Heat Transfer Analysis for Stationary Boundary Layer Slip Flow of a Power-Law Fluid in a Darcy Porous Medium with Plate Suction/Injection

In this paper, we investigate the slip effects on the boundary layer flow and heat transfer characteristics of a power-law fluid past a porous flat plate embedded in the Darcy type porous medium. The nonlinear coupled system of partial differential equations governing the flow and heat transfer of a power-law fluid is transformed into a system of nonlinear coupled ordinary differential equations by applying a suitable similarity transformation. The resulting system of ordinary differential equations is solved numerically using Matlab bvp4c solver. Numerical results are presented in the form of graphs and the effects of the power-law index, velocity and thermal slip parameters, permeability parameter, suction/injection parameter on the velocity and temperature profiles are examined.     

A second-order high resolution finite difference scheme for a structured erythropoiesis model subject to malaria infection

We develop a second-order high-resolution finite difference scheme to approximate the solution of a mathematical model describing the within-host dynamics of malaria infection. The model consists of two nonlinear partial differential equations coupled with three nonlinear ordinary differential equations. Convergence of the numerical method to the unique weak solution with bounded total variation is proved. Numerical simulations demonstrating the achievement of the designed accuracy are presented.     

Transient behavior of a chemotaxis system modelling certain types of tissue inflammation

A spatially-distributed mathematical model for the inflammatory response to bacterial invasion of tissue is proposed which includes leukocyte motility and chemotaxis behavior and chemical mediator properties explicitly. This system involves three coupled nonlinear partial differential equations and so is not amenable to analysis. Using scaling arguments and singular perturbation techniques, an approximating system of two coupled nonlinear ordinary differential equations is developed. This system now permits analysis by phase plane methods. Using the approximating model, the dependence of the dynamic behavior of the inflammatory response upon key process parameters, including leukocyte chemotaxis, is studied.This work has been supported by the Deutsche Forschungsgemeinschaft     

An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso

In this paper we present a numerical method for the bidomain model, which describes the electrical activity in the heart. The model consists of two partial differential equations (PDEs), which are coupled to systems of ordinary differential equations (ODEs) describing electrochemical reactions in the cardiac cells. Many applications require coupling these equations to a third PDE, describing the electrical fields in the torso surrounding the heart. The resulting system is challenging to solve numerically, because of its complexity and very strict resolution requirements in time and space. We propose a method based on operator splitting and a fully coupled discretization of the three PDEs. Numerical experiments show that for simple simulation cases and fine discretizations, the algorithm is second-order accurate in space and time.     

Mathematical model for transient isoelectric focusing of simple ampholytes

A mathematical model describing transient processes in isoelectric focusing (IEF) of L biprotic ampholytes is presented. The model is a generalization of our previous research on steady slate in IEF and consists of L nonlinear partial differential equations coupled with 2L+2 algebraic equations. Constraints imposed by the mode of operation, viz., constant current. voltage or power, are described. Due to the nonlinearity of the equations, analysis of the model requires computer simulation. Model equations suitable for computer implementation are derived.     

Diffusive coupling can discriminate between similar reaction mechanisms in an allosteric enzyme system

Background  

A central question for the understanding of biological reaction networks is how a particular dynamic behavior, such as bistability or oscillations, is realized at the molecular level. So far this question has been mainly addressed in well-mixed reaction systems which are conveniently described by ordinary differential equations. However, much less is known about how molecular details of a reaction mechanism can affect the dynamics in diffusively coupled systems because the resulting partial differential equations are much more difficult to analyze.     

Network Development in Biological Gels: Role in Lymphatic Vessel Development

In this paper, we present a model that explains the prepatterning of lymphatic vessel morphology in collagen gels. This model is derived using the theory of two phase rubber material due to Flory and coworkers and it consists of two coupled fourth order partial differential equations describing the evolution of the collagen volume fraction, and the evolution of the proton concentration in a collagen implant; as described in experiments of Boardman and Swartz (Circ. Res. 92, 801–808, 2003). Using linear stability analysis, we find that above a critical level of proton concentration, spatial patterns form due to small perturbations in the initially uniform steady state. Using a long wavelength reduction, we can reduce the two coupled partial differential equations to one fourth order equation that is very similar to the Cahn–Hilliard equation; however, it has more complex nonlinearities and degeneracies. We present the results of numerical simulations and discuss the biological implications of our model. This work was supported by the Royal Society (London) by the award of a University Research Fellowship.     

Double Diffusive Magnetohydrodynamic (MHD) Mixed Convective Slip Flow along a Radiating Moving Vertical Flat Plate with Convective Boundary Condition

In this study combined heat and mass transfer by mixed convective flow along a moving vertical flat plate with hydrodynamic slip and thermal convective boundary condition is investigated. Using similarity variables, the governing nonlinear partial differential equations are converted into a system of coupled nonlinear ordinary differential equations. The transformed equations are then solved using a semi-numerical/analytical method called the differential transform method and results are compared with numerical results. Close agreement is found between the present method and the numerical method. Effects of the controlling parameters, including convective heat transfer, magnetic field, buoyancy ratio, hydrodynamic slip, mixed convective, Prandtl number and Schmidt number are investigated on the dimensionless velocity, temperature and concentration profiles. In addition effects of different parameters on the skin friction factor, , local Nusselt number, , and local Sherwood number are shown and explained through tables.     

Some simple models of cannibalism

A sequence of mathematical models for a species which engages in cannibalism are investigated. The models treat the species as age-structured, and assume that adults consume the unhatched eggs of their own kind. The McKendrick or von Foerster partial differential equation model is first converted into a set of three coupled, nonlinear ordinary differential equations, and then adjusted to describe cannibalism. Some rather unusual dynamical effects are discovered. These include both a Hopf bifurcation and a catastrophic transition from an asymptotically stable equilibrium point to a stable limit cycle.     

Curvature-Driven Pore Growth in Charged Membranes during Charge-Pulse and Voltage-Clamp Experiments

We find that curvature-driven growth of pores in electrically charged membranes correctly reproduces charge-pulse experiments. Our model, consisting of a Langevin equation for the time dependence of the pore radius coupled to an ordinary differential equation for the number of pores, captures the statistics of the pore population and its effect on the membrane conductance. The calculated pore radius is a linear, and not an exponential, function of time, as observed experimentally. Two other important features of charge-pulse experiments are recovered: pores reseal for low and high voltages but grow irreversibly for intermediate values of the voltage. Our set of coupled ordinary differential equations is equivalent to the partial differential equation used previously to study pore dynamics, but permits the study of longer timescales necessary for the simulations of voltage-clamp experiments. An effective phase diagram for such experiments is obtained.     

Effect of enzyme organization on the stability of Yates-Pardee pathways

Yates-Pardee-type metabolic pathways in a heterogenous cell milieu are modeled as a system of coupled non-linear partial differential equations. A numerical solution to this systmm is described and some properties of such a physiological system are studied. Confinement with and without a membrane is considered and it is shown how confinement results in an increase in the stability of the metabolite concentrations. These results suggest that the enzyme organization may contribute to the stability of the cellular metabolism.     

Three-Dimensional Flow of an Oldroyd-B Nanofluid towards Stretching Surface with Heat Generation/Absorption

This article addresses the steady three-dimensional flow of an Oldroyd-B nanofluid over a bidirectional stretching surface with heat generation/absorption effects. Suitable similarity transformations are employed to reduce the governing partial differential equations into coupled nonlinear ordinary differential equations. These nonlinear ordinary differential equations are then solved analytically by using the homotpy analysis method (HAM). Graphically results are presented and discussed for various parameters, namely, Deborah numbers and , heat generation/absorption parameter Prandtl parameter , Brownian motion parameters, thermophoresis parameter and Lewis number . We have seen that the increasing values of the Brownian motion parameter and thermophoresis parameter leads to an increase in the temperature field and thermal boundary layer thickness while the opposite behavior is observed for concentration field and concentration boundary layer thickness. To see the validity of the present work, the numerical results are compared with the analytical solutions obtained by Homotopy analysis method and noted an excellent agreement for the limiting cases.     

Transient responses of budding yeast populations

The unequal-division model for budding yeast is used to formulate a population-balance model for the transient behavior of populations of these organisms. The model consists of linear partial differential equations coupled through algebraic equations. It is shown how the solution of this system of equations can be obtained in a systematic stepwise fashion. The special case of a population subjected to a step change in growth rate is described in some detail, and solutions for two special cases are determined for transients following an age-distribution perturbation. It is shown how experimental data on transient behavior of a cell population can yield information on single-cell mass-synthesis kinetics and on the manner in which individual cells control certain critical parameters in the cell cycle.     

An introduction to hybrid computers and their application to optimization problems

The author, who is Secretary of the Analog Section of the British Computer Society, describes the way in which some of the logic components of a parallel logic analog computer function, and goes on to discuss the use of such a computer in optimization problems, as for instance in finding the best estimates of the parameters on compartmentation analysis of isotopic tracer experiments. The increased power which is gained when an analog computer is coupled to (hybridized with) a general purpose digital computer is then discussed, as shown for example in the increased speed of random-search optimization techniques when an analog computer is used to integrate the differential equations, or the improvement in steepest ascent methods when the parallel logic can be used to keep the hill-climb path continuously on the line of steepest ascent. Finally, the use of a hybrid computer in solving partial differential equations is briefly outlined.     

Improved discretisation and linearisation of active tension in strongly coupled cardiac electro-mechanics simulations

Mathematical models of cardiac electro-mechanics typically consist of three tightly coupled parts: systems of ordinary differential equations describing electro-chemical reactions and cross-bridge dynamics in the muscle cells, a system of partial differential equations modelling the propagation of the electrical activation through the tissue and a nonlinear elasticity problem describing the mechanical deformations of the heart muscle. The complexity of the mathematical model motivates numerical methods based on operator splitting, but simple explicit splitting schemes have been shown to give severe stability problems for realistic models of cardiac electro-mechanical coupling. The stability may be improved by adopting semi-implicit schemes, but these give rise to challenges in updating and linearising the active tension. In this paper we present an operator splitting framework for strongly coupled electro-mechanical simulations and discuss alternative strategies for updating and linearising the active stress component. Numerical experiments demonstrate considerable performance increases from an update method based on a generalised Rush–Larsen scheme and a consistent linearisation of active stress based on the first elasticity tensor.     

Efficient solution of ordinary differential equations modeling electrical activity in cardiac cells

The contraction of the heart is preceded and caused by a cellular electro-chemical reaction, causing an electrical field to be generated. Performing realistic computer simulations of this process involves solving a set of partial differential equations, as well as a large number of ordinary differential equations (ODEs) characterizing the reactive behavior of the cardiac tissue. Experiments have shown that the solution of the ODEs contribute significantly to the total work of a simulation, and there is thus a strong need to utilize efficient solution methods for this part of the problem. This paper presents how an efficient implicit Runge-Kutta method may be adapted to solve a complicated cardiac cell model consisting of 31 ODEs, and how this solver may be coupled to a set of PDE solvers to provide complete simulations of the electrical activity.     

Convective Flow of Sisko Fluid over a Bidirectional Stretching Surface

The present investigation focuses the flow and heat transfer characteristics of the steady three-dimensional Sisko fluid driven by a bidirectional stretching sheet. The modeled partial differential equations are reduced to coupled ordinary differential equations by a suitable transformation. The resulting equations are solved numerically by the shooting method using adaptive Runge Kutta algorithm in combination with Newton''s method in the domain [0,∞). The numerical results for the velocity and temperature fields are graphically presented and effects of the relevant parameters are discussed in detail. Moreover, the skin-friction coefficient and local Nusselt number for different values of the power-law index and stretching ratio parameter are presented through tabulated data. The numerical results are also verified with the results obtained analytically by the homotopy analysis method (HAM). Additionally, the results are validated with previously published pertinent literature as a limiting case of the problem.     

Two-Fluid Model of Biofilm Disinfection

We consider a dynamic model of biofilm disinfection in two dimensions. The biofilm is treated as a viscous fluid immersed in a fluid of less viscosity. The bulk fluid moves due to an imposed external parabolic flow. The motion of the fluid is coupled to the biofilm inducing motion of the biofilm. Both the biofilm and the bulk fluid are dominated by viscous forces, hence the Reynolds number is negligible and the appropriate equations are Stokes equations. The governing partial differential equations are recast as boundary integral equations using a version of the Lorenz reciprocal relationship. This allows for robust treatment of the simplified fluid/biofilm motion. The transport of nutrients and antimicrobials, which depends directly on the velocities of the fluid and biofilm, is also included. Disinfection of the bacteria is considered under the assumption that the biofilm growth is overwhelmed by disinfection. Supported by NSF award DMS-0612467.     

Modelling inert gas exchange in tissue and mixed-venous blood return to the lungs

Inert gas exchange in tissue has been almost exclusively modelled by using an ordinary differential equation. The mathematical model that is used to derive this ordinary differential equation assumes that the partial pressure of an inert gas (which is proportional to the content of that gas) is a function only of time. This mathematical model does not allow for spatial variations in inert gas partial pressure. This model is also dependent only on the ratio of blood flow to tissue volume, and so does not take account of the shape of the body compartment or of the density of the capillaries that supply blood to this tissue. The partial pressure of a given inert gas in mixed-venous blood flowing back to the lungs is calculated from this ordinary differential equation. In this study, we write down the partial differential equations that allow for spatial as well as temporal variations in inert gas partial pressure in tissue. We then solve these partial differential equations and compare them to the solution of the ordinary differential equations described above. It is found that the solution of the ordinary differential equation is very different from the solution of the partial differential equation, and so the ordinary differential equation should not be used if an accurate calculation of inert gas transport to tissue is required. Further, the solution of the PDE is dependent on the shape of the body compartment and on the density of the capillaries that supply blood to this tissue. As a result, techniques that are based on the ordinary differential equation to calculate the mixed-venous blood partial pressure may be in error.     

A mechanobiological model of orthodontic tooth movement

Orthodontic tooth movement is achieved by the process of repeated alveolar bone resorption on the pressure side and new bone formation on the tension side. In order to optimize orthodontic treatment, it is important to identify and study the biological processes involved. This article presents a mechanobiological model using partial differential equations to describe cell densities, growth factor concentrations, and matrix densities occurring during orthodontic tooth movement. We hypothesize that such a model can predict tooth movement based on the mechanobiological activity of cells in the PDL. The developed model consists of nine coupled non-linear partial differential equations, and two distinct signaling pathways were modeled: the RANKL–RANK–OPG pathway regulating the communication between osteoblasts and osteoclasts and the TGF-β pathway mediating the differentiation of mesenchymal stem cells into osteoblasts. The predicted concentrations and densities were qualitatively validated by comparing the results to experiments reported in the literature. In the current form, the model supports our hypothesis, as it is capable of conceptually simulating important features of the biological interactions in the alveolar bone—PDL complex during orthodontic tooth movement.