Three-Dimensional Flow of an Oldroyd-B Fluid with Variable Thermal Conductivity and Heat Generation/Absorption

This paper looks at the series solutions of three dimensional boundary layer flow. An Oldroyd-B fluid with variable thermal conductivity is considered. The flow is induced due to stretching of a surface. Analysis has been carried out in the presence of heat generation/absorption. Homotopy analysis is implemented in developing the series solutions to the governing flow and energy equations. Graphs are presented and discussed for various parameters of interest. Comparison of present study with the existing limiting solution is shown and examined.     

General solution of cable theory with both ends sealed

General solution of the cable theory with both ends sealed when injecting an arbitrary current at an arbitrary point of the cable is presented, which is a time-dependent transient solution. The solution is an infinite series, each term of which is the product of a cosine term including a position variable only and an exponential term including a time variable only. The general solution contains almost all solutions reported hitherto as particular cases and the mutual relations among the various solutions of quite different forms are clarified by this general solution. Moreover the shorter the cable becomes, the more rapidly this solution converges, therefore it is useful for an analysis of the short cable in the case where the relative deviation error may grow large. The truncation error can also be estimated as the solution is an infinite series of simple functions.     

An axisymmetric boundary integral model for incompressible linear viscoelasticity: application to the micropipette aspiration contact problem

The micropipette aspiration test has been used extensively in recent years as a means of quantifying cellular mechanics and molecular interactions at the microscopic scale. However, previous studies have generally modeled the cell as an infinite half-space in order to develop an analytical solution for a viscoelastic solid cell. In this study, an axisymmetric boundary integral formulation of the governing equations of incompressible linear viscoelasticity is presented and used to simulate the micropipette aspiration contact problem. The cell is idealized as a homogeneous and isotropic continuum with constitutive equation given by three-parameter (E, tau 1, tau 2) standard linear viscoelasticity. The formulation is used to develop a computational model via a "correspondence principle" in which the solution is written as the sum of a homogeneous (elastic) part and a nonhomogeneous part, which depends only on past values of the solution. Via a time-marching scheme, the solution of the viscoelastic problem is obtained by employing an elastic boundary element method with modified boundary conditions. The accuracy and convergence of the time-marching scheme are verified using an analytical solution. An incremental reformulation of the scheme is presented to facilitate the simulation of micropipette aspiration, a nonlinear contact problem. In contrast to the halfspace model (Sato et al., 1990), this computational model accounts for nonlinearities in the cell response that result from a consideration of geometric factors including the finite cell dimension (radius R), curvature of the cell boundary, evolution of the cell-micropipette contact region, and curvature of the edges of the micropipette (inner radius a, edge curvature radius epsilon). Using 60 quadratic boundary elements, a micropipette aspiration creep test with ramp time t* = 0.1 s and ramp pressure p*/E = 0.8 is simulated for the cases a/R = 0.3, 0.4, 0.5 using mean parameter values for primary chondrocytes. Comparisons to the half-space model indicate that the computational model predicts an aspiration length that is less stiff during the initial ramp response (t = 0-1 s) but more stiff at equilibrium (t = 200 s). Overall, the ramp and equilibrium predictions of aspiration length by the computational model are fairly insensitive to aspect ratio a/R but can differ from the half-space model by up to 20 percent. This computational approach may be readily extended to account for more complex geometries or inhomogeneities in cellular properties.     

Two-dimensional Fourier representation used in the bioelectric forward problem.

The objective of this paper is the application of two-dimensional discrete Fourier transformation for solving the integral equation of the bioelectric forward problem. Therefore, the potential, the source term, and the integral equation kernel are assumed to be sampled at evenly spaced intervals. Thus the continuous functions of the problem domain can be expressed by their two-dimensional discrete Fourier transform in the spatial frequency domain. The method is applied to compute the surface potential generated by an eccentric dipole in a homogeneous spherical conducting medium. The integral equation for the potential is solved in the spatial frequency domain and the value of the potential at the sampling points is obtained from inverse Fourier transformation. The solution of the presented method is compared to both, an analytic solution and a solution gained from applying the boundary element method. Isoparametric quadrilateral boundary elements are used for modeling the spherical volume conductor in the boundary element solution, while in the two-dimensional Fourier transformation method the volume conductor is represented by a parametric boundary surface approximation.     

Specificities of one-dimensional dissipative magnetohydrodynamics

One-dimensional dynamics of a plane slab of cold (β ? 1) isothermal plasma accelerated by a magnetic field is studied in terms of the MHD equations with a finite constant conductivity. The passage to the limit β → 0 is analyzed in detail. It is shown that, at β = 0, the character of the solution depends substantially on the boundary condition for the electric field at the inner plasma boundary. The relationship between the boundary condition for the pressure at β > 0 and the conditions for the electric field at β = 0 is found. The stability of the solution against one-dimensional longitudinal perturbations is analyzed. It is shown that, in the limit β → 0, the stationary solution is unstable if the time during which the acoustic wave propagates across the slab is longer than the time of magnetic field diffusion. The growth rate and threshold of instability are determined, and results of numerical simulation of its nonlinear stage are presented.     

The effects of axial diffusion and permeability barriers on the transient response of tissue cylinders. I. Solution in transform space

Transient mass transfer in a Krogh tissue cylinder is described by a model taking into account axial diffusion in both blood and tissue, a localized permeability barrier at the capillary membrane and a diffusion barrier on the outer surface and at the ends of the cylinder. Radial diffusion in both blood and tissue is assumed to be infinitely fast. In contrast to previous work, which has usually relied on numerical methods for solving the equations, an exact solution is presented here in Laplace transform space. This allows calculation of the moments of the concentration at any point in the cylinder. Numerical results indicate that the moments of the residence time distribution are affected by the boundary conditions used, and that the discrepancies between the predictions using different conditions may be large in some physiological situations. Order-of-magnitude calculations are used to estimate when the use of simpler models may be feasible. The transform space solution may also be useful for parameter estimation, but it seems preferable to extend the present results to a time-domain solution for this purpose.     

Locating multiple tumors by moving shape analysis

A methodology to determine the unknown shape of an embedded tumor is proposed. A functional that represents the mismatch between a measured experimental temperature profile, which may be obtained by infrared thermography at skin surface, and the solution of an appropriate boundary problem is defined. Using the Pennes’s bioheat transfer equations, the temperature in a section of healthy tissue with a tumor region is modeled by a boundary problem. The functional is related to the shape of the tumor through the solution of the boundary problem, in such a way that finding the minimum of the functional form also means finding the unknown shape of the embedded tumor. The shape derivative of the functional is computed in each node of an approximation of the solution by the method of Finite Elements using similar methods considered by Pironneau [7]. The algorithm presented include an adaptive strategy to improve the error of the objective function. Numerical results with multiple connected tumors are considered to illustrate the potential of the proposed methodology.     

Automated statistical analysis of microbial enumeration by dilution series

Equations are formulated for the standard error and confidence interval for the MPN estimate of microbial density from a general dilution series. A statistical test of homogeneity is presented. This tests whether a handling error in the dilution series may have occurred which would invalidate the density estimate. The analysis may be automated using a Basic computer program which contains a fast algorithm for the solution of the general MPN equation. This allows the calculation of the MPN, standard error, 95% confidence interval and test statistic for any dilution series, with any degree of replication at each dilution level, with variable sample volumes at each dilution level, with variable dilution ratio between levels, and with any number of levels.     

Bipartite expressions for diffusional mass transport in biomembranes

Analytical expressions for solute diffusion through a membrane barrier for different initial and boundary conditions are available in the literature. The three commonest initial and boundary conditions are for a membrane without solute respectively immersed in a solution of constant concentration, immersed in such a solution for one side but with the other side isolated, and immersed in such a solution for one side and with the other side kept at zero concentration. The physical quantities for the first two initial and boundary conditions are concentration and average concentration (the total solute entering the membrane) with amperometric current (flux) and solute that permeates through the membrane (charge passed) for the third initial and boundary condition. Expressions for these methods in the literature are inconvenient for practical applications because of the infinite mathematical series required. An investigation of convergence of these expressions was therefore carried out. Simple but accurate bipartite expressions for these methods were constructed and provided theoretical support for studies on mass transport characterization of biomembranes. As a specific application, these expressions enabled a direct fit of the simulated observables to experimental values to obtain diffusion coefficients. For these initial and boundary conditions and corresponding physical quantities, simple one point methods for diffusion coefficient estimation are also suggested. These latter diffusion coefficients can be initial values for numerical fit methods.     

Foundations of modeling in cryobiology—II: Heat and mass transport in bulk and at cell membrane and ice-liquid interfaces

Modeling coupled heat and mass transport in biological systems is critical to the understanding of cryobiology. In Part I of this series we derived the transport equation and presented a general thermodynamic derivation of the critical components needed to use the transport equation in cryobiology. Here we refine to more cryobiologically relevant instances of a double free-boundary problem with multiple species. In particular, we present the derivation of appropriate mass and heat transport constitutive equations for a system consisting of a cell or tissue with a free external boundary, surrounded by liquid media with an encroaching free solidification front. This model consists of two parts–namely, transport in the “bulk phases” away from boundaries, and interfacial transport. Here we derive the bulk and interfacial mass, energy, and momentum balance equations and present a simplification of transport within membranes to jump conditions across them. We establish the governing equations for this cell/liquid/solid system whose solution in the case of a ternary mixture is explored in Part III of this series.     

Transport of macromolecules in arterial wallin vivo: A mathematical model and analytical solutions

A mathematical model has been developed to simulatein vivo transmural accumulation of an intravenously injected tracer in the aortic wall of experimental animals. Parameters have been included to represent the following processes that affect tracer distribution: permeation of the blood-tissue interface, diffusion through the layers of the artery wall,convective solute drag through the same, and degradation. Of particular interest for thein vivo situation situation is the inclusion of boundary conditions that account for the variation in the plasma concentration of injected tracer as a function of time. Two analytical solutions are presented. The first describes a system in which two boundaries must be delineated; it pertains if the tracer is allowed to circulate until it enters the avascular media of the artery wall both across its luminal boundary and from the capillaries in its outer layer. The second applies to shorter duration experiments in which entry across only the luminal boundary is considered. A limiting case of the solution for short circulation times is presented, compared with a previously published solution, and examined for its potential utility in parameter estimation. Because of its treatment of time-dependent boundary conditions, the model has unique application toin vivo experiments related to macromolecular transport in atherosclerosis that may otherwise elude proper interpretation. This work was supported by National Institutes of Health Grants HL-29582 and HL-07242.     

THE SERIES OF THE REDEFINED SILURIAN SYSTEM

The procedure of first defining a time boundary between the Silurian and Devonian Systems and postponing the designation of a stratotype leaves stratigraphers without guidance as to which series the Silurian should be divided into and which corresponding epochs would be practicable for reference in international work. The situation in Europe is reviewed with respect to established usage, and it is found that with present concepts Silurian series based on the succession in the Welsh Borderland and in the Barrandian can be made compatible in the critical part of the succession. A natural consequence of the establishment of a system boundary at the base of the zone with Monograptus uniformis would be the redefinition of the Downtonian as proposed by Allen & Tarlo 1963, to be used subsynonymously with the Pridolian.     

Pattern formation in generalized Turing systems

Turing's model of pattern formation has been extensively studied analytically and numerically, and there is recent experimental evidence that it may apply in certain chemical systems. The model is based on the assumption that all reacting species obey the same type of boundary condition pointwise on the boundary. We call these scalar boundary conditions. Here we study mixed or nonscalar boundary conditions, under which different species satisfy different boundary conditions at any point on the boundary, and show that qualitatively new phenomena arise in this case. For example, we show that there may be multiple solutions at arbitrarily small lengths under mixed boundary conditions, whereas the solution is unique under homogeneous scalar boundary conditions. Moreover, even when the same solution exists under scalar and mixed boundary conditions, its stability may be different in the two cases. We also show that mixed boundary conditions can reduce the sensitivity of patterns to domain changes.Supported in part by NIH Grant # GM29123     

Viscous flow through slowly expanding or contracting porous walls with low seepage Reynolds number: a model for transport of biological fluids through vessels

In this article, the problem of laminar, isothermal, incompressible and viscous flow in a rectangular domain bounded by two moving porous walls, which enable the fluid to enter or exit during successive expansions or contractions, is investigated. The governing non-linear equations and their associated boundary conditions are transformed into a highly non-linear ordinary differential equation. The series solution of the problem is obtained by utilising the homotopy perturbation method. Graphical results are presented to investigate the influence of the non-dimensional wall dilation rate and seepage Reynolds number (Re) on the velocity, normal pressure distribution and wall shear stress. Since the transport of biological fluids through contracting or expanding vessels is characterised by low seepage Res, the current study focuses on the viscous flow driven by small wall contractions and expansions of two weakly permeable walls.     

A New Trigonometric Spline Approach to Numerical Solution of Generalized Nonlinear Klien-Gordon Equation

The generalized nonlinear Klien-Gordon equation plays an important role in quantum mechanics. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline is presented for the approximate solution of this equation with Dirichlet boundary conditions. The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Several examples are discussed to exhibit the feasibility and capability of the approach. The absolute errors and error norms are also computed at different times to assess the performance of the proposed approach and the results were found to be in good agreement with known solutions and with existing schemes in literature.     

Effect of blood flow on near-the-wall mass transport of drugs and other bioactive agents: a simple formula to estimate boundary layer concentrations

Transport of bioactive agents through the blood is essential for cardiovascular regulatory processes and drug delivery. Bioactive agents and other solutes infused into the blood through the wall of a blood vessel or released into the blood from an area in the vessel wall spread downstream of the infusion/release region and form a thin boundary layer in which solute concentration is higher than in the rest of the blood. Bioactive agents distributed along the vessel wall affect endothelial cells and regulate biological processes, such as thrombus formation, atherogenesis, and vascular remodeling. To calculate the concentration of solutes in the boundary layer, researchers have generally used numerical simulations. However, to investigate the effect of blood flow, infusion rate, and vessel geometry on the concentration of different solutes, many simulations are needed, leading to a time-consuming effort. In this paper, a relatively simple formula to quantify concentrations in a tube downstream of an infusion/release region is presented. Given known blood-flow rates, tube radius, solute diffusivity, and the length of the infusion region, this formula can be used to quickly estimate solute concentrations when infusion rates are known or to estimate infusion rates when solute concentrations at a point downstream of the infusion region are known. The developed formula is based on boundary layer theory and physical principles. The formula is an approximate solution of the advection-diffusion equations in the boundary layer region when solute concentration is small (dilute solution), infusion rate is modeled as a mass flux, and there is no transport of solute through the wall or chemical reactions downstream of the infusion region. Wall concentrations calculated using the formula developed in this paper were compared to the results from finite element models. Agreement between the results was within 10%. The developed formula could be used in experimental procedures to evaluate drug efficacy, in the design of drug-eluting stents, and to calculate rates of release of bioactive substances at active surfaces using downstream concentration measurements. In addition to being simple and fast to use, the formula gives accurate quantifications of concentrations and infusion rates under steady-state and oscillatory flow conditions, and therefore can be used to estimate boundary layer concentrations under physiological conditions.     

Computation of the electrophoretic mobility of proteins.

A scheme is presented for computing the electrophoretic mobility of proteins in free solution, accounting for the details of the protein shape and charge distribution. The method of Teubner is implemented using a boundary integral formulation within which the velocity distribution, the equilibrium electrical potential around the molecule, and the potential distribution due to the applied field are solved for numerically using the boundary element method. Good agreement of the numerical result is obtained for spheres with the corresponding semi-analytical specialization of Henry's analysis. For protein systems, the method is applied to lysozyme and ribonuclease A. In both cases, the predicted mobility tensors are fairly isotropic, with the resulting scalar mobilities being significantly smaller than for spheres of equal volume and net charge. Comparisons with previously published experimental results for ribonuclease show agreement to be excellent in the presence of a net charge, but poorer at the point of zero charge. The approach may be useful for evaluating approximate methods for estimating protein electrophoretic mobilities and for using electrophoretic measurements to obtain insight into charge distributions on proteins.     

Effects of airflow and liquid temperature on ammonia mass transfer above an emission surface: Experimental study on emission rate

The present study performed a series of experiments in a wind tunnel to investigate the impact of velocity, turbulence intensity and liquid–air temperature difference on ammonia emission rates. Decreasing velocity, turbulence intensity and liquid temperature are shown to reduce the ammonia emission rates. The emission rates are more sensitive to the change of velocity at a low velocity compared to change of velocity at a higher velocity range, which corresponds with the conclusion that the boundary layer thickness of velocity increases sharply when velocity is changed from 0.2 m/s to 0.1 m/s. In addition, the emission rates are more sensitive to the change of temperature at a higher temperature than at a lower liquid temperature range. The influence of velocity and liquid–air temperature difference on boundary layer thickness is also analyzed. The relationship between the emission rate and boundary layer thickness is presented.     

Equilibrium electropotentials in electrolyte systems

The determinationof electric potentials in finite regions of symmetrical electrolyte in one-dimensional equilibrium situations requires the solution of the one-dimensional Poisson-Boltzmann equation in which the dependent variable is linearly related to the electric potential and contains unknown parameters. These require evaluation as part of the solution to a given boundary value problem. The general solution of the equation is presented. This involves elliptic functions and integrals and is sectionally isomorphic with respect to an integration parameter. The application to problems posed in terms of both initial values and two-point boundary values is discussed. The solution is used to determine the potential and concentration distributions between two flat-faced charged particles immersed in an electrolyte liquid and having interacting double layers.     

Viscous flow through slowly expanding or contracting porous walls with low seepage Reynolds number: a model for transport of biological fluids through vessels

In this article, the problem of laminar, isothermal, incompressible and viscous flow in a rectangular domain bounded by two moving porous walls, which enable the fluid to enter or exit during successive expansions or contractions, is investigated. The governing non-linear equations and their associated boundary conditions are transformed into a highly non-linear ordinary differential equation. The series solution of the problem is obtained by utilising the homotopy perturbation method. Graphical results are presented to investigate the influence of the non-dimensional wall dilation rate and seepage Reynolds number (Re) on the velocity, normal pressure distribution and wall shear stress. Since the transport of biological fluids through contracting or expanding vessels is characterised by low seepage Res, the current study focuses on the viscous flow driven by small wall contractions and expansions of two weakly permeable walls.