometric construction and properties of some families of solutions of nonlinear partial differential equations ↼I↽, J. Math. Phys., 24 (1983), pp. 510–521. jmp JMAPAQ 0022-2488 J. Math. Phys. CrossrefISIGoogle Scholar[7] Google Scholar[8] Google Scholar[9] Google ScholarKeywordsintegral surfacecomplete integralPfaff's equation Previous article Next article FiguresRelatedReferencesCited byDetails Operator splitting method for the stochastic production-inventory model equationComputers & Industrial Engineering, Vol. 25 Cross Ref Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. III. Shock and rarefaction waves in RG flows reveal limitations of the N→∞ limit in O(N) -type models13 September 2022 | Physical Review D, Vol. 106, No. 6 Cross Ref The Lagrange–Charpit Theory of the Hamilton–Jacobi Problem22 November 2021 | Mediterranean Journal of Mathematics, Vol. 19, No. 1 Cross Ref Interplay of cellular states: Role of delay as control mechanismPhysica A: Statistical Mechanics and its Applications, Vol. 572 Cross Ref Stochastic Modeling of Gene Expression14 May 2021 Cross Ref Analytical solutions to validate 3D numerical methods for solving direct ultraviolet photoionization and charge recombination of aerosol nanoparticles with laminar flow in circular ductsJournal of Applied Physics, Vol. 128, No. 8 Cross Ref Solvability of the Caldeira–Leggett modelCanadian Journal of Physics, Vol. 98, No. 7 Cross Ref Modulation of Galactic Cosmic Rays by Plasma Disturbances Propagating Through the Local Interstellar Medium in the Outer Heliosheath19 May 2020 | The Astrophysical Journal, Vol. 895, No. 1 Cross Ref Analysing Differential Equations with Uncertainties via the Liouville-Gibbs Theorem: Theory and Applications24 November 2020 Cross Ref Computers & Mathematics with Applications, Vol. 78, No. 9 Cross Ref An analytical film drainage model and breakup criterion for Taylor bubbles in slug flow in inclined round pipesInternational Journal of Multiphase Flow, Vol. 84 Cross Ref A new framework for solving partial differential equations using semi-analytical explicit RK(N)-type integratorsJournal of Computational and Applied Mathematics, Vol. 301 Cross Ref Analytical Solutions for Composition-Dependent CoagulationMathematical Problems in Engineering, Vol. 2016 Cross Ref Two Asymptotic Conditions in Queue with MMPP Arrivals and Feedback14 February 2017 Cross Ref The ghost solid methods for the elastic–plastic solid–solid interface and the ϑ-criterionJournal of Computational Physics, Vol. 302 Cross Ref FRW in Cosmological Self-creation Theory2 April 2013 | International Journal of Theoretical Physics, Vol. 52, No. 8 Cross Ref On Lie systems and Kummer-Schwarz equationsJournal of Mathematical Physics, Vol. 54, No. 3 Cross Ref Dynamic Programming Applications in Optics28 March 2013 Cross Ref A fractional characteristic method for solving fractional partial differential equationsApplied Mathematics Letters, Vol. 24, No. 7 Cross Ref Dispersion Analysis in Hypersonic Flight During Planetary Entry Using Stochastic Liouville EquationJournal of Guidance, Control, and Dynamics, Vol. 34, No. 2 Cross Ref Exact solution of a coagulation equation with a product kernel in the multicomponent casePhysica D: Nonlinear Phenomena, Vol. 239, No. 5 Cross Ref Exact solution of Smoluchowski's continuous multi-component equation with an additive kernel18 May 2007 | Europhysics Letters (EPL), Vol. 78, No. 5 Cross Ref Volume 39, Issue 2| 1997SIAM Review History Published online:02 August 2006 InformationCopyright © 1997 Society for Industrial and Applied MathematicsKeywordsintegral surfacecomplete integralPfaff's equationMSC codes35-0135F20PDF Download Article & Publication DataArticle DOI:10.1137/S0036144595293534Article page range:pp. 298-304ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

Criteria for droplet breakup while undergoing simultaneous evaporation and discharge

Criteria for droplet breakup while undergoing simultaneous evaporation and discharge

Application of Crystal Lattice Disintegration Criteria to Compute Minimum Shock induced reactive conditions in solid explosives and inert materials

AbstractA threshold particle velocity criteria derived by E.R. Fitzgerald for the beginning of crystal lattice breakup and disintegration has been applied to shocked explosives and an inert material. In shocked explosives, reactions leading to detonation occur above a certain “threshold” magnitude. The computed crystal lattice breakup shock pressures compare rather well with observed experimental “threshold” shock pressures for six high explosives. The six explosives are: Comp‐B3, Comp‐B, TNT, PBX‐9404, Tetryl, and H‐6.In addition, the crystal lattice breakup criteria provides an explanation for the observed lowering of the detonation “threshold” shock pressure as the explosives are made more porous or less dense.Finally, the shock pressures, at which output from thermocouples embedded in shocked materials (PBX‐9404 and Plexiglass) increases dramatically, compare favorably with predictions based on crystal lattice disintegration criteria.Consequently, it is tentatively concluded that crystal lattice breakup, or self‐sustained phonon fission as Fitzgerald calls it, is responsible for the initiation of detonation in shocked explosives and enhanced thermocouple output in shocked materials. It is also postulated that the lattice breakup phenomena is also responsible for phase changes, increased chemical reactivity, and anomalous electrical activity which are observed in certain inert materials under relatively low level shock loading.

Transition theory and experimental comparisons on (I) amplification into streets and (II) a strongly nonlinear break-up criterion

The two stages I, II are studied by using recent nonlinear theory and then compared with the experiments of Nishioka et al . (1979) on the transition of plane Poiseuille flow. The first stage I starts at low amplitude from warped input, which is deformed through weakly nonlinear interaction into a blow-up in amplitude and phase accompanied by spanwise focusing into streets. This leads into the strongly nonlinear stage II. It holds for a broad range of interactive boundary layers and related flows, to all of which the nonlinear break-up criterion applies. The experimental comparisons on I, II for channel flow overall show encouraging quantitative agreement, supporting recent comparisons (in the boundary-layer setting) of the description of stage I in Stewart & Smith (1992) with the experiments of Klebanoff & Tidstrom (1959) and of the break-up criterion of Smith (1988 a ) with the computations of Peridier et al (1991 a , b ).

Breakup of a liquid jet in supersonic crossflow

A theoretical study of the breakup of a circular liquid jet injected transversely into a supersonic air stream is conducted. Two different criteria for breakup are explored in the context of a previously developed model for the behavior of liquid jets in compressible crossflow (Heister et al., 1989). The local sonic point criterion first proposed by Schetz, et al. (1980) is explored, in addition to an auxiliary criterion put forth by Clark (1964) based on surface-tension stability. It is found that the local sonic point appears to provide a more reasonable approximation to the actual location of jet breakup, based on comparisons with limited experimental data.

Tidal disruption of solid bodies

This paper introduces three new considerations to the venerable problem of stress, strain, and breakup in solid satellites and stray bodies subject to tidal perturbations. A novel treatment of compressible bodies indicates that compressibility alters the stresses by several percent, more than the neglect of “Love number” or higher-order tides in most cases. Realistic failure criteria imply that stony objects generally fail by tensile fracture, but that icy bodies may fail by either tensile or shear fracture. The question of crack propagation is reexamined, and breakup criteria are given according to the size, strength, and composition of the object.

Breakup criteria for fluid particles

A simple mechanistic model is developed based on the combination of Kelvin-Helmholtz and Rayleigh-Taylor instability theory to describe the breakup of freely rising or falling fluid particles, i.e. bubbles and drops. The breakup is predicted to occur if the growth rate of interfacial waves on the leading front is faster than the rate at which waves propagate around the interface to the side of the particle. Based on this theoretical model and available experimental data, simple correlations are developed to predict the maximum size a fluid particle can reach. Predicted values of the breakup diameter are compared with experimental data for cases of freely rising bubbles, falling drops in gas and freely falling or rising drops in immiscible liquids. The results are extended to predict the maximum droplet size in a high-velocity gas field which is the most interesting case in terms of practical applications. Good agreement between predicted values and experimental data indicates that the principal mechanisms involved in the fluid particle breakup process are properly accounted for by the proposed model.

Nuclear thermometers for heavy-ion collisions.

Measurement of the ratio of deuterons to excited deuterons is prescribed asa method for determining temperatures and breakup criteria inintermediate-energy heavy-ion collisions. The sudden coalescence model iscontrasted with a thermal breakup picture where ratios are determined byBoltzmann factors.

A semiempirical model of catastrophic breakup processes

Abstract Several observable properties of asteroids, including their sizes, shapes, and rotations, are the direct outcome or are in some way connected with catastrophic breakup events caused by high-velocity impacts. On a much smaller scale, these catastrophic impacts have been investigated in the laboratory, but a comprehensive physical theory of the fragmentation phenomenon is not yet available. We have attempted to reinterpret the results of laboratory impact experiments as well as the asteroids properties in terms of a semiempirical theory which derives the observable properties of the fragments from a suitable velocity field of the target material, assumed to arise from the explosion due to the impact and from the original rotation of the target. Apart from this latter term (which is usually smaller), the field is essentially radial (though not central) and in terms of it we have defined a “local” breakup criterion and derived the size, bulk velocity, shape, and rotation of the final fragments. Numerical simulations based on the idea of studying the local fragment properties as a function of initial position in the target have allowed us to compare the results of this model (varying several free parameters) with the experimental results and, in a few cases, with asteroid properties. In particular, the fragment rotation is found to arise from the superposition of two effects, one due to the curl of the velocity field and the other to the asphericity of the fragment shapes. Some observed properties of the main asteroid families (the likely outcomes of asteroidal catastrophic collisions) are well reproduced by the model, and observable correlations among ejection velocity, size, rotation, and shape can be predicted.

On Mixed-Layer Modeling of the Stratocumulus-Topped Marine Boundary Layer

Research aircraft measurements of a well-developed marine stratocumulus cloud-topped boundary layer, made in June 1981 off the coast of California, are analyzed using the saturation point method developed by Betts. Estimates of the cloud-top entrainment rate made from the measurements permit construction of a mixing diagram in which the physics of the layer collapse to a single mixing line when diabatic processes and nonstationarity are accounted for. This is possible because the vertically-integrated (mixed-layer) budget equations balance to within measurement uncertainty. The mixing diagram allows calculation of cloud-top entrainment from the geometry of surface, mixed-layer and above-inversion parcel saturation points (corrected for diabatic processes) and the cloud-top cooling rate. This method, basically an inversion of the thermodynamic budgets, can also be used to calculate surface fluxes. It should be adaptable to routine meteorological data. While no insight is given into model parameterization of entrainment, it is concluded that, for stratocumulus layers such as the one measured in the data presented here, a mixed-layer model is likely to adequately represent the thermodynamic interactions. However, the data also indicate that the criterion for breakup of a stratocumulus deck used in such models has not been adequately developed.

Viscous heating of expanding fireballs

The effect of a finite mean free path on the expansion stage of central collisions between heavy nuclei is studied in a nonrelativistic fluid dynamical approach. A gas of point particles with localized interactions is used. Kinetic theory then provides the thermal conductivity and viscosity coefficients in terms of the nucleon-nucleon cross section. About 10% additional entropy is generated during the expansion stage of central collisions at a beam energy of 800 MeV/A due to viscous heating of the system. Surprisingly there is only a weak dependence on the mass of the colliding nuclei. These results are in substantial agreement with recent intranuclear cascade calculations, although at present the fluid dynamical results are slightly ambiguous due to an uncertainty in the breakup criterion.

Droplet breakup regimes and criteria for their existence

An analysis of experimental and theoretical studies of droplet breakup by a gas flow in shock tubes and nozzles is presented. A system of criteria defining droplet breakup regimes is developed.

Studies on Droplet Deformation and Breakup. II. Breakup of a Droplet in Nonuniform Shear Flow

The phenomenon of droplet breakup in nonuniform shear flow (e.g., in a cylindrical tube) is of practical importance in many polymer processing operations. A theoretical and experimental study was carried out on the stability of a long threadlike viscoalastic droplet suspended in a viscoelastic medium flowing through a cylindrical tube. The stability analysis was carried out using linearized dynamic equations with the Jeffreys model. A perturbation method was used for the case of small wave numbers, and the case of small Reynolds numbers was also considered. A theoretical criterion for droplet breakup was obtained in terms of the rheological properties of the fluids concerned, and the flow conditions. The results reveal that a long threadlike droplet cannot be stable under all disturbances, and that both the elastic and viscous properties affect the stability of extending droplets. Furthermore, it was found that the interfacial tension tends to destabilize the extending threadline. For the experimental study, the same apparatus and fluid systems as those described in part I of this series [H. B. Chin and C. D. Han, J. Rheol., 23(5), 557–590 (1979)] were used. The breakup patterns of droplets were recorded on both movie and still films. It was observed that the droplets were elongated greatly at the entrance region of the tube and then, under certain flow conditions, broke up into smaller droplets in the cylindrical tube section. The breakup conditions were correlated to the rheological properties of the droplet phase and the suspending medium.

Steady long slender droplets in two-dimensional straining motion

The recent analysis by Acrivos & Lo (1978) concerning the breakup of a long slender droplet in an axisymmetric straining motion is extended to the case of a two-dimensional hyperbolic flow. It is found that, although the cross-section of the droplet becomes significantly non-circular, the theoretical criterion for breakup is effectively the same as in the axisymmetric case. The theoretical predictions are in good agreement with the available experimental results.

Deformation and breakup of a single slender drop in an extensional flow

The deformation and conditions for breakup of a single slender drop placed symmetrically in a uniaxial extensional flow are examined theoretically. For the case of an inviscid drop in zero-Reynolds-number flow, Buckmaster (1972) showed, using slender-body analysis, that the shape of the drop is given byr≡εR(z)=ε(1−|z|ν)/2ν, where εεγ/Gμland γ is the interfacial tension,Gthe strength of the extensional flow, μ the viscosity of the suspending fluid andlthe drop half-length; alsov= ½P− 1, wherePis the unknown constant pressure inside the drop rendered dimensionless with respect toGμ. By requiring thatRbe analytic atz= 0, Buckmaster then concluded thatvhad to be an even integer and thereby obtained a countably infinite set of slender profiles for any (large) value of the flow strengthG. In the present work, the expression forR(z)shown above is obtained readily using the method of inner and outer expansions, the method failing when |z|[les ]O(ε) and ν is not an even integer. Thus, in general, a new solution is needed to describe the shape within the ‘singular’ region |z|[les ]O(ε). The requirement that the two solutions match in their domain of overlap then leads to the conclusion thatvcan be either equal to 2 or greater than or equal to 3. However, a stability analysis reveals that only the solution withv= 2 is stable, and hence a unique shape exists.Next, drops of low viscosity μi=O(ε2μ) are examined in zero-Reynolds-number flow. Here, again, a unique solution is obtained according to which a steady shape cannot exist if (Gμa/γ) (μi/μ)⅙> 0·148, where$a \equiv(3V4\pi)^{\frac{1}{3}}$andVis the volume of the drop. This breakup criterion is identical to that found by Taylor (1964). A similar analysis for the case of an inviscid drop in a flow with non-zero Reynolds number shows that drop breakup will occur if$(G\mu a/\gamma) (\rho a\gamma/\mu^2)^{\frac{1}{5}} > 0.284$, where ρ is the density of the suspending fluid. Finally, when μi=O(ε2μ) and inertial effects are neglected within the drop but retained in the surrounding fluid, the critical value of$(G\mu a/\gamma) (\rho a\gamma/\mu^2)^{\frac{1}{5}}$required for drop breakup is found as a function of the dimensionless group\[ (\rho a\gamma/\mu^2)^{\frac{1}{5}}(\mu/\mu_i)^{\frac{1}{6}}, \]which depends only on the physical properties of the system and the size of the drop. These last two results are the first which take into account inertial effects in determining the deformation and breakup conditions of a drop placed in a shear field.

Stability of a Submerged Frozen Crust

Consideration is given to the stability of a thin frozen crust of infinite extent growing between two different fluid materials. A linear, inviscid stability analysis is employed to obtain an estimate for the crust breakup time. The stability formulation parallels that of Rayleigh and Kelvin for the generation of waves at the plane interface between two different fluids and is used to develop formulas defining necessary conditions for frozen crust breakup. An approximate crust breakup criterion is proposed, viz, that the crust growth (or crystallization) time exceed the crust breakup time. The criterion shows reasonable agreement with experimental observations of the stability of a growing frozen crust in a gravity field alone (Rayleigh instability).