SEARCH · Search NASA
Results for “diffusion equation”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
Investigation of core transport changes in DIII-D discharges with off-axis T e profile peaks
Abstract DIII-D discharges that transition to H-mode solely with off-axis electron cyclotron heating (ECH) often exhibit strong off-axis peaking of electron temperature profiles at the heating location. Electron heat transport properties near these off-axis temperature peaks have been studied using modulated ECH. The Fourier analyzed electron temperature data have been used to infer electron thermal diffusivity. Comparisons with numerical solutions of the time-dependent electron thermal equation find that the data are consistent with a narrow region with electron diffusivity χ e an order of magnitude lower than the average value across the plasma, suggesting an electron internal transport barrier (ITB) near the ECH heating location. Detailed profile analysis and equilibrium reconstructions suggest that the formation of these ITBs are correlated with off-axis values of the safety factor q being near 1. Furthermore, the ECH driven H-mode discharges demonstrate more rapid electron heating rate near the ECH deposition location than L-mode discharges with higher auxiliary ECH heating power. Additional modeling attributes this difference to the modification of electron heat transport in the core at the L-H transition, which also sustains the off-axis electron temperature peaks.
NUMERICAL ANALYSIS OF THE FOKKER-PLANCK EQUATION WITH ADIABATIC FOCUSING: ISOTROPIC PITCH-ANGLE SCATTERING
We present numerical solutions to both the standard and modified two-dimensional Fokker-Planck equations with adiabatic focusing and isotropic pitch-angle scattering. With the numerical solution of the particle distribution function, we then discuss the related numerical issues, calculate the parallel diffusion coefficient using several different methods, and compare our numerical solutions for the parallel diffusion coefficient to the analytical forms derived earlier. We find the numerical solution to the diffusion coefficient for both the standard and modified Fokker-Planck equations agrees with that of Shalchi for the mean squared displacement method of computing the diffusion coefficient. However, we also show the numerical solution agrees with that of Litvinenko and Shalchi and Danos when calculating the diffusion coefficient via the velocity correlation function.
Interpretation and modeling of the averaged equations for a fluid-solid flow
A self-consistent derivation of the conservation laws is given for flows of a fluid-solid mixture. A unified analytical framework for obtaining constitutive relations is provided. This analysis uses a control volume/control surface approach that is widely used in fluid mechanics. All terms in the governing equations and the constitutive relations are written in terms of the mass-weighted averages except solid concentration. It is believed that the mass-weighted average is the natural bridge between micromechanics and constitutive relations. The derived momentum equations contain terms that differ from all existing models except that of Prosperetti and Jones (1984). However, their assumptions are not needed here. Special attention is given to the solid phase pressure. The physical basis of the previously assumed form for this pressure (Givler 1987) becomes clear. A number of related phenomena are also discussed. These include the anti-diffusion and anisotropic normal stresses. The energy equations are also different from existing models.
The ionization of a low-density intercloud medium by a single O star
The ionization structure of hydrogen in a cloud-containing low-density gas surrounding an O star is calculated, emphasizing how UV photons cause the ionization of hydrogen in regions which are totally obscured from the star by neutral clouds. Particular consideration is given to the ionization of such obscured regions by diffuse Lyman continuum radiation produced when H(+) recombines directly to the ground state. The assumed physical structure of the cloud-containing neighborhood of an O star is discussed, a method is outlined for expressing the emission and absorption properties of the neutral clouds and their shadows in terms of effective volume emissivities and absorption coefficients on a line of sight, and equations of radiative transfer are derived for diffuse and stellar radiation in a cloud-containing H II region. Reradiation of Lyman continuum photons from the ionized boundary layer of a neutral cloud or shadow is analyzed. The expected emission measure across a large cloud-containing H II region is determined and found to compare well with that obtained from observations of H-alpha emission around runaway O stars. It is shown that the hydrogen in cloud shadows may be completely ionized by diffuse radiation within approximately half the Stroemgren radius, depending on the extent of that radius, the average frequency of the Lyman continuum radiation from the star, and the average cloud size.
A Systematic Solution Approach for Neutron Transport Problems in Diffuse Regimes
A systematic solution approach for the neutron transport equation, based on a least-squares finite-element discretization, is presented. This approach includes the theory for the existence and uniqueness of the analytical as well as of the discrete solution, bounds for the discretization error, and guidance for the development of an efficient multigrid solver for the resulting discrete problem. To guarantee the accuracy of the discrete solution for diffusive regimes, a scaling transformation is applied to the transport operator prior to the discretization. The key result is the proof of the V-ellipticity and continuity of the scaled least-squares bilinear form with constants that are independent of the total cross section and the absorption cross section. For a variety of least-squares finite-element discretizations this leads to error bounds that remain valid in diffusive regimes. Moreover, for problems in slab geometry a full multigrid solver is presented with V(1, 1)-cycle convergence rates approximately equal to 0.1, independent of the size of the total cross section and the absorption cross section.
Unraveling the Determinant Mechanisms in Flow-Mediated Crystal Growth and Phase Behaviors
To uncover the critical mechanisms responsible for mesoscopic level development during flow-mediated crystal growth, we develop a semi-two-way hydrodynamic coupled structural phase-field crystal formalism (HXPFC-s2). The new formalism, inspired by previous attempts at coupling hydrodynamic and phase-field crystal (PFC) equations, allows for studying mesoscopic flow-mediated crystallization at diffusive timescales pertinent to industrial applications. Unlike previous efforts, the devised coupling to the structural PFC (XPFC) equations allows generalization to more complex crystal structures through explicit parameterization of the direct correlation function (DCF). Utilizing the HXPFC-s2 formalism, we seek to uncover the determinant physical mechanisms in crystallization under simple shear flows by comparing temperature-driven crystallization to flow-mediated crystallization under varying flow-strengths. Parallels and deviations of under-cooling and flow-strength effects on crystal growth are drawn using the crystal cluster-size and system ordering time evolutions. In doing so, we identify scaling behaviors with a Peclet-like number, Pe∼, a critical Peclet-like number, Pe∼*, and flow-field-crystal plane-dependent interactions. Our findings may be relevant for controlling crystal growth and phase behaviors in flow applications.
Diffusion-limited relic particle production
We examine the thermal evolution of particle number densities in the early universe when the particles have a finite diffusion length. Assuming that annihilations are impossible when the mean separation of the particles is larger than their diffusion length, we derive a version of the Boltzmann equation for freeze out in this scenario and an approximate solution, accurate to better than 2%. The effect of a finite diffusion length is to increase the final relic freeze-out abundance over its corresponding value when diffusion effects are ignored. When diffusion is limited only by scattering off of the thermal background, and the annihilation cross section is bounded by unitarity, a significant effect on the freeze-out abundance requires a scattering cross section much larger than the annihilation cross section. A similar effect is demonstrated when the relic particles are produced via the freeze-in mechanism, but in this case the finite diffusion length is due to the scattering of particles that annihilate into the relic particle of interest. For freeze in, the effect of a finite diffusion length is to reduce the final relic particle abundance. The effects of a finite diffusion length are most important when the scattering cross section or the relic mass are very large. While we have not found a particularly compelling example where this would affect previous results, with the current interest in new dark matter candidates it could become an important consideration.
Theoretical description of trapped electron diffusion in the South American anomaly.
Fokker-Planck equation for geomagnetically trapped electron distribution as function of longitude, time, energy and mirror-point field intensity
Diffusion of rays in random media.
Fokker-Planck equation applied to probability density for finding rays larger than correlation distance during propagation in random medium
Dissolution of a cylindrical disk in Hele-Shaw flow: a conformal-mapping approach
Here we apply conformal mapping to find the evolving shapes of a dissolving cylinder in a potential flow. Similar equations can be used to describe melting in a flowing liquid phase. Results are compared with microfluidic experiments and numerical simulations. Shapes predicted by conformal mapping agree almost perfectly with experimental observations, after a modest (20 %) rescaling of the time. Finite-volume simulations show that the differences with experiment are connected to the underlying assumptions of the analytical model: potential flow and diffusion-limited dissolution. Approximate solutions of the equations describing the evolution of the shape of the undissolved solid can be derived from a Laurent expansion of the mapping function from the unit circle. Asymptotic expressions for the evolution of the area of the disk and the shift in its centre of mass have been derived at low and high Péclet number. Analytic approximations to the leading-order Laurent coefficients provide additional insight into the mechanisms underlying pore-scale dissolution.
Diffusion quantum Monte Carlo calculations of SrFeO 3 and LaFeO 3
The equations of state, formation energy, and migration energy barrier of the oxygen vacancy in SrFeO 3 and LaFeO 3 were calculated in this paper with the diffusion quantum Monte Carlo (DMC) method. Calculations were also performed with various Density Functional Theory (DFT) approximations for comparison. DMC reproduces the measured cohesive energies of these materials with errors below 0.23(5) eV and the structural properties within 1% of the experimental values. The DMC formation energies of the oxygen vacancy in SrFeO 3 and LaFeO 3 under oxygen-rich conditions are 1.3(1) and 6.24(7) eV, respectively. Similar calculations with semi-local DFT approximations for LaFeO 3 yielded vacancy formation energies 1.5 eV lower. Comparison of charge density evaluated with DMC and DFT approximations shows that DFT tends to overdelocalize the electrons in defected SrFeO 3 and LaFeO 3 . Finally, calculations with DMC and local density approximation yield similar vacancy migration energy barriers, indicating that steric/electrostatic effects mainly determine migration barriers in these materials.
Square Turing patterns in reaction-diffusion systems with coupled layers
Square Turing patterns are usually unstable in reaction-diffusion systems and are rarely observed in corresponding experiments and simulations. We report here an example of spontaneous formation of square Turing patterns with the Lengyel-Epstein model of two coupled layers. The squares are found to be a result of the resonance between two supercritical Turing modes with an appropriate ratio. Besides, the spatiotemporal resonance of Turing modes resembles to the mode-locking phenomenon. Analysis of the general amplitude equations for square patterns reveals that the fixed point corresponding to square Turing patterns is stationary when the parameters adopt appropriate values.
Influences of carrier diffusion and radial mode field pattern on high speed characteristics for microring lasers
High-speed directly modulated microlasers are potential light sources for on-chip optical interconnection and photonic integrated circuits. In this Letter, dynamic characteristics are studied for microring lasers by rate equation analysis considering radial carrier hole burning and diffusion and experimentally. The coupled modes with a wide radial field pattern and the injection current focused in the edge area of microring resonator can greatly improve the high speed response curve due to the less carrier hole burning. The small-signal response curves of a microring laser connected with an output waveguide exhibit a larger 3 dB bandwidth and smaller roll-off at low frequency than that of the microdisk laser with the same radius of 15 μm, which accords with the simulation results.
A Probabilistic Approach to Interior Regularity of Fully Nonlinear Degenerate Elliptic Equations in Smooth Domains
We consider the value function of a stochastic optimal control of degenerate diffusion processes in a domain D. We study the smoothness of the value function, under the assumption of the non-degeneracy of the diffusion term along the normal to the boundary and an interior condition weaker than the non-degeneracy of the diffusion term. When the diffusion term, drift term, discount factor, running payoff and terminal payoff are all in the class of C{sup 1,1}( D-bar ) , the value function turns out to be the unique solution in the class of C{sub loc}{sup 1,1}(D) Intersection C{sup 0,1}( D-bar ) to the associated degenerate Bellman equation with Dirichlet boundary data. Our approach is probabilistic.
Multi-scale model based design of membrane reactor/separator processes for intensified hydrogen production through the water gas shift reaction
This paper aims to first quantify the impact of various diffusion models (Maxwell-Stefan, Wilke, Dusty-Gas) on the predictions of a multi-scale membrane reactor/separator mathematical model, and to then demonstrate this model's use for the design and process intensification of membrane reactor/separator systems for hydrogen production. This multi-scale model captures velocity, temperature and species' concentration profiles along the catalyst pellet's radial direction, and along the reactor's axial direction, by solving the momentum, energy, and species transport equations, accounting for convection, conduction, reaction, and diffusion mechanisms. In the first part of work, the effect of pellet-scale design parameters (mean pore diameter, volumetric porosity, tortuosity factor, etc.) and various species' flux models on the model predictions is studied. In the second part, the study focuses on the comparison, in terms of their process intensification characteristics, of various hydrogen production processes. These include a conventional high-temperature shift reactor (HTSR)/low-temperature shift reactor (LTSR) sequence, a novel HTSR/membrane separator (MS)/LTSR/MS sequence, and a process that involves low-temperature shift membrane reactors-LTSMR in a series.
Unsteady magnetohydrodynamics mixed convection flow in a rotating medium with double diffusion
Exact solutions of an unsteady Magnetohydrodynamics (MHD) flow over an impulsively started vertical plate in a rotating medium are presented. The effects of thermal radiative and thermal diffusion on the fluid flow are also considered. The governing equations are modelled and solved for velocity, temperature and concentration using Laplace transforms technique. Expressions of velocity, temperature and concentration profiles are obtained and their numerical results are presented graphically. Skin friction, Sherwood number and Nusselt number are also computed and presented in tabular forms. The determined solutions can generate a large class of solutions as special cases corresponding to different motions with technical relevance. The results obtained herein may be used to verify the validation of obtained numerical solutions for more complicated fluid flow problems.
An improved design method and experimental performance of two dimensional curved wall diffusers
A computer design program was developed to incorporate the suction slots in solving the potential flow equations with prescribed boundary conditions. Using the contour generated from this program two Griffith diffusers were tested having area ratios AR = 3 and 4. The inlet Reynolds number ranged from 600,000 to 7 million. It was found that the slot suction required for metastable operation depends on the sidewall suction applied. Values of slot suction of 8% of the inlet flow rate was required for AR = 4 with metastable condition, provided that enough sidewall suction was applied. For AR = 3, the values of slot suction was about 25% lower than those required for AR = 4. For nearly all unseparated test runs, the effectiveness was 100% and the exit flow was uniform. In addition to the Griffith diffusers, dump and cusp diffusers of comparable area ratios were built and tested. The results obtained from these diffusers were compared with those of the Griffith diffusers. Flow separation occurred in all test runs with the dump and cusp diffusers.