Self-preserving Solutions Research Articles

Topical Antibiotic Therapy in the Ocular Environment: The Benefits of Using Moxifloxacin Eyedrops.

Moxifloxacin is a fourth-generation fluoroquinolone antibiotic available for ophthalmic use. It inhibits two enzymes involved in bacterial DNA synthesis, covering Gram-positive and Gram-negative pathogens. This spectrum allows for the formulation of self-preserving bottle solutions, while its interesting pharmacological profile is distinguished by efficacy at low tissue concentrations and by an infrequent dose regimen due to its long duration on ocular tissues. This enhances patient compliance, promoting its use in children. The human eye hosts several microorganisms; this collection is called the ocular microbiota, which protects the ocular surface, assuring homeostasis. When choosing an antibiotic, it is appropriate to consider its influence on microbiota. A short dose regimen is preferred to minimize the impact of the drug. Moxifloxacin eyedrops represent an effective and safe tool to manage and prevent ocular infections. As healthcare providers face the complexity of the ocular microbiota and microbial resistance daily, the informed use of moxifloxacin is necessary to preserve its efficacy in the future. In this regard, it is well known that moxifloxacin has a lower capacity to induce resistance (an optimal WPC and MPC) compared to other quinolones, but much still needs to be explored regarding the impact that fluoroquinolones could have on the ocular microbiota.

A CFD-sectional algorithm for population balance equation coupled with multi-dimensional flow dynamics

A novel CFD-sectional algorithm is developed to address the challenges in the existing sectional simulations coupled with multi-dimensional fluid dynamics, including solution of complex sectional coefficients, serious computational burden for lots of coupled partial differential equations, and nonlinear source terms. The sectional coefficients are specified by a numerical quadrature with adaptive integration limits, which proves to be computationally efficient and accurate. The inter-equation coupling is treated by hybrid-segregated procedures and the source term is linearized by the operator splitting method. The CFD-sectional algorithm is validated against a self-preserving solution of particles undergoing Brownian coagulation. The acoustic agglomeration in a standing wave is simulated as a representative case. It has been demonstrated that the predictions regarding the particle size distribution and agglomeration process agree well with the experimental data, which verifies the capability of the developed CFD-sectional algorithm in simulating the spatially inhomogeneous population balance equation coupled with multi-dimensional flows.

Simulating aerosol chamber experiments with the particle-resolved aerosol model PartMC

ABSTRACTThis article presents a verification and validation study of the stochastic particle-resolved aerosol model PartMC. Model verification was performed against self-preserving analytical solutions, while for validation three experiments were performed where the size distribution evolution of coagulating ammonium sulfate particles was measured in a cylindrical stainless steel chamber. To compare with the chamber measurements, PartMC was extended to include the representation of fractal particle structure and wall loss. This introduced five unknown parameters to the governing equation, which were determined by a combination of scanning electron microscopy (SEM) analysis and an objective optimization procedure. Excellent agreement between modeled and measured size distributions was achieved using the same set of parameters for all three experiments. Assuming spherical particles led to model results that were inconsistent with the measurements. The best agreement between model and measurement was obtaine...

The Decay of Isotropic Turbulence Preserving a General Type of Self Similarity

In this paper, we have investigated the decay process of the kinetic energy density spectrum in a homogeneous isotropic turbulence. It has been shown that the spectral equation for the energy density spectrum E(k) with modified form of energy transfer spectrum due to Obukhov (Compt. Rend. Acad. Sci. U. R. S. S 32:22–24, 1941) and as restricted to early-period decay process, admits of a class of self-preserving solution. Such solutions are identified by their asymptotic behaviour e.g, \(E(k) \sim k^{\frac{2-3c}{c}} ( k \rightarrow 0)\) where \(c( < \frac{2}{3})\) is a parameter and \(E(k) \sim k^{-\frac{5}{3}} ( k \rightarrow \infty )\). Numerical computations of some selective spectra, corresponding to values of c e.g., \(c=\frac{1}{2}, \frac{2}{5}, \frac{1}{3}\) and \(\frac{2}{7}\) are then accomplished over the entire range of wave numbers concerned (excluding the viscous dissipation range). We attempt to find a class of non viscous self-preserving solution using the modified Obukhov form (cf Hinze An introduction to its mechanism and theory, 1975) for the spectrum function F(k, t) and compute them numerically. Finally, stability analysis is carried out for the above mentioned self-preserving spectra and it is shown that they represent different degrees of unstable situations.

Decay of Batchelor and Saffman rotating turbulence

The decay rate of isotropic and homogeneous turbulence is known to be affected by the large-scale spectrum of the initial perturbations, associated with at least two canonical self-preserving solutions of the von Kármán-Howarth equation: the so-called Batchelor and Saffman spectra. The effect of long-range correlations in the decay of anisotropic flows is less clear, and recently it has been proposed that the decay rate of rotating turbulence may be independent of the large-scale spectrum of the initial perturbations. We analyze numerical simulations of freely decaying rotating turbulence with initial energy spectra ∼k^{4} (Batchelor turbulence) and ∼k^{2} (Saffman turbulence) and show that, while a self-similar decay can not be identified for the total energy, the decay is indeed affected by long-range correlations. The decay of two- and three-dimensional modes follows distinct power laws in each case, which are consistent with predictions derived from the anisotropic von Kármán-Howarth equation, and with conservation of anisotropic integral quantities by the flow evolution.

On fully self-preserving solutions in homogeneous turbulence

Mathematical attributes of full self-preserving solutions (‘full’ meaning self-similar at all scales with non-zero viscosity) for isotropic decay as well as for homogeneous shear flow turbulence are examined from a fundamental theoretical standpoint. Fully self-preserving solutions are those wherein the two-point double and triple velocity correlations are self-similar at all scales. It is shown that the full self-preserving solutions for isotropic decay corresponds to a t −1 asymptotic power law decay, consistent with earlier studies. Fully self-preserving solutions for homogeneous shear flow correspond to a production-equals-dissipation equilibrium, with bounded turbulent kinetic energy and dissipation. It is then shown that the fully self-preserving solutions of isotropic decay and of homogeneous shear flow both require severe constraints on the behavior of the low-wavenumber energy spectra. These constraints render the full self-preserving solutions as mathematical consistent but having no physical relevance. The implications of the dependence of the full self-preserving solutions on the low-wavenumber spectra for one-point (engineering) turbulence models is discussed. Invited speaker, Santa Fe CNLS-LANL Workshop on Turbulence and Cascade.

On the structure of supercritical western boundary currents

Abstract The structure of supercritical western boundary currents is investigated using a quasi-geostrophic numerical model. The basic flow is of meridional Munk balance, and the input boundary is perturbed by the most unstable wave solution obtained from linear spatial instability calculations. Self-preserving (or equilibrium) solutions are obtained for the model runs at Re =30, 60, 90, and 120, and their energy and vorticity budgets are analyzed. In an analogy with the laboratory turbulence of wall boundary layers, the western boundary layer is divided into inner and outer layers. In the inner layer, the mean energy is dissipated via direct viscous dissipation, while in the outer layer it is converted to the eddy energy via turbulence production. The main scenario is that the mean energy is produced in the inner layer via ageostrophic pressure work divergence, and it is partly removed due to viscous action within a narrow region near the wall, defined here as viscous sub-layer. The remaining portion is converted to the eddy energy via turbulence production in the outer layer, which is in turn transported to the inner layer, then again to the viscous sub-layer where it is ultimately dissipated. In the near-wall side, the vorticity balance of the mean flow is maintained by viscous effect and Reynolds flux divergence, while in the offshore side it is maintained by beta effect and Reynolds flux divergence. The length scale of the supercritical boundary current is roughly ∼ 0.2 Re L M , where L M is the Munk length, as observed from a dimensional analysis.

The decay of homogeneous isotropic turbulence

A new theory for the decay of homogeneous, isotropic turbulence is proposed in which truly self-preserving solutions to the spectral energy equation are found that are valid at all scales of motion. The approach differs from the classical approach in that the spectrum and the nonlinear spectral transfer terms are not assumed a priori to scale with a single length and velocity scale. Like the earlier efforts, the characteristic velocity scale is defined from the turbulence kinetic energy and the characteristic length scale is shown to be the Taylor microscale, which grows as the square root of time (or distance). Unlike the earlier efforts, however, the decay rate is shown to be of power-law form, and to depend on the initial conditions so that the decay rate constants cannot be universal except possibly in the limit of infinite Reynolds number. Another consequence of the theory is that the velocity derivative skewness increases during decay, at least until a limiting value is reached. An extensive review of the experimental evidence is presented and used to evaluate the relative merits of the new theory and the more traditional views.

Decay of velocity and temperature fluctuations in grid turbulence

The dependence of the velocity and temperature fluctuation decay exponents on the initial conditions is resolved on the basis of the cascade model for isotropic turbulence. In the large-Reynolds-number limit, self-preserving solutions are obtained for the nondissipative eddies. For those solutions, the energy decay exponent n and the small-wave-number exponent s for the three-dimensional energy spectrum obey the relation n =2(s + !)/(? + 3). For temperature fluctuations, the decay exponent m and the small-wave-number spectrum exponent q satisfy m = 2(q + l)/(s + 3). Thus, the ratio r m/n of the thermal-to-mechanical time scales is completely determined by the initial spectral exponents for very large eddies, if the initial spectra are quasi-selfpreserving. These conclusions are shown to be independent of the model. The model equations are solved numerically. The ratio of thermal to mechanical integral scales is shown to depend on both n and m. The conclusions have implications in the context of exact flow simulations.

Distortion and necking of a viscous inclusion in Stokes flow

Spence &amp; Wilmott (1988) considered the deformation of a slender inclusion of highly-viscous fluid in a Stokes flow of less viscous fluid, and derived a coupled pair of equations to describe its evolution. The equations possess self-preserving solutions for elliptical inclusions, previously known from the work of Bilby et al . (1975, 1976) and Bilby &amp; Kolbuszewski (1977). In the present paper numerical solutions are presented for non-elliptic shapes. These have been obtained on a CRAY XMP-22 by use of an eigenfunction expansion suggested by the elliptic solution, leading to an initial value problem for the coefficients. The solutions show that large deformations can develop in finite time from small perturbations of the ellipse, particularly at large viscosity ratios. Special attention is directed to the phenomenon of boudinage, whereby alternate swelling and necking develops as the inclusion is stretched by the Stokes flow. This appears to characterize all solutions that depart significantly from pure elliptic shape. In the Appendix the stability of the elliptic profile to small disturbances is examined. It is found that all subharmonics of a given disturbance are excited to a finite amplitude that increases with the viscosity ratio, but that higher harmonics are not excited in linear theory, although nonlinear coupling leads to the eventual excitation of all modes.

Rotation and deformation of a viscous inclusion in Stokes flow

In a paper by Spence et al . ( Geophys J . (in the press) (1988)) the stretching and distortion of a symmetrical slender inclusion of highly viscous material along the axis of a stagnation point flow was formulated as a nonlinear evolutionary system in terms of a partial differential equation for the shape function ∆ h ( x, t ), coupled through a singular integral equation to the axial velocity of the internal flow. A self-preserving solution for an elliptic inclusion was noted. In the present paper this work is extended in two directions: (i) for inclusions with centre lines that are not straight, an additional integral equation of Cauchy type is found, coupling the evolution of the centre-line shape to that of the thickness profile; and (ii) the problem is treated in a general linear flow, with the use of rotating axes instantaneously close to the centre line of the inclusion. The analysis shows that slender inclusions rotate with the flow irrespective of the shape and viscosity ratio, while undergoing stretching and distortion that does depend on these quantities. A transformation is found that relates the evolution of the profile to that in a stagnation point flow. The particular case of an elliptic profile is examined in detail to permit comparison with the work of Bilby &amp; Kolbusczewski ( Proc. R. Soc. Lond . A 355, 335-353 (1977)). In this case a closed solution is found for the evolution of a centre line of arbitrary shape, and it appears possible for singularities to develop in finite time in a flow in which the inclusion is shrinking. The system of equations developed provides a convenient starting point for numerical investigations of flow past inclusions of arbitrary (slender) shape, for which necking and break-up might be expected.

Self-preserving solutions for turbidity currents

Subaquatic density currents and turbidity currents flowing down an incline are analyzed. By noting that at least depositing turbidity currents on a constant slope depend on the source conditions, it is proposed to consider them as self-preserving or as piecewise selfpreserving in analogy to other shear flows. The requirements for self-preservation are examined and the limiting conditions for realizable flows are derived.

Étude théorique du développement d'un panache thermique a symétrie axiale influence des propriétés thermophysiques du fluide

The influence of the variations of the physical properties with temperature on the structure of thermal plumes in air, is numerically determined. It is in that purpose that the evolution of the initial theoretical profiles is studied in the case of constant properties (giving self-preserving solutions) and in the case of variable properties. The development of the flow in these two cases is found to be significantly different. More over, the comparison with numerical calculations and our laboratory's experiment is satisfactory.

An Application of Karman’s Energy Transfer Process in Hydromagnetic Turbulence

Abstract A modified version of Chandrasekhar’s equation is derived with the aid of Karman’s energy transfer process for the decay of hydromagnetic turbulence in the limiting case of zero viscosity and infinite electrical conductivity. It is found that the asymptotic behaviour of self-preserving solutions of the aforesaid equations leads to F(k, t)≈k4 , (c = 2/7), G(k, t)≈k6 , (c = 2/9) as k→0 and F(k,t)≈k-5/3 ,G(k,t)≈k-5/3 for k→∞.

On the general theory of solid granulation

The general population balance equation for the kinetics of granulation by simultaneous coalescence and crushing and layering mechanisms is posed and self-preserving solutions are obtained. A general similarity transformation, which contains in it all the physical information concerning the granulation process, is presented. It is clearly shown that self-preserving granule size distributions are possible for granulation by simultaneous coalescence and crushing and layering.

Conditions for the size spectrum of aerosols to be self-preserving

A one-dimensional population balance governing the particle size distribution in a system where particles change their sizes due to both coagulation and individual growth is studied for the possibility of a similarity solution. It is found that when a dimensionless parameter γ is a constant, then the population balance can be transformed into an ordinary integrodifferential equation. Conditions under which the latter equation may have a self-preserving solution are established. Examples related to aerosol dynamics in the continuum regime, the free-molecule regime and the intermediate regime are discussed in detail.

Statistical mechanics of the Burgers model of turbulence

The velocity field of the Burgers one-dimensional model of turbulence at extremely large Reynolds numbers is expressed as a train of random triangular shock waves. For describing this field statistically the distributions of the intensity and the interval of the shock fronts are defined. The equations governing the distributions are derived taking into account the laws of motion of the shock fronts, and the self-preserving solutions are obtained. The number of shock fronts is found to decrease with time t as t−α, where α (0 [les ] α &lt; 1) is the rate of collision, and consequently the mean interval increases as tα. The distribution of the intensity is shown to be the exponential distribution. The distribution of the interval varies with α, but it is proved that the maximum entropy is attained by the exponential distribution which corresponds to α = ½. For α = ½, the turbulent energy is shown to decay with time as t−1, in good agreement with the numerical result of Crow &amp; Canavan (1970).

Equilibrium and relaxation in turbulent wakes

In order to study the memory of the larger eddies in turbulent shear flow, experiments have been conducted on plane turbulent wakes undergoing transition from an initial (carefully prepared) equilibrium state to a different final one, as a result of a nearly impulsive pressure gradient. It is shown that under the conditions of the experiments the equations of motion possess self-preserving solutions in the sense of Townsend (1956), but the observed behaviour of the wake is appreciably different when the pressure gradient is not very small, as the flow goes through a slow relaxation process before reaching final equilibrium. Measurements of the Reynolds stresse show that the approach to a new equilibrium state is exponential, with a relaxation length of the order of 103 momentum thicknesses. It is suggested that a flow satisfying the conditions required by a self-preservation analysis will exhibit equilibrium only if the relaxation length is small compared with a characteristic streamwise length scale of the flow.

The self-preserving particle size distribution for Brownian coagulation in the free-molecule regime

The nature of certain special classes of solutions to the equation of coagulation was explored for the case of Knudsen aerosols. Using the similarity transformation for the size distribution function, it was shown that self-preserving solutions to the coagulation equation may exist. The time decay law for the total number concentration differed in form from that calculated for coagulating particles in a continuum. By making suitable approximations in the transformed kinetic equation, the analytical form of the self-preserving distribution was obtained for extreme ranges of particle size. With numerical experiments, a solution for the complete size spectrum was determined that is consistent with the analytical results for the lower and upper extremes of the spectrum. A quantitative criterion was derived for the time to achieve the self-preserving size spectrum, and coagulation half-times were calculated. Implications for the atmospheric aerosal are discussed.

Numerical Simulation of Three-Dimensional, Shape-Preserving Convective Elements

Abstract The motions resulting from the sudden release of a fixed amount of buoyancy in an incompressible fluid are simulated. Solutions are allowed to reach a steady state in the finite computed volume by the introduction of a dynamical stretching of the coordinate system. Fully nonlinear, transformed, and finite-differenced Navier-Stokes equations are integrated in time over a three-dimensional grid. It is shown that a steady-state solution to the transformed equations is equivalent to a self-preserving solution in real space. Physically realistic results are presented for a range of Reynolds numbers between 10 and 100. In a strongly diffusive regime the simulation agrees with an existing theoretical solution. Reynolds numbers of order 50 are sufficient to reproduce many of the features of laboratory experiments.