Steady-state Analytical Solutions Research Articles

Two-dimensional description of surface-bounded exospheres with application to the migration of water molecules on the Moon.

On the Moon, water molecules and other volatiles are thought to migrate along ballistic trajectories. Here, this migration process is described in terms of a two-dimensional partial differential equation for the surface concentration, based on the probability distribution of thermal ballistic hops. A random-walk model, a corresponding diffusion coefficient, and a continuum description are provided. In other words, a surface-bounded exosphere is described purely in terms of quantities on the surface, which can provide computational and conceptual advantages. The derived continuum equation can be used to calculate the steady-state distribution of the surface concentration of volatile water molecules. An analytic steady-state solution is obtained for an equatorial ring; it reveals the width and mass of the pileup of molecules at the morning terminator.

Steady-state analytical models for performance assessment of landfill composite liners.

One-dimensional mathematical models were developed for organic contaminant transport through landfill composite liners consisting of a geomembrane (GM) and a geosynthetic clay liner (GCL) or a GM and a compacted clay liner (CCL). The combined effect of leakage through GM defects, diffusion in GM and the underlying soil liners, and degradation in soil liners were considered. Steady state analytical solutions were provided for the proposed mathematical models, which consider the different combinations of advection, diffusion, and degradation. The analytical solutions of the time lag for contaminant transport in the composite liners were also derived. The performance of GM/GCL and GM/CCL was analyzed. For GM/GCL, the bottom flux can be reduced by a factor of 4 when the leachate head decreases from 10 to 0.3 m. The influence of degradation can be ignored for GM/GCL. For GM/CCL, when the leachate head decreases from 10 to 0.3 m, the bottom flux decreases by a factor of 2-4. Leachate head has greater influence on bottom flux in case of larger degradation rate (e.g., half-life = 1 year) compared to the case with lower degradation rate (e.g., half-life = 10 years). As contaminant half-life in soil liner decreases from 10 to 1 year, bottom flux decreases by approximately 2.7 magnitudes of orders. It is indicated that degradation may have greater influence on time lag of composite liner than leachate head. As leachate head increases from zero to 10 m, time lag for GM/CCL can be reduced by 5-6 years. Time lag for the same composite liner can be reduced by 10-11 years as contaminant half-life decreases from 10 to 1 year. Reducing leachate head acting on composite liners and increasing the degradation capacity of the soil liner would be the effective methods to improve the performance of the composite liners. The proposed analytical solutions are relatively simple and can be used for preliminary design and performance assessment of composite liners.

Exploration of the critical depth hypothesis with a simple NPZ model

AbstractThe critical depth hypothesis (CDH) is a predictive criteria for the onset of phytoplankton blooms that comes from the steady-state analytical solution of a simple mathematical model for phytoplankton growth presented by Sverdrup in 1953. Sverdrup's phytoplankton-only model is very elementary compared with state-of-the-art ecosystem models whose numerical solution in a time-varying environment do not systematically conform to the CDH. To highlight which model ingredients make the bloom onset deviate from the CDH, the complexity of Sverdrup's model is incrementally increased, and the impact that each new level of complexity introduced is analysed. Complexity is added both to the ecosystem model and to the parameterization of physical forcing. In the most complete experiment, the model is a one-dimensional Nutrient-Phytoplankton-Zooplankton model that includes seasonally varying mixed layer depth and surface irradiance, light and nutrient limitation, variable grazing, self-shading, export, and remineralization. When complexity is added to the ecosystem model, it is found that the model solution only marginally deviates from the CDH. But when the physical forcing is also changed, the model solution can conform to two competing theories for the onset of phytoplankton blooms—the critical turbulence hypothesis and the disturbance recovery hypothesis. The key roles of three physical ingredients on the bloom onset are highlighted: the intensity of vertical mixing at the end of winter, the seasonal evolution of the mixed-layer depth from the previous summer, and the seasonal evolution of surface irradiance.

Heat transfer through fractal-like porous media: A tessellated continuum approach

This paper tests the hypothesis that the analysis of heat transfer through a cellular structure represented by a fractal or a pre-fractal can be achieved through analysis on a tessellated continuum. A transport theory for fractals is introduced, which is coupled to a hole-fill mapping strategy, to facilitate the analysis of transport problems on a tessellated continuum. The hole-fill maps are constructed by means of an iterated function scheme similar to that applied in the fractal generation process. The method enables known analytical and numerical continuum solutions to be immediately transferred to the fractal medium. A feature of the approach is that complex fractal structures result in continuum transport equations with material properties that are inhomogeneous, anisotropic and discontinuous.It is demonstrated that a measurable temperature is possible on fractal structures, along with finite measures of heat flux and energy. Analytical steady-state thermal solutions are presented incorporating convective heat transfer.

Magnetoelectroluminescence of organic heterostructures: Analytical theory and spectrally resolved measurements

The effect of a magnetic field on the electroluminescence of organic light emitting devices originates from the hyperfine interaction between the electron/hole polarons and the hydrogen nuclei of the host molecules. In this paper, we present an analytical theory of magnetoelectroluminescence for organic semiconductors. To be specific, we focus on bilayer heterostructure devices. In the case we are considering, light generation at the interface of the donor and acceptor layers results from the formation and recombination of exciplexes. The spin physics is described by a stochastic Liouville equation for the electron/hole spin density matrix. By finding the steady-state analytical solution using Bloch-Wangsness-Redfield theory, we explore how the singlet/triplet exciplex ratio is affected by the hyperfine interaction strength and by the external magnetic field. To validate the theory, spectrally resolved electroluminescence experiments on BPhen/m-MTDATA devices are analyzed. With increasing emission wavelength, the width of the magnetic field modulation curve of the electroluminescence increases while its depth decreases. These observations are consistent with the model.

Generalized semi-analytical solution of advection–diffusion–reaction in finite and semi-infinite cylindrical ducts

Abstract A semi-analytical solution for the transient advection–diffusion–reaction problem within finite and semi-infinite ducts is derived. The solution allows for general radial- and time-dependent inlet/outlet conditions, complex boundary conditions on the duct wall including adsorption and decay, and arbitrary velocity profiles of the transporting fluid. The only numerical step of the solution is the inverse Laplace transform in the time variable. Therefore, the approach also produces fully analytical steady-state solutions. The solution is verified against computational fluid dynamics (CFD) simulations under various boundary conditions and velocity profiles (Newtonian and power-law), and in all cases good agreement is obtained. Although theoretically applicable to all regimes, the solution is computationally difficult at very high Peclet numbers and very early times due to numerical instabilities as a result of finite precision arithmetic of computers. A convergence analysis is conducted to delineate the boundaries of this limit for two important cases. The solution was derived using a new approach for solving two-dimensional partial differential equations (PDEs) with non-constant coefficients which parallels the Frobenius and power series methods for solving ordinary differential equations (ODEs). The approach reduces the original PDE to a single infinite-order ODE with constant coefficients. The approach is suspected to provide solutions to a large class of PDEs of this type. The solution may find applications in a number of engineering and/or biomedical fields, it can be used to verify numerical simulators, and serve as a simple and easy-to-implement alternative where access to numerical simulators is not available.

Detection and characterization of multi-filament evolution during resistive switching

We report resistive switching data in TaOx memristors displaying signatures of multi-filament switching modes and present a technique which enables the characterization of the evolution of multiple filaments within a single device during switching, including their temperature, heat flow, conductivity, and time evolving areas. Using a geometrically defined equivalent circuit, we resolve the individual current/voltage values of each filament and demonstrate that the switching curves of each filament collapse onto a common curve determined by the analytical steady-state resistive switching solution for filamentary switching. Finally, we discuss operational modes which may limit the formation of additional conducting filaments, potentially leading to increased device endurance.

Combined analytical and numerical solution for an elastically supported Timoshenko beam to a moving load

A combined analytical and numerical method is presented to get a response for an elastically supported Timoshenko beam to a moving load. The analytical steady-state solution as a particular solution is established and summed with the numerically calculated homogeneous solution. A steady-state solution is sought analytically through the direct application of the Fourier transform to the moving step load. It is shown to be a compact formula composed of exponential and sinusoidal functions depending on the load velocity. The homogeneous solution is established numerically to cancel out the discontinuities and the inconsistent boundary and initial conditions of the steady-state solution. The discontinuities produced by the steady-state solution are removed using the physical characteristics related to the bar wave. Some response curves are shown to compare the beam motions at different load velocities.

Analysis and optimization of an M/G/1 machine repair problem with multiple imperfect coverage

This paper examines an M/G/1 machine repair problem with multiple imperfect coverage. When an operating machine fails, it is immediately detected and located with coverage probability c. We first use a recursive method and supplementary variable technique to develop steady-state analytic solutions. Various system performance measures are developed based on the steady-state probability distributions. We then construct the expected cost function per machine per unit time and formulate an optimization problem to determine the minimum cost. Direct search and Quasi-Newton methods are implemented to search for the optimal number of operating machines, the optimal repair rate, and the optimal coverage factor. Sensitivity analysis and numerical illustrations are also provided.

On the Air-filled Effective Porosity Parameter of Rogers and Nielson’s (1991) Bulk Radon Diffusion Coefficient in Unsaturated Soils

The radon exhalation rate at the earth's surface from soil or rock with radium as its source is the main mechanism behind the radon activity concentrations observed in both indoor and outdoor environments. During the last two decades, many subsurface radon transport models have used Rogers and Nielson's formula for modeling the unsaturated soil bulk radon diffusion coefficient. This formula uses an "air-filled effective porosity" to account for radon adsorption and radon dissolution in the groundwater. This formula is reviewed here, and its hypotheses are examined for accuracy in dealing with subsurface radon transport problems. The author shows its limitations by comparing one dimensional steady-state analytical solutions of the two-phase (air/water) transport equation (Fick's law) with Rogers and Nielson's formula. For radon diffusion-dominated transport, the calculated Rogers and Nielson's radon exhalation rate is shown to be unrealistic as it is independent of the values of the radon adsorption and groundwater dissolution coefficients. For convective and diffusive transport, radon exhalation rates calculated using Fick's law and this formula agree only for high values of gas-phase velocity and groundwater saturation. However, these conditions are not usually met in most shallow subsurface environments where radon migration takes place under low gas phase velocities and low water saturation.

Nonlinear vibration of a rotating laminated composite circular cylindrical shell: traveling wave vibration

In this paper, the large-amplitude (geometrically nonlinear) vibrations of rotating, laminated composite circular cylindrical shells subjected to radial harmonic excitation in the neighborhood of the lowest resonances are investigated. Nonlinearities due to large-amplitude shell motion are considered using the Donnell’s nonlinear shallow-shell theory, with account taken of the effect of viscous structure damping. The dynamic Young’s modulus which varies with vibrational frequency of the laminated composite shell is considered. An improved nonlinear model, which needs not to introduce the Airy stress function, is employed to study the nonlinear forced vibrations of the present shells. The system is discretized by Galerkin’s method while a model involving two degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the forced vibration responses of the two-degrees-of-freedom system. The stability of analytical steady-state solutions is analyzed. Results obtained with analytical method are compared with numerical simulation. The agreement between them bespeaks the validity of the method developed in this paper. The effects of rotating speed and some other parameters on the nonlinear dynamic response of the system are also investigated.

Analytical solutions for two‐phase subsurface flow to a leaky fault considering vertical flow effects and fault properties

AbstractConductive faults in geologic formations can serve as leakage pathways for fluids such as CO2 and methane, which would otherwise remain trapped beneath low‐permeability layers. To estimate leakage rates and the associated pressure effects on adjacent aquifers, modeling of the flow in the vicinity of leaky faults must be performed. The flow from an aquifer to a leaky fault is controlled by aquifer properties as well as fault properties, including the fault permeability, fault width, and anisotropy in the fault permeability. Depending on aquifer properties and leakage rates, vertical flow equilibrium may not be reached in the aquifer in the vicinity of the leaky fault. Therefore, analytical solutions for leakage through faults that allow for vertical flow need to be developed. To do this, the vertically integrated form of the two‐phase flow formulation based on vertical equilibrium is expanded by assuming a linearly structured vertical flow field, and steady state analytical solutions for single‐phase and two‐phase leakage are derived. In the case of two‐phase leakage, anisotropic aquifer permeabilities are shown to produce stronger vertical flow effects than in the case of single‐phase leakage. Using numerical simulations, a model representing fault properties that is compatible with the analytical solutions is developed. The combination of the fault model and the analytical solutions captures the effects of leakage through faults with different properties and vertical flow effects. The incorporation of this fault model/analytical solution in larger basin‐wide multiscale models can enable subgrid‐scale effects due to leakage through faults to be captured with improved efficiency.

An analytical solution for steady state three dimensional thermoelasticity of functionally graded circular plates due to axisymmetric loads

Abstract In this paper, exact solution of steady state thermo-elastic problem of three dimensional circular plate made of functionally graded material is developed. The thermal and mechanical loads on upper and lower surfaces are assumed to apply axisymmetric and a simply supported boundary conditions on the lateral surface of the plate is considered, where no limiting assumption is used. The material properties, except Poisson's ratio, are assumed to varying exponentially across the thickness direction. A full analytical method is used, and an exact analytical solution is presented.

Heat Diffusion in an Anisotropic Medium with Central Heat Source

Anisotropy is a property of some materials in which the physical properties vary with different crystallographic orientations. In this study, the analytical steady state solution to the anisotropic heat conduction in an orthotropic sphere is presented. There is a point heat source in the center of the sphere which supply the constant thermal energy. Using a transformation can change the anisotropic problem to an isotropic one which can be solved by the change of variables. The results showed that the isothermal surfaces are concentric ellipsoids which approach spheres while the conductivities in different directions are close to each other. The analytical results were validated with the 2D experimental tests performed for quartz as an orthotropic material. Despite the isotropic materials, it can be seen that the vectors of temperature gradients T ∇ and the heat flux q are not parallel.

Modelling and analysis of gene regulatory networks based on the G-network

G-networks are a class of stochastic models that have had a broad range of applications ranging from the performance analysis of computer systems and networks to the modelling of gene regulatory networks. Gene regulatory networks consist of thousands of genes and proteins which are dynamically interacting with each other. Once these regulatory structures are revealed, it is necessary to understand their dynamical behaviours since pathway activities could be changed by their given conditions. This review mainly focuses on a stochastic GRN modelling techniques based on G-networks which provide the analytical steady-state solution of a system for efficient GRN dynamics modelling. Three applications of the G-network model to GRNs show that this novel approach can serve to detect abnormalities from protein expression data, and that they can help to explicit the behaviour of complicated GRN models by dividing the gene regulatory processes into DNA and protein layers.

Cellular senescence in the Penna model of aging

Cellular senescence is thought to play a major role in age-related diseases, which cause nearly 67% of all human deaths worldwide. Recent research in mice showed that exercising mice had higher levels of telomerase, an enzyme that helps maintain telomere length, than nonexercising mice. A commonly used model for biological aging was proposed by Penna. I propose a modification of the Penna model that incorporates cellular senescence and find an analytical steady-state solution following Coe, Mao, and Cates [Phys. Rev. Lett. 89, 288103 (2002)]. I find that models corresponding to delayed cellular senescence have younger populations that live longer. I fit the model to the United Kingdom's death distribution, which the original Penna model cannot do.

Unified solutions for steady-state ground-coupled heat transfer

Unified analytical steady-state solutions are presented for uninsulated two-dimensional ground-coupled heat transfer. Using Schwarz–Christoffel transformations, the unified solutions have been applied to obtain analytical expressions for several typical ground-coupled heat transfer problems which include floors on a flat ground surface, near a sloping ground, floors above a constant-temperature water table and basements. It is demonstrated that several well-known ground-coupled heat transfer expressions can be derived as the special cases of the unified solutions. A method has also been provided for extending solutions for a single-zone building to multi-zone ground-coupled heat transfer with varying ground surface temperatures.

Phenomenological model of bacterial aerotaxis with a negative feedback

A phenomenological model for the suspension of the aerotactic swimming microorganisms placed in a chamber with its upper surface open to air is presented. The model was constructed to embody some complexity of the aerotaxis phenomenon, especially, changes in the average bacteria drift velocity under changing environmental conditions. It was assumed that effective forces applied to the cell (gravitational, drag, and thrust) should be essential for the overall system dynamics; and that bacterial propulsion force, but not their swimming velocity, is proportional to the gradient of the oxygen concentration. Mathematically, the model consists of three coupled equations for the oxygen dynamics; for the cell conservation; and for the balance of forces acting on bacteria. An analytical steady-state solution is given for the shallow and deep layers and numerical results are given for the steady-state and initial value problems which are compared with corresponding ones to the Keller–Segel model.

The DREAM: An Integrated Photonic Thresholder

We propose a novel all-optical integrated thresholder called the dual resonator enhanced asymmetric Mach-Zehnder interferometer (DREAM). Unlike prior integrated photonic devices, the DREAM exhibits properties of stable binary decision making, outputting a constant “one” power value for signals above a certain power level and “zero” for signals of lower powers. This thresholding shape arises from the interference of complementary nonlinear effects of two microring resonators (MRR), one in each arm of a Mach-Zehnder interferometer (MZI). The proposed device performs several orders of magnitude better in size, decision latency, energy efficiency, and stability compared to fiber-based methods of optical thresholding. It is best suited for application in densely integrated systems where rapid conversion between analog and digital signal domains is ubiquitous, such as hybrid analog-digital and neuromorphic processing architectures. We derive analytical steady-state solutions to the nonlinear MRR, which enable design simulation, optimization, and automation of a continuous signal thresholder about three orders of magnitude faster than with numerical simulation. Additional numerical simulations indicate the possibility of a 50 GHz pulse thresholder with a 380 pJ switching threshold in a silicon-on-insulator (SOI) platform. The proposed circuit design techniques are potentially applicable to a wide range of materials, waveguide platforms, and resonator types, but for concreteness, we limit the focus of this paper to MRRs in SOI.

Impacts of Aerosol Shortwave Radiation Absorption on the Dynamics of an Idealized Convective Atmospheric Boundary Layer

We investigated the impact of aerosol heat absorption on convective atmospheric boundary-layer (CBL) dynamics. Numerical experiments using a large-eddy simulation model enabled us to study the changes in the structure of a dry and shearless CBL in depth-equilibrium for different vertical profiles of aerosol heating rates. Our results indicated that aerosol heat absorption decreased the depth of the CBL due to a combination of factors: (i) surface shadowing, reducing the sensible heat flux at the surface and, (ii) the development of a deeper inversion layer, stabilizing the upper CBL depending on the vertical aerosol distribution. Steady-state analytical solutions for CBL depth and potential temperature jump, derived using zero-order mixed-layer theory, agreed well with the large-eddy simulations. An analysis of the entrainment zone heat budget showed that, although the entrainment flux was controlled by the reduction in surface flux, the entrainment zone became deeper and less stably stratified. Therefore, the vertical profile of the aerosol heating rate promoted changes in both the structure and evolution of the CBL. More specifically, when absorbing aerosols were present only at the top of the CBL, we found that stratification at lower levels was the mechanism responsible for a reduction in the vertical velocity and a steeper decay of the turbulent kinetic energy throughout the CBL. The increase in the depth of the inversion layer also modified the potential temperature variance. When aerosols were present we observed that the potential temperature variance became significant already around $$0.7z_i$$ (where $$z_i$$ is the CBL height) but less intense at the entrainment zone due to the smoother potential temperature vertical gradient.