This paper is concerned with the numerical simulation of a thermodynamically compatible viscoelastic shear-thinning fluid model, particularly well suited to describe the rheological response of blood, under physiological conditions. Numerical simulations are performed in two idealized three-dimensional geometries, a stenosis and a curved vessel, to investigate the combined effects of flow inertia, viscosity and viscoelasticity in these geometries. The aim of this work is to provide new insights into the modeling and simulation of homogeneous rheological models for blood and a basis for further developments in modeling and prediction.

One of the most intriguing questions in life science is how living organisms develop and maintain their predominant form and shape via the cascade of the processes of differentiation starting from the single cell. Mathematical modeling of these developmental processes could be a very important tool to properly describe the complex processes of evolution and geometry of morphogenesis in time and space. Here, we summarize the most important biological knowledge on plant development, exploring the different layers of investigation in developmental processes such as plant morphology, genetics, plant physiology, molecular biology and epigenetics. As knowledge on the fundamentals of plant embryogenesis, growth and development is constantly improving, we gather here the latest data on genetic, molecular and hormonal regulation of plant development together with the basic background knowledge. Special emphasis is placed on the regulation of cell cycle progression, on the role of the signal molecules phytohormones in plant development and on the details of plant meristems (loci containing plant stem cells) function. We also explore several proposed biological models regarding regulating plant development. The information presented here could be used as a basis for mathematical modeling and computer simulation of developmental processes in plants.

This special issue of Mathematical Modelling of Natural Phenomena on biomathematics education shares the work of fifteen groups at as many different institutions that have developed beautiful biological applications of mathematics that are different in three ways from much of what is currently available. First, many of these selections utilize current research in biomathematics rather than the well-known textbook examples that are at least a half-century old. Second, the selections focus on modules that are intended for instant classroom adoption, adaptation, and implementation. Instead of focusing on how to overcome the challenges of implementing mathematics into biology or biology into mathematics or on educational research on the effectiveness of some small implementation, the authors develop individual biological models sufficiently well such that they can be easily adopted and adapted for use in both mathematics and biology classrooms. A third difference in this collection is the substantive inclusion of discrete mathematics and innovative pedagogies. Because calculus-based models have received the majority of the biomathematics modeling attention until very recently, the focus on discrete models may seem surprising. The examples range from DNA nanostructures through viral capsids to neuronal processes and ecosystem problems. Furthermore, a taxonomy of quantitative reasoning and the role of modeling per se as a different practice are contextualized in contemporary biomathematics education.

The experimentally known phenomenon of oscillatory instability in convective burning of porous explosives is discussed. A simple phenomenological model accounting for the ejection of unburned particles from the consolidated charge is formulated and analyzed. It is shown that the post-front hydraulic resistance induced by the ejected particles provides a mechanism for the oscillatory burning.

We derive a biomembrane model consisting of a fluid enclosed by a lipid membrane. The membrane is characterized by its Canham-Helfrich energy (Willmore energy with area constraint) and acts as a boundary force on the Navier-Stokes system modeling an incompressible fluid. We give a concise description of the model and of the associated numerical scheme. We provide numerical simulations with emphasis on the comparisons between different types of flow: the geometric model which does not take into account the bulk fluid and the biomembrane model for two different regimes of parameters.

Over the last three decades Computational Fluid Dynamics (CFD) has gradually joined the wind tunnel and flight test as a primary flow analysis tool for aerodynamic designers. CFD has had its most favorable impact on the aerodynamic design of the high-speed cruise configuration of a transport. This success has raised expectations among aerodynamicists that the applicability of CFD can be extended to the full flight envelope. However, the complex nature of the flows and geometries involved places substantially increased demands on the solution methodology and resources required. Currently most simulations involve Reynolds-Averaged Navier-Stokes (RANS) codes although Large Eddy Simulation (LES) and Detached Eddy Suimulation (DES) codes are occasionally used for component analysis or theoretical studies. Despite simplified underlying assumptions, current RANS turbulence models have been spectacularly successful for analyzing attached, transonic flows. Whether or not these same models are applicable to complex flows with smooth surface separation is an open question. A prerequisite for answering this question is absolute confidence that the CFD codes employed reliably solve the continuous equations involved. Too often, failure to agree with experiment is mistakenly ascribed to the turbulence model rather than inadequate numerics. Grid convergence in three dimensions is rarely achieved. Even residual convergence on a given grid is often inadequate. This paper discusses issues involved in residual and especially grid convergence.

Artists have long recognized that trees are self-similar across enormous differences in magnitudes; i.e., they share a common fractal structure - a trunk subdivides into branches which subdivide into more branches which eventually terminate in leaves, flowers, fruits, etc. Artistid Lindenmayer (1971, 1975, 1989, 1990) invented a mathematics based on graph grammar rewriting systems to describe such iteratively branching structures; these were named in honor of him and are referred to as L-systems. With the advent of fractals into computer graphics, numerous artists have similarly produced a wide variety of software packages to illustrate the beauty of fractal/L-system generated plants. Some tree visualizations such as L-Peach (Allen , 2005) do depend very explicitly upon a complex set of precise measurements of a single species of tree. Nonetheless, we felt that there is a need to build a package that allowed scientists (and students) to collect data from actual specimens in the field or laboratory, insert these measurements into an L-system package, and then visually compare actual trees to the computer generated image with such specimens. Furthermore, the effect of variance in parameters helps users evaluate the developmental plasticity both within and between species and varieties. We have developed 3D FractaL Tree (the L is capitalized in honor of Lindemayer) to generate trees based upon measurement of (1) relative lengths of two successive segments averaged over several iterations, (2) the angle theta between bifurcating limbs at successive joints, (3) the number of steps in branching that one must follow to find a branch extending at the same angle as the first one under consideration to determine the phyllotactic angle phi, (4) the average of the summed areas (determined from measurement of diameters) of bifurcations compared to the trunk to determine whether area of flow is preserved (and to consider Poiseuille’s/Murray’s law of laminar flow in a fractal network), (5) the total number of iterative branching from the base to the tip of tree averaged over several counts based on following out different major limbs, (6) an editable L-system rule chosen from a library of branching patterns that roughly correspond to a specimen under consideration, and (7) a degree of stochasticity applied to the above rules to represent some variation over the course of a lifetime. Of course, turned upside down, the computer imagery could be used to represent root structure instead of above ground growth or the bronchial system of a lung, for example. The measurements are recorded and analyzed in a series of worksheets in Microsoft Excel and the results are entered into the graphics engine in a Java application. 3D FractaL Tree produces a rotatable three-dimensional image of the tree which is helpful for examining such characters as self-avoidance (entanglement and breakage), reception of and penetration of sunlight, distances that small herbivores (such as caterpillars) would have to traverse to go from one tip to another, allometric relationships between the convex hull of the crown (as perceived in a top-down projection of the tree) and the trunk’s diameter, and convex hull of the volume distribution of biomass on different subsections of a tree which have been discussed in the Adaptive Geometry of Trees (Horn, 1971) and subsequent research for the past four decades. Besides being able to rotate the three dimensional tree in the x-y, y-z, and x-z planes as well as zoom-in and zoom-out, three different representations are available in 3D FractaL Tree images: wire frame, solid, and transparent. Easy options for editing L-system rules and saving and exporting images are included. 3D FractaL-Tree is published with a Creative Commons license so that it is freely available for downloading, use, and extending with attribution from our Biological ESTEEM Project (http://bioquest.org/esteem).

A hybrid model of red blood cell production, where cells are considered as discrete objects while intra-cellular proteins and extra-cellular biochemical substances are described with continuous models, is proposed. Spatial organization and regulation of red blood cell production (erythropoiesis) are investigated. Normal erythropoiesis is simulated in two dimensions, and the influence on the output of the model of some parameters involved in cell fate (differentiation, self-renewal, and death by apoptosis) is studied.

One interesting example of a discrete mathematical model used in biology is a food web. The first biology courses in high school and in college present the fundamental nature of a food web, one that is understandable by students at all levels. But food webs as part of a larger system are often not addressed. This paper presents materials that can be used in undergraduate classes in biology (and mathematics) and provides students with the opportunity to explore mathematical models of predator-prey relationships, determine trophic levels, dominant species, stability of the ecosystem, competition graphs, interval graphs, and even confront problems that would appear to have logical answers that are as yet unsolved.

We perform a numerical study of the fluctuations of the rescaled hydrodynamic transverse velocity field during the cooling state of a homogeneous granular gas. We are interested in the role of Molecular Chaos for the amplitude of the hydrodynamic noise and its relaxation in time. For this purpose we compare the results of Molecular Dynamics (MD, deterministic dynamics) with those from Direct Simulation Monte Carlo (DSMC, random process), where Molecular Chaos can be directly controlled. It is seen that the large time decay of the fluctuation’s autocorrelation is always dictated by the viscosity coefficient predicted by granular hydrodynamics, independently of the numerical scheme (MD or DSMC). On the other side, the noise amplitude in Molecular Dynamics, which is known to violate the equilibrium Fluctuation-Dissipation relation, is not always accurately reproduced in a DSMC scheme. The agreement between the two models improves if the probability of recollision (controlling Molecular Chaos) is reduced by increasing the number of virtual particles per cells in the DSMC. This result suggests that DSMC is not necessarily more efficient than MD, if the real number of particles is small (~103 ± 104) and if one is interested in accurately reproduce fluctuations. An open question remains about the small-times behavior of the autocorrelation function in the DSMC, which in MD and in kinetic theory predictions is not a straight exponential.

Atmospheric flow equations govern the time evolution of chemical concentrations in the atmosphere. When considering gas and particle phases, the underlying partial differential equations involve advection and diffusion operators, coagulation effects, and evaporation and condensation phenomena between the aerosol particles and the gas phase. Operator splitting techniques are generally used in global air quality models. When considering organic aerosol particles, the modeling of the thermodynamic equilibrium of each particle leads to the determination of the convex envelope of the energy function. Two strategies are proposed to address the computation of convex envelopes. The first one is based on a primal-dual interior-point method, while the second one relies on a variational formulation, an appropriate augmented Lagrangian, an Uzawa iterative algorithm, and finite element techniques. Numerical experiments are presented for chemical systems of atmospheric interest, in order to compute convex envelopes in various space dimensions.

This paper proposes a quantitative model of the reaction-diffusion type to examine the distribution of interferon-α (IFNα) in a lymph node (LN). The numerical treatment of the model is based on using an original unstructured mesh generation software Ani3D and nonlinear finite volume method for diffusion equations. The study results in suggestion that due to the variations in hydraulic conductivity of various zones of the secondary lymphoid organs the spatial stationary distribution of IFNα is essentially heterogeneous across the organs. Highly protected domains such as sinuses, conduits, co-exist with the regions in which where the stationary concentration of IFNα is lower by about 100-fold. This is the first study where the spatial distribution of soluble immune factors in secondary lymphoid organs is modelled for a realistic three-dimensional geometry.

Plants and animals have highly ordered structure both in time and in space, and one of the main questions of modern developmental biology is the transformation of genetic information into the regular structure of organism. Any multicellular plant begins its development from the universal unicellular state and acquire own species-specific structure in the course of cell divisions, cell growth and death, according to own developmental program. However the cellular mechanisms of plant development are still unknown. The aim of this work was to elaborate and verify the formalistic approach, which would allow to describe and analyze the large data of cellular architecture obtained from the real plants and to reveal the cellular mechanisms of their morphogenesis. Two multicellular embryos of Calla palustris L. (Araceae) was used as a model for the verification of our approach. The cellular architecture of the embryos was reconstructed from the stack of optical and serial sections in three dimensions and described as graphs of genealogy and space adjacency of cells. In result of the comparative analysis of these graphs, a set of regular cell types and highly conservative pattern of cell divisions during five cell generations were found. This mechanism of cellular development of the embryos could be considered as a developmental program, set of rules or grammars applied to the zygote. Also during the comparative analysis the finite plasticity in cell adjacency was described. The structural equivalence and the same morphogenetic potencies of some cells of the embryos were considered as the space-temporal symmetries. The symmetries were represented as a set of regular cell type permutations in the program of development of the embryo cellular architecture. Two groups of cell type permutations were revealed, each was composed of two elements and could be interpreted as the mirror and rotational space symmetries. The results obtained as well as the developed approach can be used in plant tissue modelling based on the real, large and complex structural data.

The newly developed unifying discontinuous formulation named the correction procedure via reconstruction (CPR) for conservation laws is extended to solve the Navier-Stokes equations for 3D mixed grids. In the current development, tetrahedrons and triangular prisms are considered. The CPR method can unify several popular high order methods including the discontinuous Galerkin and the spectral volume methods into a more efficient differential form. By selecting the solution points to coincide with the flux points, solution reconstruction can be completely avoided. Accuracy studies confirmed that the optimal order of accuracy can be achieved with the method. Several benchmark test cases are computed by solving the Euler and compressible Navier-Stokes equations to demonstrate its performance.

A computer-aided method for accurately carrying out the Chapman-Enskog expansion of the Boltzmann equation, including its inelastic variant, is presented and employed to derive a hydrodynamic description of a dilute binary mixture of smooth inelastic spheres. Constitutive relations, formally valid for all physical values of the coefficients of restitution, are calculated by carrying out the pertinent Chapman-Enskog expansion to sufficient high orders in the Sonine polynomials to ensure numerical convergence. The resulting hydrodynamic description is applied to the analysis of a vertically vibrated binary mixture of particles (under gravity) differing only in their respective coefficients of restitution. It is shown that even with this “minor”difference the mixture partly segregates, its steady state exhibiting a sandwich-like configuration.

Continuum mechanics (e.g., hydrodynamics, elasticity theory) is based on the assumption that a small set of fields provides a closed description on large space and time scales. Conditions governing the choice for these fields are discussed in the context of granular fluids and multi-component fluids. In the first case, the relevance of temperature or energy as a hydrodynamic field is justified. For mixtures, the use of a total temperature and single flow velocity is compared with the use of multiple species temperatures and velocities.