Free convection in a triangular enclosure with fluid-saturated porous medium and internal heat generation

Unsteady natural convection inside a triangular cavity has been studied in this study. The cavity is filled with a saturated porous medium with non-isothermal left inclined wall while the bottom surface is isothermally heated and the right inclined surface is isothermally cooled. An internal heat generation is also considered which is dependent on the fluid temperature. The governing equations are solved numerically by finite volume method. The Prandtl number, Pr of the fluid is considered as 0.7 (air) while the aspect ratio and the Rayleigh number, Ra are considered as 0.5 and 105 respectively. The effect of heat generation on the fluid flow and heat transfer have been presented as a form of streamlines and isotherms. The rate of heat transfer through three surfaces of the enclosure is also presented.

Accuracy of rate estimation using relaxed-clock models with a critical focus on the early metazoan radiation

In recent years, a number of phylogenetic methods have been developed for estimating molecular rates and divergence dates under models that relax the molecular clock constraint by allowing rate change throughout the tree. These methods are being used with increasing frequency, but there have been few studies into their accuracy. We tested the accuracy of several relaxed-clock methods (penalized likelihood and Bayesian inference using various models of rate change) using nucleotide sequences simulated on a nine-taxon tree. When the sequences evolved with a constant rate, the methods were able to infer rates accurately, but estimates were more precise when a molecular clock was assumed. When the sequences evolved under a model of autocorrelated rate change, rates were accurately estimated using penalized likelihood and by Bayesian inference using lognormal and exponential models of rate change, while other models did not perform as well. When the sequences evolved under a model of uncorrelated rate change, only Bayesian inference using an exponential rate model performed well. Collectively, the results provide a strong recommendation for using the exponential model of rate change if a conservative approach to divergence time estimation is required. A case study is presented in which we use a simulation-based approach to examine the hypothesis of elevated rates in the Cambrian period, and it is found that these high rate estimates might be an artifact of the rate estimation method. If this bias is present, then the ages of metazoan divergences would be systematically underestimated. The results of this study have implications for studies of molecular rates and divergence dates.

Time dependency of molecular rate estimates and systematic overestimation of recent divergence times

Studies of molecular evolutionary rates have yielded a wide range of rate estimates for various genes and taxa. Recent studies based on population-level and pedigree data have produced remarkably high estimates of mutation rate, which strongly contrast with substitution rates inferred in phylogenetic (species-level) studies. Using Bayesian analysis with a relaxed-clock model, we estimated rates for three groups of mitochondrial data: avian protein-coding genes, primate protein-coding genes, and primate d-loop sequences. In all three cases, we found a measurable transition between the high, short-term (<1–2 Myr) mutation rate and the low, long-term substitution rate. The relationship between the age of the calibration and the rate of change can be described by a vertically translated exponential decay curve, which may be used for correcting molecular date estimates. The phylogenetic substitution rates in mitochondria are approximately 0.5% per million years for avian protein-coding sequences and 1.5% per million years for primate protein-coding and d-loop sequences. Further analyses showed that purifying selection offers the most convincing explanation for the observed relationship between the estimated rate and the depth of the calibration. We rule out the possibility that it is a spurious result arising from sequence errors, and find it unlikely that the apparent decline in rates over time is caused by mutational saturation. Using a rate curve estimated from the d-loop data, several dates for last common ancestors were calculated: modern humans and Neandertals (354 ka; 222–705 ka), Neandertals (108 ka; 70–156 ka), and modern humans (76 ka; 47–110 ka). If the rate curve for a particular taxonomic group can be accurately estimated, it can be a useful tool for correcting divergence date estimates by taking the rate decay into account. Our results show that it is invalid to extrapolate molecular rates of change across different evolutionary timescales, which has important consequences for studies of populations, domestication, conservation genetics, and human evolution.

Evidence for time dependency of molecular rate estimates

Long-term changes in the genetic composition of a population occur by the fixation of new mutations, a process known as substitution. The rate at which mutations arise in a population and the rate at which they are fixed are expected to be equal under neutral conditions (Kimura, 1968). Between the appearance of a new mutation and its eventual fate of fixation or loss, there will be a period in which it exists as a transient polymorphism in the population (Kimura and Ohta, 1971). If the majority of mutations are deleterious (and nonlethal), the fixation probabilities of these transient polymorphisms are reduced and the mutation rate will exceed the substitution rate (Kimura, 1983). Consequently, different apparent rates may be observed on different time scales of the molecular evolutionary process (Penny, 2005; Penny and Holmes, 2001). The substitution rate of the mitochondrial protein-coding genes of birds and mammals has been traditionally recognized to be about 0.01 substitutions/site/million years (Myr) (Brown et al., 1979; Ho, 2007; Irwin et al., 1991; Shields and Wilson, 1987), with the noncoding D-loop evolving several times more quickly (e.g., Pesole et al., 1992; Quinn, 1992). Over the past decade, there has been mounting evidence that instantaneous mutation rates substantially exceed substitution rates, in a range of organisms (e.g., Denver et al., 2000; Howell et al., 2003; Lambert et al., 2002; Mao et al., 2006; Mumm et al., 1997; Parsons et al., 1997; Santos et al., 2005). The immediate reaction to the first of these findings was that the polymorphisms generated by the elevated mutation rate are short-lived, perhaps extending back only a few hundred years (Gibbons, 1998; Macaulay et al., 1997). That is, purifying selection was thought to remove these polymorphisms very rapidly.

Fatigue crack growth and fatigue life evaluation for welded steel bridge members with initial crack

Initial crack widely exists in the welded members of steel bridge induced by the welding procedure or by the fatigue damage crack initiation. The behavior of crack growth with a view to fatigue damage accumulation on the tip of cracks is discussed. Fatigue life of welded components with initial crack in bridges under traffic loading is investigated. Based on existing fatigue experiment results of welded members with initial crack and the fatigue experiment results of welded bridge members under constant stress cycles, the crack would keep semi-elliptical shape with variable ratio of a/c during the crack propagation. Based on the concept of continuum damage accumulated on the tip of fatigue cracks,the fatigue damage law suitable for steel bridge members under traffic loading is modified to consider the crack growth.The virtual crack growth method and the semi-elliptical crack shape assumption are proposed in this paper to deduce a new model of fatigue crack growth rate for welded bridge members under traffic loading. And the calculated method of the stress intensity factor necessary for evaluation of the fatigue life of welded bridge members with cracks is discussed.The proposed fatigue crack growth model is then applied to calculate the crack growth and the fatigue life of existing welded members with fatigue experimental results. The fatigue crack propagation computation results show that the ratio of crack depth to the half crack surface length a/c is variable during crack propagation process and the stress cycle increases with the increase of a0/c0 with certain a0/t0 .The calculated and measured fatigue lives are generally in good agreement,at some initial conditions of cracking, for welded members widely used in steel bridges.

Efficient preconditioning of the method of lines for solving nonlinear two-sided space-fractional diffusion equations

A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the non-local nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear two-sided space-fractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we show how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

Use of Bayesian MUNE to show differing rate of loss of motor units in subgroups of ALS

Objectives To evaluate differences among patients with different clinical features of ALS, we used our Bayesian method of motor unit number estimation (MUNE). Methods We performed serial MUNE studies on 42 subjects who fulfilled the diagnostic criteria for ALS during the course of their illness. Subjects were classified into three subgroups according to whether they had typical ALS (with upper and lower motor neurone signs) or had predominantly upper motor neurone weakness with only minor LMN signs, or predominantly lower motor neurone weakness with only minor UMN signs. In all subjects we calculated the half life of MUs, defined as the expected time for the number of MUs to halve, in one or more of the abductor digiti minimi (ADM), abductor pollicis brevis (APB) and extensor digitorum brevis (EDB) muscles. Results The mean half life of MUs was less in subjects who had typical ALS with both upper and lower motor neurone signs than in those with predominantly upper motor neurone weakness or predominantly lower motor neurone weakness. In 18 subjects we analysed the estimated size of the MUs and demonstrated the appearance of large MUs in subjects with upper or lower motor neurone predominant weakness. We found that the appearance of large MUs was correlated with the half life of MUs. Conclusions Patients with different clinical features of ALS have different rates of loss and different sizes of MUs. Significance: These findings could indicate differences in disease pathogenesis.

Double diffusive magneto-convection fluid flow in a strong cross magnetic field with uniform surface heat and mass flux

In this study, magnetohydrodynamic natural convection boundary layer flow of an electrically conducting and viscous incompressible fluid along a heated vertical flat plate with uniform heat and mass flux in the presence of strong cross magnetic field has been investigated. For smooth integrations the boundary layer equations are transformed in to a convenient dimensionless form by using stream function formulation as well as the free variable formulation. The nonsimilar parabolic partial differential equations are integrated numerically for Pr ≪1 that is appropriate for liquid metals against the local Hartmann parameter ξ . Further, asymptotic solutions are obtained near the leading edge using regular perturbation method for smaller values of ξ . Solutions for values of ξ ≫ 1 are also obtained by employing the matched asymptotic technique. The results obtained for small, large and all ξ regimes are examined in terms of shear stress, τw, rate of heat transfer, qw, and rate of mass transfer, mw, for important physical parameter. Attention has been given to the influence of Schmidt number, Sc, buoyancy ratio parameter, N and local Hartmann parameter, ξ on velocity, temperature and concentration distributions and noted that velocity and temperature of the fluid achieve their asymptotic profiles for Sc ≥ 10:0.

Modeling ion channel dynamics through reflected stochastic differential equations

Ion channels are membrane proteins that open and close at random and play a vital role in the electrical dynamics of excitable cells. The stochastic nature of the conformational changes these proteins undergo can be significant, however current stochastic modeling methodologies limit the ability to study such systems. Discrete-state Markov chain models are seen as the "gold standard," but are computationally intensive, restricting investigation of stochastic effects to the single-cell level. Continuous stochastic methods that use stochastic differential equations (SDEs) to model the system are more efficient but can lead to simulations that have no biological meaning. In this paper we show that modeling the behavior of ion channel dynamics by a reflected SDE ensures biologically realistic simulations, and we argue that this model follows from the continuous approximation of the discrete-state Markov chain model. Open channel and action potential statistics from simulations of ion channel dynamics using the reflected SDE are compared with those of a discrete-state Markov chain method. Results show that the reflected SDE simulations are in good agreement with the discrete-state approach. The reflected SDE model therefore provides a computationally efficient method to simulate ion channel dynamics while preserving the distributional properties of the discrete-state Markov chain model and also ensuring biologically realistic solutions. This framework could easily be extended to other biochemical reaction networks. © 2012 American Physical Society.

Mixed convection flow of micropolar fluid in an open ended arc-shape cavity

Numerical simulations for mixed convection of micropolar fluid in an open ended arc-shape cavity have been carried out in this study. Computation is performed using the Alternate Direct Implicit (ADI) method together with the Successive Over Relaxation (SOR) technique for the solution of governing partial differential equations. The flow phenomenon is examined for a range of values of Rayleigh number, 102 ≤ Ra ≤ 106, Prandtl number, 7 ≤ Pr ≤ 50, and Reynolds number, 10 ≤ Re ≤ 100. The study is mainly focused on how the micropolar fluid parameters affect the fluid properties in the flow domain. It was found that despite the reduction of flow in the core region, the heat transfer rate increases, whereas the skin friction and microrotation decrease with the increase in the vortex viscosity parameter, Δ.

Anomalous subdiffusion equations by the meshless collocation method

This paper aims to develop an implicit meshless collocation technique based on the moving least squares approximation for numerical simulation of the anomalous subdiffusion equation(ASDE). The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach related to the time discretization are investigated theoretically and numerically. The numerical examples with regular and irregular nodal distributions are used to the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling of ASDEs.

Analytical solutions of the multi-term modified power law wave equations in a finite domain

Fractional partial differential equations with more than one fractional derivative term in time, such as the Szabo wave equation, or the power law wave equation, describe important physical phenomena. However, studies of these multi-term time-space or time fractional wave equations are still under development. In this paper, multi-term modified power law wave equations in a finite domain are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals (1, 2], [2, 3), [2, 4) or (0, n) (n > 2), respectively. Analytical solutions of the multi-term modified power law wave equations are derived. These new techniques are based on Luchko’s Theorem, a spectral representation of the Laplacian operator, a method of separating variables and fractional derivative techniques. Then these general methods are applied to the special cases of the Szabo wave equation and the power law wave equation. These methods and techniques can also be extended to other kinds of the multi term time-space fractional models including fractional Laplacian.

Numerical methods for solving the multi-term time-fractional wave-diffusion equations

In this paper, the multi-term time-fractional wave diffusion equations are considered. The multiterm time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

Analytical solutions for the two- and three-dimensional time-fractional telegraph equations by the method of separating variables

In this paper, a method of separating variables is effectively implemented for solving a time-fractional telegraph equation (TFTE) in two and three dimensions. We discuss and derive the analytical solution of the TFTE in two and three dimensions with nonhomogeneous Dirichlet boundary condition. This method can be extended to other kinds of the boundary conditions.

Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain

Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time-space Caputo-Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0, 1], [1, 2] and [0, 2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchko's Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations. © 2012.