11 results
Ion evaporation from Taylor cones of propylene carbonate mixed with ionic liquids
- I. GUERRERO, R. BOCANEGRA, F. J. HIGUERA, J. FERNANDEZ DE LA MORA
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- Journal:
- Journal of Fluid Mechanics / Volume 591 / 25 November 2007
- Published online by Cambridge University Press:
- 30 October 2007, pp. 437-459
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A combined experimental and numerical approach is used to extract information on the kinetics of ion evaporation from the region of high electric field around the tip of a Taylor cone of the neutral solvent propylene carbonate (PC) mixed with two ionic liquids. On the numerical side, the electric field on the surface of the liquid is computed in the absence of evaporation by solving the electrohydrodynamic problem in this region within the framework of the leaky dielectric model. These computations justify the approximate (2% max error) scaling Emax = β Ek for the maximum electric field on the surface, with Ek = γ1/2 ϵ0−2/3 (K/Q)1/6 for 0.111 < K < 0.888 S m−1 and a numerical value of β ≈ 0.76. Here γ is the surface tension of PC, ϵ0 is the electrical permittivity of vacuum, and K and Q are the liquid electrical conductivity and flow rate. On the experimental side, 16 different propylene carbonate solutions with either of the ionic liquids 1-ethyl-3-methylimidazolium tetrafluoroborate (EMI-BF4) or EMI-bis(trifluoro-methylsulfonyl)imide (EMI-Im) are electrosprayed in a vacuum from a single Taylor cone, and their emissions of charged drops and ions are analysed by time-of-flight mass spectrometry at varying liquid flow rates Q. The sprays contain exclusively drops at large Q, both for small and for large electrical conductivities K, but enter a mixed ion–drop regime at sufficiently large K and small Q. Interestingly, the mixtures containing 10% and 15% (vol) EMI-Im exhibit no measurable ion currents at high Q, but approach a purely ionic regime (almost no drops) at small Q. The charge/mass ratio for the drops produced in these two mixtures increases continuously with decreasing Q, and gets very close to ionic values. Measured ion currents are represented versus computed maximum electric fields Emax on the liquid surface to infer ion evaporation kinetics. Comparison of measured ion currents with predictions from ion evaporation theory yields an anomalously low activation energy (~1.1 eV). This paradox appears to be due to alteration of the pure conj–eet electric field in the scaling laws used for the pure cone–jet regime, due to the substantial ion current density arising even when the ion current is relatively small. Elimination of this interference would require future ion current measurements in the 10–100 pA level. The electrical propulsion characteristics of the emissions from these liquids are determined and found to be excellent, particularly for 10% and 15% (vol) EMI-Im.
Brownian motion far from equilibrium: a hypersonic approach
- P. Riesco-Chueca, J. Fernández de la Mora
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- Journal:
- Journal of Fluid Mechanics / Volume 214 / May 1990
- Published online by Cambridge University Press:
- 26 April 2006, pp. 639-663
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An investigation is carried out on the effects of Brownian agitation in the motion of small particles in a carrier gas in situations far from equilibrium. Although the standard near-equilibrium closure of the hydrodynamic equations is not valid for the heavy particles, the smallness of their speed of thermal agitation allows an alternative systematic hypersonic closure. The hypersonic equations are solved for two known instances where the kinetic Fokker–Planck equation describing the non-equilibrium particle distribution function admits exact solutions. These problems are characterized by a null or a spatially constant value for the gradient of the velocity field in the carrier gas, both being free from boundary surfaces. In the first case, where the background velocity is uniform, a fundamental solution (expressed as an integral) is obtained for the steady flow of particles from a point source; this result has obvious applications for the description of the Brownian broadening of particle streamlines. An asymptotic integration of the fundamental solution yields analytical expressions for the particle hydrodynamic properties valid everywhere except near the source, where a direct integration of the Vlasov equation completes the description. The exact solution for the second example, where the background velocity field gradient is uniform, is taken from the literature. Once these reference solutions have been established, the hypersonic equations are attacked by a variety of methods. In particular, for the uniform steady flow, a boundary-layer analysis yields analytical results identical to those obtained from the asymptotic evaluation of the kinetic fundamental solution. In both problems, the agreement found between kinetic and hydrodynamic solutions is excellent even for values of order one of the inverse particle Mach number, the expansion parameter of the hypersonic theory.
The current emitted by highly conducting Taylor cones
- J. Fernández De La Mora, I. G. Loscertales
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- Journal:
- Journal of Fluid Mechanics / Volume 260 / 10 February 1994
- Published online by Cambridge University Press:
- 26 April 2006, pp. 155-184
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When a liquid meniscus held at the exit of a metallic capillary tube is charged to a high voltage V, the free surface often takes the form of a cone whose apex emits a steady microjet, and thus injects a certain charge I and liquid volume Q per unit time into the surrounding gas. This work deals with liquids with relatively large conductivities K, for which the jet diameter dj is much smaller than the diameter dn of the capillary tube. In the limit dj/dn → 0, the structure of the jet (dj and I, in particular) becomes independent of electrostatic parameters such as V or the electrode configuration, being governed mostly by the liquid properties and flow rate Q. Furthermore, the measured current is given approximately by I = f(ε) (γQK/ε)½ for a wide variety of liquids and conditions (ε, and γ are, respectively, the dielectric constant of the liquid and the coefficient of interfacial tension; f(ε) is shown in figure 11). The following explanation is proposed for this behaviour. Convection associated with the liquid flow Q transports the net surface charge towards the cone tip. This upsets the electrostatic surface charge distribution slightly at distances r from the apex large compared to a certain charge relaxation length λ, but substantially when r ∼ λ. When the fluid motion is modelled as a sink flow, λ is of the order of r* = (Qεε0/K)$\frac13$ (ε0 is the electrical permittivity of vacuum). If, in addition, the surface charge density is described through Taylor's theory, the corresponding surface current convected towards the apex scales as Is ∼ (γQK/ε)½, as observed for the spray current. The sink flow hypothesis is shown to be realistic for sufficiently small jet Reynolds numbers. In a few photographs of ethylene glycol cone jets, we find the rough scaling dj ∼ 0.4r* for the jet diameter, which shows that the jet forms as soon as charge relaxation effects set in. In the limit ε [Gt ] 1, an upper bound is found for the convected current at the virtual cone apex, which accounts for only one-quarter of the total measured spray current. The rest of the charge must accordingly reach the head of the jet by conduction through the bulk.
Solution breakdown in a family of self-similar nearly inviscid axisymmetric vortices
- R. Fernandez-Feria, J. Fernandez de la mora, A. Barrero
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- Journal:
- Journal of Fluid Mechanics / Volume 305 / 25 December 1995
- Published online by Cambridge University Press:
- 26 April 2006, pp. 77-91
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Many axisymmetric vortex cores are found to have an external azimuthal velocity v, which diverges with a negative power of the distance r to their axis of symmetry. This singularity can be regularized through a near-axis boundary layer approximation to the Navier-Stokes equations, as first done by Long for the case of a vortex with potential swirl, v∼r−1. The present work considers the more general situation of a family of self-similar inviscid vortices for which v∼rm−2, where m is in the range 0 n< m < 2. This includes Longs Vortex for the case m =1. The corresponding solutions also exhibit self-similar structure, and have the interesting property of losing existence when the ratio of the inviscid near-axis swirl to axial velocity (the swirl parameter) is either larger (when 1m < 2) or smaller (when 0m < 1) than an m-dependent critical value. This behaviour shows that viscosity plays a key role in the existence or lack of existence of these particular nearly inviscid vortices and supports the theory proposed by Hall and others on vortex breakdown. Comparison of both the critical swirl parameter and the viscous core structure for the present family of vortices with several experimental results under conditions near the onset of vortex breakdown show a good agreement for values of m slightly larger than 1. These results differ strongly from those in the highly degenerate case m =1.
The effect of charge emission from electrified liquid cones
- J. Fernández De La Mora
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- Journal:
- Journal of Fluid Mechanics / Volume 243 / October 1992
- Published online by Cambridge University Press:
- 26 April 2006, pp. 561-574
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The formation of stable cones in electrified liquid interfaces was explained by Taylor as a balance between electrical and capillary tensions, where the electrostatic potential varies as ϕ ∼ r½ with the distance r from the cone tip. Although Taylor's predictions for the dependence of the onset voltage for cone formation on the liquid surface tension γ and the cone dimensions agree with observed trends, his conclusion that the cone semiangle α can only take the value α = αT = 49.3° does not. A more general theory free from this paradox is constructed for highly conducting fluids by accounting for the space charge of the droplets emanating from the cone apex, whose potential has the remarkable property of also obeying Taylor's r½ law. In this formulation, where the apex of a conical meniscus of semiangle α emits an angularly uniform opposed coaxial conical spray of semiangle π—β, both β and the spray current I turn out to be fixed as functions of α; namely, β = β(α), and I = 2πγKqG(α), where Kq and q are the droplet's electrical mobility and total charge, respectively. In experiments with 5% H2SO4 in 1-octanol, the observed sprays are approximately conical with an apex nearly touching the meniscus tip. The measured and predicted β(α) relations are in reasonable agreement in the range 46° > α > 32°, where the liquid cone is stable and the spray is visible, though the data fall clearly below the theoretical curve. The predicted spray current I is also in rough agreement with preliminary experiments. The analysis applies neither to sprays of large droplets with significant inertia, nor to liquid cones in vacuo.
Shock wave structure in gas mixtures with large mass disparity
- R. Fernández-Feria, J. Fernández De La Mora
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- Journal:
- Journal of Fluid Mechanics / Volume 179 / June 1987
- Published online by Cambridge University Press:
- 21 April 2006, pp. 21-40
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The structure of normal shock waves is considered when the ratio of molecular masses mp/m of a binary mixture of inert monatomic gases is large and the density ratio ρp/ρ is of order unity or below. Generalized hydrodynamic equations, valid for arbitrary intermolecular potentials, are obtained from a hypersonic closure of the kinetic equation for the heavy gas and a near-equilibrium closure for the light component. Because the Prandtl number of the light gas and the Schmidt number of the mixture are nearly constant, the only independent transport coefficient arising in the model is the viscosity μ of the light gas, which is absorbed into a new independent position variable s. Knowledge of μ as a function of temperature thus determines the shock structure independently from the details of the intermolecular potential, allowing comparison with experiments in the complete absence of free parameters. In terms of the ratio M (frozen Mach number) between the speed of propagation and the sound speed of the light gas in the unperturbed medium, one finds that: (i) When M > 1, the behaviour is similar to that of a ‘dusty gas’, with a broad relaxation layer (outer solution) following a sharp boundary layer through which the speed of the heavy gas is almost constant (a shock within a shock). (ii) When (1 + ρp/ρ)s−½ < M < 1, the boundary layer disappears, yielding a so-called ‘fully dispersed wave’. (iii) Because the internal energy of the heavy gas is negligible, the present problem differs from previous shock studies in that, for the first time, the structure of the relaxation region is obtained algebraically in phase space, thus permitting an exhaustive study of the behaviour. From it, the overshooting solution found by Sherman (1960) is related to the unphysical degenerate branch of the outer solution arising when M > 1, showing a failure of the Chapman–Enskog theory, even for weak shocks, when the heavy gas is dilute. Also, an algebraic explanation arises for the ‘double hump structure’ observed in He–Xe shocks. (iv) When M is nearly unity, the initial boundary layer spreads out, and the structure must be obtained by integration of a numerically unstable system of three differential equations. However, the reduction of order brought about by the weak variation of the light-gas entropy at the head of the shock, results in a stable system of equations that we integrate numerically. Excellent phase-space agreement with recent shock-tube experiments of Tarczynski, Herczynski & Walenta (1986) is found for both weak and strong shocks.
Two-fluid Euler theory of sound dispersion in gas mixtures of disparate masses
- J. Fernandez De La Mora, A. Puri
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- Journal:
- Journal of Fluid Mechanics / Volume 168 / July 1986
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- 21 April 2006, pp. 369-382
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The suitability of an Euler-level two-fluid theory to describe the behaviour of gas mixtures with disparate masses is explored for the problem of sound propagation at frequencies high enough that dispersion effects are important. The determination of the speed of propagation is reduced to solving a quadratic equation in the complex plane. The model leads to small errors of the order of the molecular mass ratio M when the molar fraction xp of the heavy gas is small (xp = O(M)), becoming increasingly inaccurate at larger values of xp. Yet agreement with He-Xe experiments is excellent for the whole range of frequencies tested, up to values of xp = 0.4. For values of xp above 0.5 our quantitative results become poorer but they still agree qualitatively with experiments, predicting small and negative dispersion coefficients and the presence of a bifurcation at critical values of the frequency and the composition. It is concluded that this generalized Euler theory provides an excellent framework within which to develop a two-fluid boundary-layer description of the peculiar dynamics of disparate-mass mixtures in the region of parameters of greatest industrial interest.
Aerodynamic focusing of particles in a carrier gas
- J. Fernández De La Mora, P. Riesco-Chueca
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- Journal:
- Journal of Fluid Mechanics / Volume 195 / October 1988
- Published online by Cambridge University Press:
- 21 April 2006, pp. 1-21
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The problem of whether a stream of microscopic particles may be concentrated into a focal point by entrainment within a carrier gas is considered for dilute particles linearly coupled to the velocity field of an incompressible gas. Typically, the dynamical behaviour of the particles is governed by a so-called Stokes number S, the product of their relaxation time and a characteristic value of the velocity gradient in the suspending fluid. An inequality due to Robinson (1956) is used to illustrate the natural tendency of potential flows to concentrate the particles. For geometries with planar or axial symmetry, with errors cubic in their initial distance to the axis, the trajectories of identical particles originating near an axis of symmetry are shown to cross it at a common focal point provided they have some initial convergence and their Stokes number is larger than a critical value S*. The position of the focal point of supercritical particles depends on their Stokes number, tending to infinity as S approaches S*. Particle trajectories originating far from the axis of symmetry are seen to cross the centreline at defocused positions, in analogy with the optical geometric aberration effect. The focusing phenomenon is illustrated numerically for two-dimensional potential flows through nozzles of several geometries and also analysed in the proximity of the axis of symmetry. For these examples, the threshold value S* of the Stokes number for focusing is of order one, over an order of magnitude larger than typical values of the familiar critical Stokes number marking the onset of particle impaction on solid surfaces. The focal width may be made over two orders of magnitude smaller than the nozzle diameter by restricting the region where particles are seeded to a moderate angle away from the axis. This angle may be higher than ¼π for the case of a jet exiting through a slit in an infinitely thin plate. There is also some discussion of the use of high-resolution focusing instruments.
Variational determination of the coefficient of sound dispersion in binary gas mixtures
- P. Riesco-Chueca, J. Fernandez De La Mora
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- Journal:
- Journal of Fluid Mechanics / Volume 188 / March 1988
- Published online by Cambridge University Press:
- 21 April 2006, pp. 205-221
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The propagation of plane linear acoustic wave in a mixture of inert gases is considered by means of a variational formulation of the Boltzmann equations, through which the sound speed c is expressed with errors of order ε2 in terms of trial functions determined with errors of order ε. This feature allows the exact determination of the coefficient of sound dispersion d2 ≡ [dc/dω2] at zero frequency (ω = 0), in terms of trial functions known from the Chapman—Enskog theory. Explicit results for d2 are given for all combinations of noble gases from He to Xe, assumed to interact through the Lennard—Jones potential. Comparison with previous approximate descriptions and with experiments is made.
Effects of inertia on the diffusional deposition of small particles to spheres and cylinders at low Reynolds numbers
- J. Fernandez De La Mora, D. E. Rosner
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- Journal:
- Journal of Fluid Mechanics / Volume 125 / December 1982
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- 20 April 2006, pp. 379-395
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A formalism that accounts for inertial and diffusive effects in the dynamics of a dilute gas-particle suspension is introduced. The treatment is purely deterministic away from a very thin Brownian diffusion sublayer, while, within the sublayer, inertial effects are small, permitting a near-equilibrium expansion in powers of the Stokes number (particle relaxation time divided by flow characteristic residence time). This expansion provides phenomenological expressions for the particle velocity including two terms : the standard Brownian diffusion, and an additional inertial drift velocity which is closely related to the pressure diffusion term of the Chapman-Enskog expansion. As an example, the general formalism is applied in detail to the case of Stokes flow about a sphere, and sketched for the similar case of a cylinder. Two competing mechanisms are seen to affect the total rate of particle capture by the sphere : (if the stagnation-point region is considerably enriched in particles owing to the high compressibility of the particle phase, which leads to locally enhanced deposition; (ii) centrifugal forces tend to deplete the Brownian diffusion sublayer of particles, reducing diffusion rates away from the stagnation point to the surface. The first effect is seen to dominate over the second except in a very narrow zone of small Stokes numbers. Our method bridges the gap between Levich's solution for the ‘pure-diffusion’ limit and Michael's treatment in the ‘pure-inertia’ limit.
Inertial impaction of heavy molecules
- J. Fernández De La Mora, B. L. Halpern, J. A. Wilson
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- Journal:
- Journal of Fluid Mechanics / Volume 149 / December 1984
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- 20 April 2006, pp. 217-233
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The transition from diffusion-dominated to inertia-dominated behaviour in the transport of condensable heavy molecules carried in a continuum subsonic He jet that impinges on a solid surface is studied experimentally. The Stokes number S, or ratio between the heavy-molecule relaxation time and the fluid-dynamic time, is varied in the interval 0 [lsim ] S [lsim ] 1 by changing the jet Mach number at a constant value of the Reynolds number. Although the heavy species departs considerably from equilibrium at all but the smallest values of S, the helium jet is always near equilibrium conditions. At values of S of order unity the observed rate of deposition at the stagnation point asymptotes to a value some six times greater than in the diffusion region (where S → 0), implying that the process is governed by the large inertia of the heavy species, very much like in aerosol impactors. As a result, it is argued that the concept of pressure diffusion is unsuitable to explain the observed behaviour. An approximate theoretical description of the transport process is given for the region S [Lt ] 1 where the kinetic problem is amenable to a hydrodynamic treatment. Finally, the analogy with the inertia-dominated behaviour of aerosols is used to assess the relative merits of various aerodynamics schemes aiming at separating isotopes.