Application of the Lambert W function to steady shearing flows of the Papanastasiou model

Using the Lambert W function, the constitutive relation of the Papanastasiou model is inverted so that the second invariant of the first Rivlin-Ericksen tensor can be expressed as a function of the second invariant of the extra stress tensor. In steady shearing flows, this results in the magnitude of the shear rate becoming a function of the magnitude of the shear stress. Since the distribution of the latter is known explicitly in channel, Poiseuille and Couette flows, one can investigate the nature of analytical solutions in these flows. It is shown that explicit answers are found for channel and Poiseuille flows only, with the Couette flow requiring a numerical solution in general. From the channel flow results, it is obvious that there is a great amount of congruence between the predictions of the Papanastasiou model and the Bingham fluid. In turn, this lends further confidence to the application of the Papanastasiou model to study the flows of Bingham fluids.     

Strong fragmentation and coagulation with power-law rates

Existence of global classical solutions to fragmentation and coagulation equations with unbounded coagulation rates has been recently proved for initial conditions with finite higher-order moments. These results cannot be directly generalized to the most natural space of solutions with finite mass and number of particles due to the lack of precise characterization of the domain of the generator of the fragmentation semigroup. In this paper we show that such a generalization is possible in the case when fragmentation is described by power-law rates, which are commonly used in engineering practice. This is achieved through direct estimates of the resolvent of the fragmentation operator, which in this case is explicitly known, proving that it is sectorial and carefully intertwining the corresponding intermediate spaces with appropriate weighted L ₁ spaces.     

Analysis of linear resisted projectile motion using the Lambert W function

We solve the linear resisted projectile motion problem using the Lambert W function. In the problem, the launching point is higher than the landing point. The explicit solutions for the range and optimal angle of elevation are expressed in terms of the Lambert W function. The two conclusions can be made for the same projectile: (i) The higher the launching point, the larger the range is. (ii) The higher the launching point, the smaller the optimal angle is.     

Eine allgemeine Berechnungsmethode für Wärmeübertragerschaltungen

A general method for calculating any heat exchanger network is presented. As heat exchanger networks can be considered as coupled heat exchangers, that are supplied in a predetermined order, it is convenient to consider even a single heat exchanger with complicated flow patterns as a network composed of several ideally-supplied, coupled devices. The advantage of this method lies in its higher accuracy and the use of a simpler calculation formula for an ideal device model. These advantages far outweigh the disadvantages of the arduous iterative type of calculation used to establish the temperature between ideal devices. The presented method allows, for the first time, without using the iterative method, a calculation of heat exchanger networks based upon the principle of coupled matrix equations. These equations are for any given heat exchanger network simply allocated parameters, and therefore this method is especially suited for the calculation of any network modification. For the application of this method is no special software necessary. The solution of the model equations is obtained very easy and clear with mathematical standard software, e.g. Mathcad or Maple.     

Hall effects on the unsteady hydromagnetic flows of an Oldroyd-B fluid

The influence of Hall currents and rotation on the oscillatory flows of an infinite plate is investigated. Exact solutions for the two problems are obtained.The fluid considered is a homogeneous Oldroyd-B. During the mathematical analysis it is found that governing differential equation for steady flow in an Oldroyd-B fluid is identical to that of viscous fluid. Further, it is observed that in absence of the strength of transverse magnetic field (B₀) the solution in resonance case does not satisfy the boundary condition at infinity. Physical significance of mathematical results is also discussed.     

Meshfree natural vibration analysis of 2D structures

Determination of resonance frequencies and vibration modes of mechanical structures is one of the most important tasks in the product design procedure. The main goal of this paper is to describe a pioneering application of the solution structure method (SSM) to 2D structural natural vibration analysis problems and investigate the numerical properties of the method. SSM is a meshfree method which enables construction of the solutions to the engineering problems that satisfy exactly all prescribed boundary conditions. This method is capable of using spatial meshes that do not conform to the shape of a geometric model. Instead of using the grid nodes to enforce boundary conditions, it employs distance fields to the geometric boundaries and combines them with the basis functions and prescribed boundary conditions at run time. This defines unprecedented geometric flexibility of the SSM as well as the complete automation of the solution procedure. In the paper we will explain the key points of the SSM as well as investigate the accuracy and convergence of the proposed approach by comparing our results with the ones obtained using analytical methods or traditional finite element analysis. Despite in this paper we are dealing with 2D in-plane vibrations, the proposed approach has a straightforward generalization to model vibrations of 3D structures.     

On estimating stress intensity factors and modulus of cohesion for fractal cracks

Existing solutions for the singular stress field in the vicinity of a fractal crack tip have been adapted for a somewhat modified problem. Since the integration along the fractal curve is prohibitive and does not lend itself to the presently available mathematical treatments, a simplified one has replaced the original problem. The latter involves a smooth crack embedded in a singular stress field, for which the order of singularity is adjusted to match exactly the one obtained from the analyses pertaining to the fractal crack. Of course, this is only an approximation, and we may only hope that it leads toward correct results, at least in a cursory sense. The advantage of such an approach becomes obvious when one inspects the final closed-form solutions for (a) the stress intensity factor in mode I fractal fracture, and (b) cohesion modulus, which results from the cohesive zone model and serves as a measure of the material resistance to crack propagation. As expected for the fractal geometry employed here, our results are strongly dependent on the fractal dimension D (or roughness exponent H).     

A mathematical model of adult subventricular neurogenesis

Neurogenesis has been the subject of active research in recent years and many authors have explored the phenomenology of the process, its regulation and its purported purpose. Recent developments in bioluminescent imaging (BLI) allow direct in vivo imaging of neurogenesis, and in order to interpret the experimental results, mathematical models are necessary. This study proposes such a mathematical model that describes adult mammalian neurogenesis occurring in the subventricular zone and the subsequent migration of cells through the rostral migratory stream to the olfactory bulb (OB). This model assumes that a single chemoattractant is responsible for cell migration, secreted both by the OB and in an endocrine fashion by the cells involved in neurogenesis. The solutions to the system of partial differential equations are compared with the physiological rodent process, as previously documented in the literature and quantified through the use of BLI, and a parameter space is described, the corresponding solution to which matches that of the rodent model. A sensitivity analysis shows that this parameter space is stable to perturbation and furthermore that the system as a whole is sloppy. A large number of parameter sets are stochastically generated, and it is found that parameter spaces corresponding to physiologically plausible solutions generally obey constraints similar to the conditions reported in vivo. This further corroborates the model and its underlying assumptions based on the current understanding of the investigated phenomenon. Concomitantly, this leaves room for further quantitative predictions pertinent to the design of future proposed experiments.     

Integrating the Lambert W Function to a Tolerance Optimization Problem

This paper explores the integration of the Lambert W function to a tolerance optimization problem with the assessment of costs incurred by both the customer and a manufacturer. By trading off manufacturing and rejection costs, and a quality loss, this paper shows how the Lambert W function, widely used in physics, can be efficiently applied to the tolerance optimization problem, which may be the first attempt in the literature related to tolerance optimization and synthesis. Using the concept of the Lambert W function, a closed‐form solution is derived, which may serve as a means for quality practitioners to make a quick decision on their optimal tolerance without resorting to rigorous optimization procedures using numerical methods. A numerical example is illustrated and a sensitivity analysis is performed. Copyright © 2005 John Wiley & Sons, Ltd.     

Orthogonal projectors and the method of symmetrical components for multiphase electromagnetic systems

A method of constructing orthogonal operator projectors for expanding a vector-function (for example, the voltages and currents of a multiphase electromagnetic system) into orthogonal components, symmetrical with respect to one of the elements of a finite multiplicative cyclic group of linear operators, is proposed. This method is a mathematical generalization of the method of current and voltage symmetrical components. Translated from Izmeritel’naya Tekhnika, No. 1, pp. 42–47, January, 2009.     

Tolerance optimization using the Lambert W function: an empirical approach

This paper explores the integration of the Lambert W function to a tolerance optimization problem with two unique features. First, the Taguchi loss function has been a popular tool for quantifying a quality loss incurred by the customer. This paper utilizes an empirical approach based on a well-established regression analysis, which may be more appealing to engineers and better capture the customer's perception of product performance. Second, by trading off manufacturing and rejection costs incurred by a manufacturer and quality loss incurred by the customer, this paper shows how the Lambert W function, widely used in physics, can be efficiently applied, which is perhaps the first attempt in the literature related to tolerance optimization and synthesis. Using the concept of the Lambert W function, this paper derives a closed-form solution, which may serve as a means for quality practitioners to make a quick decision on their optimal tolerances without resorting to rigorous optimization procedures using numerical methods. A numerical example is illustrated and a sensitivity analysis is performed.     

Gas evolution from a solid following de-trapping and diffusion during tempering

The gas evolution rate from the surface of a solid is determined as a function of temperature during controlled heating of the solid, for the case where it is assumed that gas atoms are initially trapped at centres requiring an activation energy E for stimulation into a diffused mode. Diffusion then proceeds with a different activation energy Q. The initial location of atoms is assumed to be a plane, depth pλ below the surface and the heating function is assumed to be 1/T = 1/T_o ? bt. Even in this simplified case it is shown that although the release rate can be expressed as a formal mathematical function of temperature, numerical evaluation is necessary for each value of E, Q, b and pλ.     

Kritische Anmerkungen zu einem weitverbreiteten Konzept: der Wärmeübergangskoeffizient α

The widely used concept of heat transfer coefficient h is critically analyzed on the background of mathematical/physical modelling. Several examples demonstrate that it can be misleading if used uncritically. Alltogether the limitations of the concept are shown as clearly as possible. On the other hand its potential, with the physics accounted for properly, is demonstrated.     

Applications of the Heisenberg magnetic model in nanoscience

The theoretical Heisenberg magnet model and its solution given by Bethe and Hulthén (B.H.) known as Bethe Ansatz (BA) is widely applied in physics (solid state physics, quantum dots, statistical physics, high-temperatures superconductivity, low-dimensional systems, etc.), chemistry (polymers, organic metals and magnets), biology (biological molecular arrays and chains), etc. In most of the applications, the Heisenberg model is applied to infinite chains (asymptotic case), which is a good reality approximation for objects of macroscopic size. In such a case, the solutions of the model are well known. However, for objects of nanoscale size, one has to find solutions of the Heisenberg model of a finite chain consisting of N nodes. For such a chain, the problem of solving of B.H. equations is very complicated (because of the strange nonlinearity of these equations) even for very small objects N<20. Along with an increase in the length of the chain, mathematical difficulties in solving the equations increase combinatorially as 2^N (combinatorial explosion). In such cases, even numerical methods are helpless. In our paper, we propose an approach in which numerical methods could be adapted to such a large numerical problem, as B.H. solutions for objects consisting of N>100, which responds to nanoscale physical or biological objects. This method is based on the ‘experimental’ observation that B.H. solutions change in a quasi-continuous way with respect to N.     

Exact solutions for free vibrations of orthotropic rectangular Mindlin plates

Exact closed-form solutions are obtained for free vibrations of orthotropic rectangular Mindlin plates by using the separation of variables method although it is difficult to solve them. The plates have two opposite edges simply supported and all possible combinations of classical boundary conditions at the other two edges. The exact solutions of orthotropic rectangular Mindlin plates are compared with those of isotropic ones and their differences are discussed. The exact solutions are validated through both mathematical proof and numerical comparisons with available p-Ritz solutions and the differential quadrature finite element method solutions calculated by the authors.     

Maximizing protein translation rate in the non-homogeneous ribosome flow model: a convex optimization approach

Translation is an important stage in gene expression. During this stage, macro-molecules called ribosomes travel along the mRNA strand linking amino acids together in a specific order to create a functioning protein. An important question, related to many biomedical disciplines, is how to maximize protein production. Indeed, translation is known to be one of the most energy-consuming processes in the cell, and it is natural to assume that evolution shaped this process so that it maximizes the protein production rate. If this is indeed so then one can estimate various parameters of the translation machinery by solving an appropriate mathematical optimization problem. The same problem also arises in the context of synthetic biology, namely, re-engineer heterologous genes in order to maximize their translation rate in a host organism. We consider the problem of maximizing the protein production rate using a computational model for translation–elongation called the ribosome flow model (RFM). This model describes the flow of the ribosomes along an mRNA chain of length n using a set of n first-order nonlinear ordinary differential equations. It also includes n + 1 positive parameters: the ribosomal initiation rate into the mRNA chain, and n elongation rates along the chain sites. We show that the steady-state translation rate in the RFM is a strictly concave function of its parameters. This means that the problem of maximizing the translation rate under a suitable constraint always admits a unique solution, and that this solution can be determined using highly efficient algorithms for solving convex optimization problems even for large values of n. Furthermore, our analysis shows that the optimal translation rate can be computed based only on the optimal initiation rate and the elongation rate of the codons near the beginning of the ORF. We discuss some applications of the theoretical results to synthetic biology, molecular evolution, and functional genomics.     

Tables of the Negative Binomial Probability Distribution

An age replacement policy requires that a unit be replaced when it attains a specified age, or at failure, whichever occurs first. The mathematical solution to the problem of what is the optimal age for replacement is well known. This paper adapts this general mathematical solution to the cases when a truncated normal, Gamma, or Weibull distribution can be assumed. Graphs are presented from which one can readily ascertain the optimal solutions for these models.     

Frequency response analyses in vibroacoustics using the method of fundamental solutions

The method of fundamental solutions, one of the promising boundary-type meshless methods, is proposed as a direct procedure to formulate and analyze the vibroacoustic problem. The coupled system discussed in this study is composed of an acoustic-cavity and excited by an external force or an internal sound source harmonically. The wall of cavity consists of the beam or the plate components, respectively, in two- and three-dimensional problems. The two independent sub-systems interact at the interface simultaneously by satisfying the necessary equilibrium and compatibility conditions. The mathematical formulations described by the presented meshless method demonstrate straightforwardly the frequency responses of the vibroacoustic problems with no boundary integrals. General characteristics of the dynamic coupling effect are displayed, based on the systematic natural frequencies and mode shapes. Feasible results simulated by the presented numerical scheme are validated through meshless numerical experiments including the acoustic-wave propagation problems and the vibroacoustic problems.     

On the Naghdi-Hsu type solution in elastodynamics

Summary The purpose of this note is to point out that the mathematical argument presented for a dynamic generalization of a known elastostatic solution representation in [1] involves faulty steps.     

Effect of the W addition content on valence electron structure and properties of MoSi2-based solid solution alloys

Based on the empirical electron theory (EET) of solids and molecules, the valence electron structures (VES) of MoSi₂-based solid solution alloys have been analyzed using the average atom model. The results showed that with the increase of the W addition content, the hybridization steps of Mo and Si atom of the alloys occurred in C3 and 1, respectively. The hybridization step of W was always C5. The bond energy of the main bond branch, the covalence electron number on the strongest bond and the percentage of the total covalent electron numbers accounting for the total valence electron number of (Mo_1−x, W_x)Si₂ solid solutions increased with the increase of W addition content. These suggested that the addition of W would increase the melting point, hardness and strength and decrease the fracture toughness of (Mo_1−x, W_x)Si₂ solid solutions. Based on those results, MoSi₂-based solid solution alloys were manufactured, and the results of the experiments were in accordance with those of the theory.