Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1994"
TL;DR: In this paper, the authors present analytical tip region solutions for fracture width and pressure when a power law fluid drives a plane strain fracture in an impermeable linear elastic solid, where the tip region stress is dominated by a singularity which is particular to the hydraulic fracturing problem.
Abstract: We present analytical tip region solutions for fracture width and pressure when a power law fluid drives a plane strain fracture in an impermeable linear elastic solid. Our main result is an intermediate asymptotic solution in which the tip region stress is dominated by a singularity which is particular to the hydraulic fracturing problem. Moreover this singularity is weaker than the inverse square root singularity of linear elastic fracture mechanics. We also show how the solution for a semi-infinite crack may be exploited to obtain a useful approximation for the finite case.
288 citations
TL;DR: In this paper, the authors used a first-order theory and derived asymptotic results for the maximum run-up within the validity of the theory for different types of N -waves.
Abstract: Anecdotal reports of tsunamis climbing up coastlines have often described the shoreline receding significantly before the tsunami waves run-up on the beach. These waves are caused by tsunamigenic earthquakes close to the shoreline, when the generated wave does not have sufficient propagation distance to evolve into leading-elevation waves or a series of solitary waves. Yet all previous run-up investigations have modelled periodic waves or solitary waves which initially only run-up on the beach. In our studies of these initially receding shorelines, we have found a class of N -shaped waves with very interesting and counterintuitive behaviour which may lead to a new paradigm for the studies of tsunami run-up. We will use a first-order theory and we will derive asymptotic results for the maximum run-up within the validity of the theory for different types of N -waves. We have observed that leading depression N -waves run-up higher than leading elevation N -waves, suggesting that perhaps the solitary wave model may not be adequate for predicting an upper limit for the run-up of near-shore generated tsunamis.
286 citations
TL;DR: In this paper, the authors present a study of some basic modes of precise deposition and solidification of molten microdrops, and experimental results and theoretical analyses are given for various basic deposition modes, including columnar deposition at both low and high frequencies, sweep deposition of continuous beads on flat surfaces, and repeated sweep deposition for buildup of larger objects or materials.
Abstract: Objects, materials or components may be built up by precise deposition of molten microdrops under controlled thermal conditions. This provides a means of ‘digital microfabrication’, or fabrication of 3D objects microdrop by microdrop under complete computer control much in the same way as 2D hard copy is obtained by ink-jet printing. In this paper we present a study of some basic modes of precise deposition and solidification of molten microdrops. The conditions required for controlled deposition are discussed, and experimental results and theoretical analyses are given for various basic deposition modes. These include columnar (i. e. drop-on-drop) deposition at both low and high frequencies, sweep deposition of continuous beads on flat surfaces, and repeated sweep deposition for buildup of larger objects or materials. The theory provides a means for generalizing our particular experimental results, which were obtained with hard waxes, to other melts. An important parameter in the theory is the solidification angle, that is, the apparent contact angle of the solidified melt. Our study indicates that in microscale deposition this angle appears under some conditions to be a property of the melt material, the target material and the characteristic temperatures involved, independent of the spreading dynamics.
202 citations
TL;DR: In this article, the first sharp diffraction peak (FSDP) observed in the structure factor of many liquid and glassy materials is approached by treating this peak as a distinct feature, and the FSDP confers a marked oscillatory character of periodicity 2π/k 1 (where k 1 is the position) on the IRO when the local structural units, which give rise to the density fluctuations on the intermediate range order, exist as stable entities for a timescale τ ≫ 5 × 10 -12 s.
Abstract: The problem of those discernible features of the intermediate range order (IRO) which can be attributed to the first sharp diffraction peak (FSDP) observed in the structure factor of many liquid and glassy materials is approached by treating this peak as a distinct feature. It is found, by considering the measured partial structure factors, S αβ ( k ), for molten ZnCl 2 , GeSe 2 , MgCl 2 , NiBr 2 and Nil 2 and the measured total structure factors, F ( k ), for glassy SiO 2 , PS 4 and liquid CCl 4 , that the propensity of the FSDP to have a prominent effect on the underlying features of the IRO depends noticeably on the system type. Specifically, the FSDP confers a marked oscillatory character of periodicity 2π/ k 1 (where k 1 is the FSDP position) on the IRO when the local structural units, which give rise to the density fluctuations on the IRO scale, exist as stable entities for a timescale τ ≫ 5 × 10 -12 s. The FSDP therefore accounts for the discernible features of the underlying IRO for the viscous glass forming liquids ZnCl 2 and GeSe 2 , for the glasses SiO 2 and PS 4 , and for the molecular liquid CCl 4 . The influence of the FSDP on the IRO is less pronounced for molten MgCl 2 and is negligible for molten NiBr 2 and Nil 2 , both of which have a high cation mobility which leads to a relative instability of the Ni 2+ centred structural units. The effect on the FSDP of temperature and pressure are briefly considered as are the development of the FSDP in molten ZnX 2 (when X is changed from Cl to I to Br) and the minimum size of r -space model which is required if the FSDP is to be accurately predicted.
185 citations
TL;DR: In this article, the differential constraint method is applied to prove that a parabolic evolution equation admits infinitely many characteristic second order reductions, but admits a non-characteristic second order reduction if and only if it is linearizable.
Abstract: Direct reductions of partial differential equations to systems of ordinary differential equations are in one-to-one correspondence with compatible differential constraints. The differential constraint method is applied to prove that a parabolic evolution equation admits infinitely many characteristic second order reductions, but admits a non-characteristic second order reduction if and only if it is linearizable.
154 citations
TL;DR: In this article, conditions for the existence of a centre in two-dimensional systems are considered along the lines of Darboux, and they are used in the search for maximal numbers of bifurcating limit cycles.
Abstract: Conditions for the existence of a centre in two-dimensional systems are considered along the lines of Darboux. We show how these methods can be used in the search for maximal numbers of bifurcating limit cycles. We also extend the method to include more degenerate cases such as are encountered in less generic systems. These lead to new classes of integrals. In particular, the Kukles system is considered, and new centre conditions for this system are obtained.
146 citations
TL;DR: In this article, the radiation pressure exerted by an axisymmetric sound field on a rigid sphere suspended freely in a viscous fluid is calculated, the sphere being considered as having an arbitrary radius relative to the sound and viscous wavelengths.
Abstract: The acoustic radiation pressure exerted by an axisymmetric sound field on a rigid sphere suspended freely in a viscous fluid is calculated, the sphere being considered as having an arbitrary radius relative to the sound and viscous wavelengths. The limiting cases of special interest are then investigated, namely, the acoustic radiation force due to a plane progressive and plane standing wave is examined for the limiting cases when the sound wavelength is much more than both the sphere radius and the viscous wavelength and the sphere radius is, in its turn, small or large compared with the viscous wavelength. It is shown that the influence of the viscosity of the fluid surrounding the sphere on the radiation force can be quite considerable in both a quantitative and a qualitative sense. The case of a fastened sphere is also considered.
137 citations
TL;DR: In this paper, it was shown that the nonlinear interaction terms between cross-and longshore currents represent a dispersive mechanism that has an effect similar to the required mixing, and that the dispersion effect is at least an order of magnitude larger than the turbulent mixing.
Abstract: Longshore currents have in the past been analysed assuming that the lateral mixing could be attributed to turbulent processes. It is found, however, that the mixing that can be justified by assuming an eddy viscosity v t = l√k where l is the turbulent length scale, k the turbulent kinetic energy, combined with reasonable estimates for l and k is at least an order of magnitude smaller than required to explain the measured cross-shore variations of longshore currents. In this paper, it is shown that the nonlinear interaction terms between cross-and longshore currents represent a dispersive mechanism that has an effect similar to the required mixing. The mechanism is a generalization of the one-dimensional dispersion effect in a pipe discovered by Taylor (1954) and the three-dimensional dispersion in ocean currents on the continental shelf found by Fischer (1978). Numerical results are given for the dispersion effect, for the ensuing cross-shore variation of the longshore current and for the vertical profiles of the longshore currents inside as well as outside the surf zone. It is found that the dispersion effect is at least an order of magnitude larger than the turbulent mixing and that the characteristics of the results are in agreement with the sparse experimental data material available.
137 citations
TL;DR: In this article, two phase-field models for solidification of a eutectic alloy were discussed, one based on a regular solution model for the solid with a chemical miscibility gap and another based on two order parameters to distinguish the liquid phase and the two solid phases.
Abstract: In this paper we discuss two phase-field models for solidification of a eutectic alloy, a situation in which a liquid may transform into two distinct solid phases. The first is based on a regular solution model for the solid with a chemical miscibility gap. This model suffers from the deficiency that, in the sharp interface limit, it approximates a free-boundary problem in which the surface energy of the solid-solid interface is zero and consequently mechanical equilibrium at a trijunction requires that the solid-solid interface has zero dihedral angle (locally planar). We propose a second model which uses two order parameters to distinguish the liquid phase and the two solid phases. We provide a thermodynamically consistent derivation of this phase-field model which ensures that the local entropy production is positive. We conduct a sharp interface asymptotic analysis of the liquid-solid phase transition and show it is governed by a free-boundary problem in which both surface energy and interface kinetics are present. Finally, we consider a sharp interface asymptotic analysis of a stationary trijunction between the two solid phases and the liquid phase, from which we recover the condition that the interfacial surface tensions are in mechanical equilibrium (Young's equation). This sharp interface analysis compares favourably with numerical solutions of the phase-field model appropriate to a trijunction.
137 citations
TL;DR: In this article, a single soft asperity is modelled by a blunt copper wedge in sliding contact with a flat hard steel surface under conditions of boundary lubrication, and two ways in which the process can be driven have been identified: (i) pummelling of the soft surface by the asperities of the hard surface and (ii) cyclic stress concentrations which occur at the edges of a hard slider.
Abstract: Many researchers have observed metallic wear debris in the form of very thin platelets. In particular Akagaki & Kato (1987) revealed how such debris can be formed by progressive plastic extrusion from the edges of the irregularities on the softer of two sliding surfaces. This behaviour has been reproduced in experiments reported here, in which a single soft asperity is modelled by a blunt copper wedge in sliding contact with a flat hard steel surface under conditions of boundary lubrication. This progressive extrusion with continuous sliding is attributed to ‘plastic ratchetting’ and two ways in which the process can be driven have been identified: (i) pummelling of the soft surface by the asperities of the hard surface and (ii) cyclic stressing of the soft surface by the stress concentrations which occur at the edges of a hard slider. The kinematical shakedown theorem from the theory of plasticity is used to determine the asperity contact pressure necessary to drive these ratchetting processes. A significant feature of this mechanism is that it can occur under frictionless conditions.
135 citations
TL;DR: In this article, the authors considered the Cauchy problem for the quasilinear heat equation where σ > 0 is a fixed constant, with the critical exponent in the source term β = βc = σ + 1 + (σ + 2)/N.
Abstract: We consider the Cauchy problem for the quasilinear heat equationwhere σ > 0 is a fixed constant, with the critical exponent in the source term β = βc = σ + 1 + 2/N. It is well-known that if β ∈(1,βc) then any non-negative weak solution u(x, t)≢0 blows up in a finite time. For the semilinear heat equation (HE) with σ = 0, the above result was proved by H. Fujita [4].In the present paper we prove that u ≢ 0 blows up in the critical case β = σ + 1 + 2/N with σ > 0. A similar result is valid for the equation with gradient-dependent diffusivitywith σ > 0, and the critical exponent β = σ + 1 + (σ + 2)/N.
TL;DR: It is shown that this family of mixed wavelets with the structure wm, n(x) provides a complete set of orthogonal basis functions for signal analysis and provides greater frequency discrimination than is possible with harmonic wavelets whose frequency interval is always an octave.
Abstract: The concept of a harmonic wavelet is generalized to describe a family of mixed wavelets with the structure w m, n (x) = {exp (i n 2π x ) – exp (i m 2π x )}/i( n – m ) 2π x . It is shown that this family provides a complete set of orthogonal basis functions for signal analysis. By choosing the (real) numbers m and n (not necessarily integers) appropriately, wavelets whose frequency content ascends according to the musical scale can be generated. These musical wavelets provide greater frequency discrimination than is possible with harmonic wavelets whose frequency interval is always an octave. An example of the wavelet analysis of music illustrates possible applications.
TL;DR: In this article, the asymptotic behavior of the solutions to a nonlocal evolution equation which arises in models of phase separation is investigated, where stationary spatially nonhomogeneous solutions exist, representing the interface profile between stable phases.
Abstract: The paper is concerned with the asymptotic behaviour of the solutions to a nonlocal evolution equation which arises in models of phase separation. As in the Allen–Cahn equations, stationary spatially nonhomogeneous solutions exist, which represent the interface profile between stable phases. Local stability of these interface profiles is proved.
TL;DR: The primary quantum state diffusion (PSD) theory as mentioned in this paper is an alternative quantum theory from which classical dynamics, quantum dynamics and localization dynamics are derived, based on four principles, that a system is represented by an operator, its state by a normalized state vector, the state vector satisfies a Langevin-Ito state diffusion equation, and the resultant density operator for an ensemble must satisfy an equation of elementary Lindblad form.
Abstract: Primary quantum state diffusion (PSD) theory is an alternative quantum theory from which classical dynamics, quantum dynamics and localization dynamics are derived. It is based on four principles, that a system is represented by an operator, its state by a normalized state vector, the state vector satisfies a Langevin-Ito state diffusion equation, and the resultant density operator for an ensemble must satisfy an equation of elementary Lindblad form. There are three conditions. The ז 0 first determines the operator, to within an undetermined universal time constant ז 0 . The second and third conditions put opposing bounds on ז 0 . Dissipation of coherence is distinguished from destruction of coherence. The state diffusion destroys coherence and produces the localization or reduction that makes classical dynamics possible. PSD theory is a development of the environmental quantum state diffusion theory of Gisin and Percival and particularly resembles earlier proposals by Gisin and by Milburn. It is also related to the spontaneous localization theories of Ghirardi, Rimini and Weber, of Diosi and of Pearle. The non-relativistic PSD theory is of value only for systems which occupy small regions of space. Special relativity is needed for more extended systems even when they contain only slowly moving massive particles. Experiments on coherence lifetimes and matter interferometry are proposed which either measure ז 0 or put bounds on it, and which might distinguish between PSD and ordinary quantum mechanics.
TL;DR: In this paper, a procedure is described to generate fundamental solutions or Green's functions for time harmonic point forces and sources, where the linearity of the field equations permits the Green's function to be represented as an integral over the surface of a unit sphere, and the integrand is the solution of a one-dimensional impulse response problem.
Abstract: A procedure is described to generate fundamental solutions or Green's functions for time harmonic point forces and sources. The linearity of the field equations permits the Green's function to be represented as an integral over the surface of a unit sphere, where the integrand is the solution of a one-dimensional impulse response problem. The method is demonstrated for the theories of piezoelectricity, thermoelasticity, and poroelasticity. Time domain analogues are discussed and compared with known expressions for anisotropic elasticity.
TL;DR: This article measured the kinematic viscosity of glycerol-water mixtures in the temperature range 10-50 °C using a series of Ubbelohde viscometers.
Abstract: We have measured the kinematic viscosity of glycerol-water mixtures, for glycerol mass fractions ranging from 0 to 1, in the temperature range 10-50 °C. The measurements were made by using a series of Ubbelohde viscometers. Apart from comprehensiveness and comparative accuracy the present measurements expose serious errors in the limited data that were earlier available on such mixtures. It is shown that all the data can be reasonably represented by the empirical correlation (In ν m - In ν w )/(In ν g - In ν w ) = x g [1 + (1 - x g ) { a + bx g + cx g 2 }], where ν w , v g and ν m are the kinematic viscosities of water, glycerol and the mixture respectively and x g is the mass fraction of glycerol in the mixture. The constants a, b and c are tabulated in the paper as functions of temperature. This correlation can now be used at a given temperature to tailor make a mixture of prescribed kinematic viscosity. While this paper is addressed, principally, to fluid dynamicists these results should be of interest to physicists studying the liquid state and physical chemists interested in mixtures.
TL;DR: In this paper, a stochastic model for rainfall at a single site, in which storms arrive in a Poisson process, each storm generating a cluster of rain cells, with each cell having a random duration and a random intensity.
Abstract: This paper further develops a stochastic model for rainfall at a single site, in which storms arrive in a Poisson process, each storm generating a cluster of rain cells, with each cell having a random duration and a random intensity. The model is generalized by allowing each generated cell to be of n types. The duration of each cell is an exponential random variable that has parameter dependent on the cell type. The distribution of cell intensity is also dependent on the cell type, so that the generalized model provides a correlation between the intensities and durations of the generated rain cells. The case for two cell types is considered in some detail. The cells are categorized as either ‘heavy’ or ‘light9, where the heavy cells have a shorter expected lifetime than the light cells, which agrees with observational studies on precipitation fields. Properties are derived and used to fit the model to rainfall data at a single site. The adequacy of fit is assessed by considering properties not used in the fitting procedure but which are of interest in applications, e. g. extreme values. Further properties are derived which enable the model to be fitted to multi-site rainfall data.
TL;DR: For positive solutions of a class of quasilinear elliptic equations in a domain Ω⊂ℝN via a generalisation of Serrin's sweeping principle, it is shown in this article that the solution is radially symmetric.
Abstract: Existence and uniqueness results are proved for positive solutions of a class of quasilinear elliptic equations in a domain Ω⊂ℝN via a generalisation of Serrin's sweeping principle. In the case when Ω is an annulus, it is shown that the solution is radially symmetric.
TL;DR: In this paper, the authors consider the question of whether a set of matrices with no rank-one connections can support a nontrivial Young measure limit of gradients with mean value 0.
Abstract: We consider the following question: given a set of matrices ⊁ with no rank-one connections, does it support a nontrivial Young measure limit of gradients? Our main results are these: (a) a Young measure can be supported on four incompatible matrices; (b) in two space dimensions, a Young measure cannot be supported on finitely many incompatible elastic wells; (c) in three or more space dimensions, a Young measure can be supported on three incompatible elastic wells; and (d) if ⊁ supports a nontrivial Young measure with mean value 0, then the linear span of ⊁ must contain a matrix of rank one.
TL;DR: In this paper, the authors studied all the stationary solutions of the form u(r)einθ to the complex-valued Ginzburg-Landau equation on the complex plane.
Abstract: In this paper, we study all the stationary solutions of the form u(r)einθ to the complex-valued Ginzburg–Landau equation on the complex plane: here (r, θ) are the polar coordinates, and n is any real number. In particular, we show that there exists a unique solution which approaches to a nonzero constant as r → ∞.
TL;DR: In this paper, the authors obtained the Γ(L1(Ώ))-limit of the sequence where Ee is the family of anisotropic perturbationsof the nonconvex functional of vector-valued functions.
Abstract: We obtain the Γ(L1(Ώ))-limit of the sequencewhere Ee is the family of anisotropic perturbationsof the nonconvex functional of vector-valued functionsThe proof relies on the blow-up argument introduced by Fonseca and Muller.
TL;DR: In this paper, the authors examined the agreement between laboratory experiments performed with three-dimensional granular avalanches moving along a partly curved surface and their numerical predictions and showed that the numerical results fit the experimental data surprisingly well.
Abstract: In this paper the agreement between laboratory experiments performed with three-dimensional granular avalanches moving along a partly curved surface and their numerical predictions shall be examined. First, the most important elements of the theory describing the flow of a cohesionless granular material down a rough bed are presented. Based on the depth-averaged model equations, an advanced numerical integration scheme is developped by making use of a Lagrangian representation (i. e., the grid moves with the deforming pile) and a finite difference approximation that handles the numerically two-dimensional problem accurately. Second, experiments are described that were conducted with a finite mass of granular material moving down, respectively, an inclined plane and a surface consisting of an inclined and a horizontal plane connected by a curved transition area; the initial geometry of the avalanche is generated by a spherical cap. Third, for a number of different experiments a comparison is carried out between the experimentally determined positions of the granular avalanche during its motion and the numerical prediction of these positions. It shows that the numerical results fit the experimental data surprisingly well.
TL;DR: In this article, a new formulation of recursion operators is presented which eliminates diffi-culties associated with integro-differential operators and treats recursion operator and their inverses on an equal footing.
Abstract: A new formulation of recursion operators is presented which eliminates difficulties associated with integro-differential operators. This interpretation treats recursion operators and their inverses on an equal footing. Efficient techniques for constructing non-local symmetries of differential equations result.
TL;DR: In this paper, exact continuum forms of balance (for mass, momentum, and energy) are established as relations between weighted space-time averages of molecular quantities computed at any supra-atomic length-time scales.
Abstract: Exact continuum forms of balance (for mass, momentum, and energy) are established as relations between weighted space-time averages of molecular quantities computed at any supra-atomic length-time scales. The choice of weighting function, and the physical interpretation of all terms, are discussed.
TL;DR: In this paper, the authors studied radially symmetric solutions of a nonlinear elliptic partial differential equation in R 2 with critical Sobolev growth, i.e., the nonlinearity is of exponential type.
Abstract: We study radially symmetric solutions of a nonlinear elliptic partial differential equation in R 2 with critical Sobolev growth, i. e. the nonlinearity is of exponential type. This problem arises from a wide variety of important areas in theoretical physics including superconductivity and cosmology. Our results lead to many interesting implications for the physical problems considered. For example, for the self-dual Chern–Simons theory, we are able to conclude that the electric charge, magnetic flux, or energy of a non-topological N -vortex solution may assume any prescribed value above an explicit lower bound. For the Einstein-matter-gauge equations, we find a necessary and sufficient condition for the existence of a self-dual cosmic string solution. Such a condition imposes an obstruction for the winding number of a string in terms of the universal gravitational constant.
TL;DR: In this article, a three-dimensional extension of the two-dimensional Savage-Hutter model for granular avalanches is presented, where the avalanche is described as a threedimensional incompressible continuum obeying a Coulomb dry friction law at the base and a Mohr-Coulomb plastic yield criterion in the interior.
Abstract: This paper deals with three-dimensional gravity driven free surface flows of piles of granular materials along bottom profiles that are weakly curved downward and plane laterally. We present in detail a three-dimensional extension of the two-dimensional Savage-Hutter model for such granular avalanches. In this extended model, the avalanche is described as a three-dimensional incompressible continuum obeying a Coulomb dry friction law at the base and a Mohr-Coulomb plastic yield criterion in the interior. Based on this, the balance laws of mass and linear momentum and kinematic and stress boundary conditions at the free surface and the base are used to derive depth-averaged dynamic equations that describe the temporal evolution of the height and the depth-averaged horizontal velocity components as functions of position and time. A computation is performed for a pile of granular material with an initial spherical cap geometry moving down an inclined plane.
TL;DR: In this paper, the first two Dirichlet eigenvalues of a planar domain D of unit area were determined numerically, and the problem of minimizing the n th eigenvalue when the minimizing domain is diconnected was studied.
Abstract: For each planar domain D of unit area, the first two Dirichlet eigenvalues of —∆ on D determine a point (λ 1 ( D ), λ 2 ( D ) in the (λ 1 , λ 2 ) plane. As D varies over all such domains, this point varies over a set R which we determine. Its boundary consists of two semi-infinite straight lines and a curve connecting their endpoints. This curve is found numerially. We also show how to minimize the n th eigenvalue when the minimizing domain is diconnected. For n = 3 we show that the minimizing domain is connected and that λ 3 is a local minimum for D a circular disc.
TL;DR: In this paper, the authors investigate numerically the flow of an electrically conducting fluid in a differentially rotating spherical shell, in the presence of an imposed magnetic field, and show that the interior flow consists essentially of a solid-body rotation, with the precise rate determined by a torque balance between the inner and outer Ekman-Hartmann boundary layers.
Abstract: I investigate numerically the flow of an electrically conducting fluid in a differentially rotating spherical shell, in the presence of an imposed magnetic field. For a very weak field the flow is seen to consist of an Ekman layer on the inner and outer spherical boundaries, and a Stewartson layer on the cylinder circumscribing the inner sphere and parallel to the axis of rotation, in agreement with the classical non-magnetic analysis. As the field strength is increased, the non-magnetic Ekman layers merge smoothly into magnetic Ekman-Hartmann layers, and the Stewartson layer is suppressed. In the fully magnetic regime the interior flow consists essentially of a solid-body rotation, with the precise rate determined by a torque balance between the inner and outer Ekman-Hartmann boundary layers.
TL;DR: In this paper, the authors explain why in the non-simply laced case, the dual affine Weyl group is needed and why in this case it is necessary to use monodromic systems (certain local systems defined on subvarieties of a line bundle) over an affine flag manifold.
Abstract: There has been recently substantial progress in the programme of expressing the characters of irreducible modular representations of a semisimple group over a field of positive characteristic in terms of combinatorics of affine Hecke algebras (see Andersen et al . 1992; Kazhdan & Lusztig 1979). This paper is a further contribution to this programme: I explain why in the non-simply laced case, the ‘dual’ affine Weyl group is needed and why in this case it is necessary to use monodromic systems (certain local systems defined on subvarieties of a line bundle) over an affine flag manifold.
TL;DR: In this article, a detailed account of the origin of induced air motion within spray jets is given, and this lays the basis for a new one-dimensional model for predicting the induced axial air velocity.
Abstract: A study of the fundamental mechanics of the droplet and gas motion in liquid sprays is presented in this paper. Only vertical sprays without any externally applied gas flow are considered. First a detailed account of the origin of induced air motion within spray jets is given, and this lays the basis for a new one-dimensional model for predicting the induced axial air velocity. Two main flow zones (zone I and zone II) are identified, where the droplet velocity is much greater and of the same order as the induced air velocity respectively. Within zone I there is a near-sub-zone I , close to the nozzle where the droplet velocities deviate little from their initial values, and it is found that the air velocity decreases or increases to a maximum value, depending on whether its initial value is greater or less than a critical value, which itself is a function of the drag coefficient, the initial spray radius and the droplet velocity. In this zone the average induced air velocity decays more slowly, as z -½ ( z being the downstream distance) than the rate of decay, as z -1 , in regular unforced jets. Further downstream in the adjacent forced jet sub-zone , the drag of the faster moving droplets forces an air jet to develop with a rate of growth that is determined by the turbulence if the angle of the spray droplets is small or by the angle of the spray if the angle is large. In this second sub-zone, which typically extends to the stopping distance of the droplets, the flow is largely independent of the flow in the near sub-zone. The 1D model was applied to a rose-head axisymmetric spray and a flat-fan agricultural spray. The calculations agree closely with experimental observations. To calculate the radial variation of the air velocity a 2D axisymmetric model was developed where the air velocity was obtained in the form of a similarity solution. The predictions were in good agreement with the measurements of Binark & Ranz (1958). Finally it is shown that the 1D and the 2D models are consistent with each other.