Evolution of a Polydisperse Ensemble of Spherical Particles in a Metastable Medium with Allowance for Heat and Mass Exchange with the Environment

Motivated by a wide range of applications in various fields of physics and materials science, we consider a generalized approach to the evolution of a polydisperse ensemble of spherical particles in metastable media. An integrodifferential system of governing equations, consisting of a kinetic equation for the particle-size distribution function (Fokker–Planck type equation) and a balance equation for the temperature (concentration) of a metastable medium, is formulated. The kinetic equation takes into account fluctuations in the growth/reduction rates of individual particles, the velocity of particles in a spatial direction, the withdrawal of particles of a given size from the metastable medium, and their source/sink term. The heat (mass) balance equation takes into account the growth/reduction of particles in a metastable system as well as heat (mass) exchange with the environment. A generalized system of equations describes various physical and chemical processes of phase transformations, such as the growth and dissolution of crystals, the evaporation of droplets, the boiling of liquids and the combustion of a polydisperse fuel. The ways of analytical solution of the formulated integrodifferential system of equations based on the saddle-point technique and the separation of variables method are considered. The theory can be applied when describing the evolution of an ensemble of particles at the initial and intermediate stages of phase transformation when the distances between the particles are large enough, and interactions between them can be neglected. © 2022 by the authors. Licensee MDPI, Basel, Switzerland.Russian Foundation for Basic Research, РФФИ: 20-08-00199; Ministry of Education and Science of the Russian Federation, Minobrnauka: FEUZ-2020-0057Funding: This paper comprises two parts of research studies, including (i) the model generalization and discussion of governing equations, analytical solutions, analysis of the results obtained, and (ii) checking the mathematics, numerical calculations, visualization, and discussion of the functions obtained. Two parts of this review article were supported by two financial sources. The first part was supported by the Russian Foundation for Basic Research (grant no. 20-08-00199). At the same time, the authors are grateful to the Ministry of Science and Higher Education of the Russian Federation (grant no. FEUZ-2020-0057) for the support of the second part of the research studies

DIFFERENTIAL DIAGNOSIS OF RADICULOPATHY IN LYME BORRELIOSISAND DYSTROPHIC LESIONS OF THE VERTEBRAL COLUMN

Subjects and methods. Thirty patients, including 9 (30.0%) men and 21 (70.0%) women, with signs of radiculopathy (RP) in late-stages Lyme borreliosis (LB) were examined. A control group comprised 30 patients with vertebrogenic RP in the presence of dystrophic changes in the vertebral column. Results. 56.7% of the patients with LB were observed to have its primary chronic course in the absence of the acute period of LB. The latter with the signs of RP showed a topic association between the pain location and the tick bite site in 43.3% of the patients. A gradual disease development was more frequently (63.3%) observed in LB while the periods of remission and exacerbation were more typical (56.7%) in vertebrogenic RP. In the patients with LB, pain syndrome depended on posture and physical exercise less frequently (30.0%) than in those with vertebrogenic RP (96.7%). Bilateral pain irradiation was more characteristic of RP in LB than in dystrophic lesions of the vertebral column. The symptoms of tonic muscle tension and limited movement volume in the afflicted part of the vertebral column were significantly less common in the patients with LB than in those with vertebrogenic RP. LB was marked by a concomitance of radicular and polyneuritic disorders (83.3%) and vertebrogenic RP was characterized by a preponderance of the radicular-type of sensitivity disorder (100%). Systemic inflammatory syndrome and polysystemacy of manifestations were more characteristic of LB. The benefits of nonsteriodal antiinflammatory drugs in LB patients with RP were significantly worse than in those with vertebrogenic RP; regression of symptoms in LB was seen only after a course of specific antibiotic therapy

Traveling waves in a profile of phase field: Exact analytical solutions of a hyperbolic Allen-Cahn equation

To obtain solutions of the hyperbolic Allen-Calm equation, the first integral method, which follows from well-known Hilbert Null-theorem, is used. Exact analytical solutions are obtained in a form of traveling waves, which define complete class of the hyperbolic Allen-Calm equation. It is shown that two subclasses of solutions exist within this complete class. The first subclass exhibits continual solutions and the second subclass is represented by solutions with singularity at the origin of coordinate system. Such non-uniqueness of solutions stands a question about stable attractor, i. e., about the traveling wave to which non-stationary solutions may attract. The obtained solutions include earlier solutions for the parabolic Allen-Calm equation in a form of finite number of tanh-functions