Mathematical modeling of gas lubrication problems

After a brief review of the origin and development of gas lubrication as a branch of modern engineering, the principles of mathematical modeling of gas lubrication theory are formulated. It is shown that the gas lubrication theory is underlain by the boundary layer equations for gases, although the boundary conditions differ substantially from those used in external flow problems. Numerical solutions are constructed for stationary and nonstationary gas lubrication problems, including those taking into account rarefaction and the elastic properties of the boundary surfaces. Numerical examples are given.     

Graphical solution of axially symmetric problems of plastic flow

Résumé Une méthode graphique de solution des problèmes dans le cas de symétrie axiale a été proposée pour des corps rigides, parfaitement plastiques, obéissants au critère d'écoulement de Coulomb-Tresca et à l'hypothèse du potentiel plastique. Deux cas ont été considérés: d'une part des régimes de Haar-Kármán pour lesquels la contrainte circonférencielle est égale à l'une des contraintes principales contenues dans le plan axial, et d'autre part des régimes duor lesquels l'une des vitesses de déformation principales dans le plan axial est nulle.     

Mathematical problems on the fluid-solute-heat flow through prous media

A degenerate parabolic system arising from the fluid-solute-heat flow through unsaturated porous media is considered. The existence of weak solutions to the initial boundary value problem of this system is established by parabolic regularization. The regularity is proved as well, i. e. the weak solutions satisfy the equations and initial boundary conditions pointwise in the region where the moisture content is positive.This project is supported by National Natural Science Foundation of China.     

Mathematical problems on the fluid-solute-heat flow through porous media

A degenerate parabolic system arising from the fluid-solute-heat flow through partially saturated porous media is considered. The existence of weak solutions to the initial boundary value problem of this system is established by time discretization, and the continuity of the weak solutions is discussed.     

Mathematical modeling of delamination and nonmonotone friction problems by hemivariational inequalities

The paper deals with approximations and the numerical realization of a class of hemivariational inequalities used for modeling of delamination and nonmonotone friction problems. Assumptions guaranteeing convergence of discrete models are verified and numerical results of several model examples computed by a nonsmooth variant of Newton method are presented.The research was realized in the frame of the bilateral cooperation between Charles University, Prague and Aristotle University, Thessaloniki. The second author also acknowledges the support of the grant no. IAA1075402 of the Grant Agency of the Academy of Sciences of the Czech Republic and MSM 113200007.This revised version was published online in April 2005 with a corrected issue number.     

Mathematical modeling of pulsatile flow of non-Newtonian fluid in stenosed arteries

The pulsatile flow of blood through mild stenosed artery is studied. The effects of pulsatility, stenosis and non-Newtonian behavior of blood, treating the blood as Herschel–Bulkley fluid, are simultaneously considered. A perturbation method is used to analyze the flow. The expressions for the shear stress, velocity, flow rate, wall shear stress, longitudinal impedance and the plug core radius have been obtained. The variations of these flow quantities with different parameters of the fluid have been analyzed. It is found that, the plug core radius, pressure drop and wall shear stress increase with the increase of yield stress or the stenosis height. The velocity and the wall shear stress increase considerably with the increase in the amplitude of the pressure drop. It is clear that for a given value of stenosis height and for the increasing values of the stenosis shape parameter from 3 to 6, there is a sharp increase in the impedance of the flow and also the plots are skewed to the right-hand side. It is observed that the estimates of the increase in the longitudinal impedance increase with the increase of the axial distance or with the increase of the stenosis height. The present study also brings out the effects of asymmetric of the stenosis on the flow quantities.     

Mathematical modeling of the stress state of a transverse plastic layer in a round rod

We construct mathematical models of plastic deformations of a continuous round rod containing a transverse (less strong) inhomogeneous layer under an axial load. We thoroughly study the local strengthening of such layers by involving the base material of the rod in the plastic deformation process. We obtain explicit formulas for the critical stress states in the layer and the critical axial load on the rod.     

Mathematical modeling and computation of fire induced turbulent flow in partial enclosures

The purpose of this paper is to understand the growth and spread of fires in ceiling vented enclosures. The transport phenomena due to fire have been modeled as buoyancy-induced turbulent flow in partial enclosures. The governing equations comprises the Reynolds averaged Navier–Stokes (RANS) equations with k–?$k''–''?$ turbulence model in stream function–vorticity formulation approach. The governing equations are solved by high accuracy compact finite difference schemes with four stage Runge–Kutta method for time integration. Results are reported for Grashof numbers varied from 10⁸${10}^8$ to 10¹⁰${10}^{10}$. The effects of multiple heat sources in rectangular enclosure and ceiling vent aspect ratio in square enclosure are investigated. The thermal plume growth rate, ambient entrainment flow rate and the oscillatory nature at the vent opening are reported. As the Grashof number increases the effect of entrained ambient air is significant with higher volume flow rates through ceiling vent. A bidirectional flow is visualized at the ceiling opening. The distance between two heat sources governs the unified and independent behavior of thermal plumes. Present results are matching very well with the numerical and experimental results available in literature.     

Bilinear modeling solution approach for fixed charge network flow problems

We present a continuous, bilinear formulation for the fixed charge network flow problem. This formulation is used to derive an exact algorithm for the fixed charge network flow problem converging in a finite number of steps. Some preliminary computational experiments are reported to show the performance of the algorithm.     

Mathematical modeling of the movement of suspended particles subjected to acoustic and flow fields

When particles are subjected to an acoustic field particle trajectories depend on the particle and fluid compressibility and density values. Hence a combination of acoustic and flow fields on particles can be used to deflect and trap, or to segregate and/or fractionate fine particles in fluid suspensions. Using particle physics in an acoustic field, a mathematical model was developed to calculate trajectories of deflected particles due to the application of acoustic standing waves. The resulting second order ordinary differential equation was quite stiff and hence difficult to solve numerically and did not have a closed form solution. The analysis of the above equation showed that the basic problems with numerical solutions could not be ameliorated through the use of standard rescaling techniques. A combination of phase space and asymptotic analysis turns out to be far more useful in obtaining approximate solutions. An approximate solution was derived which enabled the calculation of the particle trajectories and concentration at collection planes in the acoustic field. Analysis of the solution showed that all the particles move toward the pressure node to which the particles are supposed to move. Particles with 2 μm diameter took approximately 20 s to reach that node. Then at the bench scale, the above technology was implemented by building a flow chamber with two transducers at opposite ends to generate an acoustic standing wave. SiC particle trajectories were tracked using captured digital images from a high-resolution microscope. The displacements of SiC particles due to an acoustic force were compared with the mathematical model predictions. For input power levels between 3.0 and 5.0 W, the experimental data were comparable to mathematical model predictions. Hence it was concluded that the proposed approximate solution was both quantitatively and qualitatively closer to experimental results than the simplified form ignoring the second order term reported in the literature.     

Mathematical modeling of turbulent fiber suspension and successive iteration solution in the channel flow

The modified Reynolds mean motion equation of turbulent fiber suspension and the equation of probability distribution function for mean fiber orientation are firstly derived. A new successive iteration method is developed to calculate the mean orientation distribution of fiber, and the mean and fluctuation-correlated quantities of suspension in a turbulent channel flow. The derived equations and successive iteration method are verified by comparing the computational results with the experimental ones. The obtained results show that the flow rate of the fiber suspension is large under the same pressure drop in comparison with the rate of Newtonian fluid in the absence of fiber suspension. Fibers play a significant role in the drag reduction. The amount of drag reduction augments with increasing of the fiber mass concentration. The relative turbulent intensity and the Reynolds stress in the fiber suspension are smaller than those in the Newtonian flow, which illustrates that the fibers have an effect on suppressing the turbulence. The amount of suppression is also directly proportional to the fiber mass concentration.     

Mathematical modeling of church growth

The possibility of using mathematics to model church growth is investigated using ideas from population modeling. It is proposed that a major mechanism of growth is through contact between religious enthusiasts and unbelievers, where the enthusiasts are only enthusiastic for a limited period. After that period they remain church members but less effective in recruitment. This leads to the general epidemic model which is applied to a variety of church growth situations. Results show that even a simple model like this can help understand the way in which churches grow, particularly in times of religious revival.     

Mathematical modeling of plant morphogenesis

Current trends in biology call for analysis of the bulk information accumulated through mathematical simulations of biological processes aiming at revealing certain regularities and verifying hypotheses and predictions. Evolution of organisms is of special interest to mathematical modeling because it integrates a great body of different processes varying over time and over space. In this paper, models of biological processes as related to plant development are reviewed. The models are classified, and approaches to problems that are most intricate from the simulation standpoint, as well as relevant mathematical methods, are described.     

Mathematical modeling of real-world images

The problem of finding appropriate mathematical objects to model images is considered. Using the notion of acompleted graph of a bounded function, which is a closed and bounded point set in the three-dimensional Euclidean spaceR ₃, and exploring theHausdorff distance between these point sets, a metric spaceIM _D of functions is defined. The main purpose is to show that the functionsf∈IM _D, defined on the squareD=[0,1]², are appropriate mathematical models of real world images. The properties of the metric spaceIM _D are studied and methods of approximation for the purpose of image compression are presented. The metric spaceIM _D contains the so-calledpixel functions which are produced through digitizing images. It is proved that every functionf∈IM _D may be digitized and represented by a pixel functionp _n, withn pixels, in such a way that the distance betweenf andp _n is no greater than 2n ^?1/2. It is advocated that the Hausdorff distance is the most natural one to measure the difference between two pixel representations of a given image. This gives a natural mathematical measure of the quality of the compression produced through different methods.     

Mathematical modeling of demographic processes

The article examines a mathematical model that describes the dynamics of the total population and its age structure. Time-dependent birth and death rates are assumed. The mathematical model is a first-order partial differential equation. The analytical solution makes it possible to determine the age distribution at each time instant depending on the birth and death functions and the initial distribution. The model can be used for demographic planning and forecasting. It has been applied to analyze the demographics of Russia. Translated from Prikladnaya Matematika i Informatika, No. 28, pp. 50–65, 2008.