Search NASA⌕ Search

SEARCH · Search NASA

Results for “Diffusion equations”

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.

At least 1,009 records · Page 56
All sourcesNASA NTRSDOE OSTI
NASA NTRS1986

Transitions to chaos in two-dimensional double-diffusive convection

The partial differential equations governing two-dimensional thermosolutal convection in a Boussinesq fluid with free boundary conditions have been solved numerically in a regime where oscillatory solutions can be found. A systematic study of the transition from nonlinear periodic oscillations to temporal chaos has revealed sequences of period-doubling bifurcations. Overstability occurs if the ratio of the solutal to the thermal diffusivity tau is less than 1 and the solutal Rayleigh number Rs is sufficiently large. Solutions have been obtained for two representative values of tau. For tau = 0.316, R(s) = 10,000, symmetrical oscillations undergo a bifurcation to asymmetry, followed by a cascade of period-doubling bifurcations leading to aperiodicity, as the thermal Rayleigh number R(T) is increased. At higher values of R(T), the bifurcation sequence is repeated in reverse, restoring simple periodic solutions. As R(T) is further increased more period-doubling cascades, followed by chaos, can be identified. Within the chaotic regions there are narrow periodic windows, and multiple branches of oscillatory solutions coexist. Eventually the oscillatory branch ends and only steady solutions can be found. The development of chaos has been investigated for tau = 0.1 by varying R(T) for several different values of R(s). When R(s) is sufficiently small there are periodic solutions whose period becomes infinite at the end of the oscillatory branch. As R(s) is increased, chaos appears in the neighborhood of these heteroclinic orbits. At higher values of R(s), chaos is found for a broader range in R(T). A truncated fifth-order model suggest that the appearance of chaos is associated with heteroclinic bifurcations.

Knobloch, E.↗
NASA NTRS2002

Volcanic Plume Heights on Mars: Limits of Validity for Convective Models

Previous studies have overestimated volcanic plume heights on Mars. In this work, we demonstrate that volcanic plume rise models, as currently formulated, have only limited validity in any environment. These limits are easily violated in the current Mars environment and may also be violated for terrestrial and early Mars conditions. We indicate some of the shortcomings of the model with emphasis on the limited applicability to current Mars conditions. Specifically, basic model assumptions are violated when (1) vertical velocities exceed the speed of sound, (2) radial expansion rates exceed the speed of sound, (3) radial expansion rates approach or exceed the vertical velocity, or (4) plume radius grossly exceeds plume height. All of these criteria are violated for the typical Mars example given here. Solutions imply that the convective rise, model is only valid to a height of approximately 10 kilometers. The reason for the model breakdown is hat the current Mars atmosphere is not of sufficient density to satisfy the conservation equations. It is likely that diffusion and other effects governed by higher-order differential equations are important within the first few kilometers of rise. When the same criteria are applied to eruptions into a higher-density early Mars atmosphere, we find that eruption rates higher than 1.4 x 10(exp 9) kilograms per second also violate model assumptions. This implies a maximum extent of approximately 65 kilometers for convective plumes on early Mars. The estimated plume heights for both current and early Mars are significantly lower than those previously predicted in the literature. Therefore, global-scale distribution of ash seems implausible.

Glaze, Lori S.↗
DOE OSTI2015

Insights into cadmium diffusion mechanisms in two-stage diffusion profiles in solar-grade Cu(In,Ga)Se{sub 2} thin films

Cadmium diffusion experiments were performed on polished copper indium gallium diselenide (Cu(In,Ga)Se{sub 2} or CIGS) samples with resulting cadmium diffusion profiles measured by time-of-flight secondary ion mass spectroscopy. Experiments done in the annealing temperature range between 275 °C and 425 °C reveal two-stage cadmium diffusion profiles which may be indicative of multiple diffusion mechanisms. Each stage can be described by the standard solutions of Fick's second law. The slower cadmium diffusion in the first stage can be described by the Arrhenius equation D{sub 1} = 3 × 10{sup −4} exp (− 1.53 eV/k{sub B}T) cm{sup 2} s{sup −1}, possibly representing vacancy-meditated diffusion. The faster second-stage diffusion coefficients determined in these experiments match the previously reported cadmium diffusion Arrhenius equation of D{sub 2} = 4.8 × 10{sup −4} exp (−1.04 eV/k{sub B}T) cm{sup 2} s{sup −1}, suggesting an interstitial-based mechanism.

36 MATERIALS SCIENCE↗
DOE OSTI2018

Nonlinear optimization for a tumor invasion PDE model

In this work, we introduce a methodology to approximate unknown parameters that appear on a non-linear reaction–diffusion model of tumor invasion. These equations consider that tumor-induced alteration of micro-environmental pH furnishes a mechanism for cancer invasion. A coupled system reaction–diffusion explaining this model is given by three partial differential equations for the non-dimensional spatial distribution and temporal evolution of the density of normal tissue, the neoplastic tissue growth and the excess concentration of H{sup +} ions. The tumor model parameters have a corresponding biological meaning: the reabsorption rate, the destructive influence of H{sup +} ions in the healthy tissue, the growth rate of tumor tissue and the diffusion coefficient. We propose to solve the direct problem using the Finite Element Method (FEM) and minimize an appropriate functional including both the real data (obtained via in-vitro experiments and fluorescence ratio imaging microscopy) and the numerical solution. The gradient of the functional is computed by the adjoint method.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2018

27 Al Pulsed Field Gradient, Diffusion–NMR Spectroscopy of Solvation Dynamics and Ion Pairing in Alkaline Aluminate Solutions

Here, pulsed field gradient nuclear magnetic resonance (PFG-NMR) measurements were successfully applied to the 27 Al (I = 5/2) nucleus in concentrated electrolytes to investigate the diffusion of aluminate ions [Al(OH) 4 – ] in simulant high-level nuclear waste (3 M NaOH) between 25 and 85 °C. The temperature-dependent diffusion coefficients obtained from 1 H, 23 Na, and 27 Al PFG-NMR were well fit by a Vogel–Fulcher–Tammann model and a power law equation. Comparison of 27 Al diffusion coefficients of 0.1 M Al(OH) 4 – in ~3 M MOH (where M = Na + , K + , (CH 3 ) 4 N + ) at room temperature varied in agreement with the expected changes in solution viscosity via Stokes–Einstein relationship, confirming that the dominant Al species at these conditions are Al(OH) 4 – monomers. This 27 Al PFG-NMR study extends an established methodology to a previously unexplored nucleus enabling this experimental technique to be leverage for exploring ion transport, speciation, and solution structure in concentrated electrolytes.

Diffusion↗
NASA NTRS1987

On the acceleration of charged particles at relativistic shock fronts

The diffusive acceleration of highly relativistic particles at a shock is reconsidered. Using the same physical assumptions as Blandford and Ostriker (1978), but dropping the restriction to nonrelativistic shock velocities, the authors find approximate solutions of the particle kinetic equation by generalizing the diffusion approximation to higher order terms in the anisotropy of the particle distribution. The general solution of the transport equation on either side of the shock is constructed, which involves the solution of an eigenvalue problem. By matching the two solutions at the shock, the spectral index of the resulting power law is found by taking into account a sufficiently large number of eigenfunctions. Low-order truncation corresponds to the standard diffusion approximation and to a somewhat more general method described by Peacock (1981). In addition to the energy spectrum, the method yields the angular distribution of the particles and its spatial dependence.

Kirk, J. G.↗
NASA NTRS1994

Microwave heating and joining of ceramic cylinders: A mathematical model

A thin cylindrical ceramic sample is placed in a single mode microwave applicator in such a way that the electric field strength is allowed to vary along its axis. The sample can either be a single rod or two rods butted together. We present a simple mathematical model which describes the microwave heating process. It is built on the assumption that the Biot number of the material is small, and that the electric field is known and uniform throughout the cylinder's cross-section. The model takes the form of a nonlinear parabolic equation of reaction-diffusion type, with a spatially varying reaction term that corresponds to the spatial variation of the electromagnetic field strength in the waveguide. The equation is analyzed and a solution is found which develops a hot spot near the center of the cylindrical sample and which then propagates outwards until it stabilizes. The propagation and stabilization phenomenon concentrates the microwave energy in a localized region about the center where elevated temperatures may be desirable.

Booty, Michael R.↗
DOE OSTI2016

A model for the compositions of non-stoichiometric intermediate phases formed by diffusion reactions, and its application to Nb 3 Sn superconductors

In this work we explore the compositions of non-stoichiometric intermediate phases formed by diffusion reactions: a mathematical framework is developed and tested against the specific case of Nb 3 Sn superconductors. In the first part, the governing equations for the bulk diffusion and interphase interface reactions during the growth of a compound are derived, numerical solutions to which give both the composition profile and growth rate of the compound layer. The analytic solutions are obtained with certain approximations made. In the second part, we explain an effect that the composition characteristics of compounds can be quite different depending on whether it is the bulk diffusion or grain boundary diffusion that dominates in the compounds, and that “frozen” bulk diffusion leads to unique composition characteristics that the bulk composition of a compound layer remains unchanged after its initial formation instead of varying with the diffusion reaction system; here the model is modified for the case of grain boundary diffusion. Lastly, we apply this model to the Nb 3 Sn superconductors and propose approaches to control their compositions.

36 MATERIALS SCIENCE↗
DOE OSTI2017

CRPropa 3.1—a low energy extension based on stochastic differential equations

The propagation of charged cosmic rays through the Galactic environment influences all aspects of the observation at Earth. Energy spectrum, composition and arrival directions are changed due to deflections in magnetic fields and interactions with the interstellar medium. Today the transport is simulated with different simulation methods either based on the solution of a transport equation (multi-particle picture) or a solution of an equation of motion (single-particle picture). We developed a new module for the publicly available propagation software CRPropa 3.1, where we implemented an algorithm to solve the transport equation using stochastic differential equations. This technique allows us to use a diffusion tensor which is anisotropic with respect to an arbitrary magnetic background field. The source code of CRPropa is written in C++ with python steering via SWIG which makes it easy to use and computationally fast. In this paper, we present the new low-energy propagation code together with validation procedures that are developed to proof the accuracy of the new implementation. Furthermore, we show first examples of the cosmic ray density evolution, which depends strongly on the ratio of the parallel κ{sub ∥} and perpendicular κ{sub ⊥} diffusion coefficients. This dependency is systematically examined as well the influence of the particle rigidity on the diffusion process.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Analytical closure to the spatially-filtered Euler equations for shock-dominated flows

To ensure numerical stability in the vicinity of shocks, a variety of methods have been used, including shock-capturing schemes such as weighted essentially non-oscillatory schemes, as well as the addition of artificial diffusivities to the governing equations. Centered finite difference schemes are often avoided near discontinuities due to the tendency for significant oscillations. However, such schemes have desirable conservation properties compared to many shock-capturing schemes. The objective of this work is to derive all necessary viscous/diffusion terms from first principles and then demonstrate the performance of these analytical terms within a centered differencing framework. The physical Euler equations are spatially-filtered with a Gaussian-like filter. Sub-filter scale (SFS) terms arise in the momentum and energy equations. Analytical closure is provided for each of them by leveraging the jump conditions for a shock. No SFS terms are present in the continuity or species equations. Here, this approach is tested for several problems involving shocks in one and two dimensions. Implemented within a centered difference code, the SFS terms perform well for a range of flow conditions without introducing excessive diffusion.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Pore-scale prediction of transport properties in reconstructed nanostructures of organic matter in shales

Size, morphology and distributions of pores in organic matter of shale matrix are discussed based on high resolution images from experiments in the literature. 250 nanoscale structures of the organic matter are then reconstructed by randomly placing pore spheres with different diameters and overlap tolerances. Effects of porosity, the mean diameter and the overlap tolerance on void space connectivity and pore size distribution are studied. Furthermore, a pore-scale model based on the lattice Boltzmann method developed in a previous study is used to predict the Knudsen diffusivity and permeability of the reconstructed organic matter. The simulation results show that the mean pore diameter and overlap tolerance significantly affect the transport properties. The predicted Knudsen effective diffusivity is compared with Bruggeman equation and it is found that this equation underestimates the tortuosity. A modified Bruggeman equation is proposed based on the simulation results. The predicted intrinsic permeability is in acceptable agreement with Kozeny–Carman (KC) equation. In addition, the apparent permeability is determined based on Knudsen diffusivity and intrinsic permeability predicted. The apparent permeability is compared with that obtained with various correlations in the literature. Lastly, Knudsen’s correlations match best with our numerical results and are recommended for calculating the apparent permeability.

04 OIL SHALES AND TAR SANDS↗
NASA NTRS1978

Effects of wave-induced diffusion on thermospheric acoustic-gravity waves

When wave-induced diffusion is included in the equations governing acoustic-gravity wave propagation in an atmosphere in diffusive equilibrium, it is found that phase and amplitude relationships between the density fluctuations of individual atmospheric constituents agree with AE-C satellite observations of wavelike structure only over a narrow range of wave periods. For a 400-km horizotal wavelength, the observations are consistent with internal gravity waves of periods 11.5-22 minutes propagating at phase speeds 300-600 m/sec.

Del Genio, A. D.↗
NASA NTRS2002

Magnetohydrodynamic MACH Code Used to Simulate Magnetoplasmadynamic Thrusters

The On-Board Propulsion program at the NASA Glenn Research Center is utilizing a state of-the-art numerical simulation to model the performance of high-power electromagnetic plasma thrusters. Such thrusters are envisioned for use in lunar and Mars cargo transport, piloted interplanetary expeditions, and deep-space robotic exploration of the solar system. The experimental portion of this program is described in reference 1. This article describes the numerical modeling program used to guide the experimental research. The synergistic use of numerical simulations and experimental research has spurred the rapid advancement of high-power thruster technologies for a variety of bold new NASA missions. From its inception as a U.S. Department of Defense code in the mid-1980's, the Multiblock Arbitrary Coordinate Hydromagnetic (MACH) simulation tool has been used by the plasma physics community to model a diverse range of plasma problems--including plasma opening switches, inertial confinement fusion concepts, compact toroid formation and acceleration, z-pinch implosion physics, laser-target interactions, and a variety of plasma thrusters. The MACH2 code used at Glenn is a time-dependent, two-dimensional, axisymmetric, multimaterial code with a multiblock structure. MACH3, a more recent three-dimensional version of the code, is currently undergoing beta tests. The MACH computational mesh moves in an arbitrary Lagrangian-Eulerian (ALE) fashion that allows the simulation of diffusive-dominated and dispersive-dominated problems, and the mesh can be refined via a variety of adaptive schemes to capture regions of varying characteristic scale. The mass continuity and momentum equations model a compressible viscous fluid, and three energy equations are used to simulate nonthermal equilibrium between electrons, ions, and the radiation field. Magnetic fields are modeled by an induction equation that includes resistive diffusion, the Hall effect, and a thermal source for magnetic fields. Various models of plasma resistivity are included, along with ablation models and multiport circuit solvers. The set of equations is closed using either an ideal gas or real equation of state.

Mikellides, Pavlos G.↗
NASA NTRS1973

An approximate analysis of the diffusing flow in a self-controlled heat pipe.

Constant-density two-dimensional axisymmetric equations are presented for the diffusing flow of a class of self-controlled heat pipes. The analysis is restricted to the vapor space. Condensation of the vapor is related to its mass fraction at the wall by the gas kinetic formula. The Karman-Pohlhausen integral method is applied to obtain approximate solutions. Solutions are presented for a water heat pipe with neon control gas.

Somogyi, D.↗
DOE OSTI2021

Discrete ion stochastic continuum overdamped solvent algorithm for modeling electrolytes

In this paper we develop a methodology for the mesoscale simulation of strong electrolytes. The methodology is an extension of the fluctuating immersed-boundary approach that treats a solute as discrete Lagrangian particles that interact with Eulerian hydrodynamic and electrostatic fields. In both algorithms the immersed-boundary method of Peskin is used for particle-field coupling. Hydrodynamic interactions are taken to be overdamped, with thermal noise incorporated using the fluctuating Stokes equation, including a "dry diffusion" Brownian motion to account for scales not resolved by the coarse-grained model of the solvent. Long-range electrostatic interactions are computed by solving the Poisson equation, with short-range corrections included using an immersed-boundary variant of the classical particle-particle particle-mesh technique. Also included is a short-range repulsive force based on the Weeks-Chandler-Andersen potential. This methodology is validated by comparison to Debye-Hückel theory for ion-ion pair correlation functions, and Debye-Hückel-Onsager theory for conductivity, including the Wien effect for strong electric fields. In each case, good agreement is observed, provided that hydrodynamic interactions at the typical ion-ion separation are resolved by the fluid grid.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

Stochastic Modeling in a Multimaterial Continuum Mixture Shock Physics Code

Stochastic modelling approaches are presented to capture random effects at multiple time and length scales. Random processes that occur at the microscale produce nondeterministic effects at the macroscale. Here we present three stochastic modeling approaches that describe random processes at microscopic length scales and map these processes to the macroscopic length scale. The first stochastic modeling approach is based upon a particle based numerical technique to solve a Stochastic Differential Equation (SDE) using an arbitrary diffusion process to capture random processes at the microstructural level. The second approach prescribes a Probability Density Function (PDF) for the drift and diffusion of the random variable derived using the forward and backward Kolmogorov equations. This method requires mean and drift evolution PDF transport equations. The third approach is the coupling of multiple random variables which are dependent on each other. The relationship of the PDFs and a coupling function, known as a copula, produces a Joint Probability Density Function (JPDF). These stochastic modeling approaches are implemented into a Multiple Component (MC) shock physics computational code and used to model statistical fracture and reactive flow applications.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

The equilibrium-diffusion limit for radiation hydrodynamics

The equilibrium-diffusion approximation (EDA) is used to describe certain radiation-hydrodynamic (RH) environments. When this is done the RH equations reduce to a simplified set of equations. The EDA can be derived by asymptotically analyzing the full set of RH equations in the equilibrium-diffusion limit. Here, we derive the EDA this way and show that it and the associated set of simplified equations are both first-order accurate with transport corrections occurring at second order. Having established the EDA’s first-order accuracy we then analyze the grey nonequilibrium-diffusion approximation and the grey Eddington approximation and show that they both preserve this first-order accuracy. Further, these approximations preserve the EDA’s first-order accuracy when made in either the comoving-frame (CMF) or the lab-frame (LF). And while analyzing the Eddington approximation, we found that the CMF and LF radiation-source equations are equivalent when neglecting O(β 2 ) terms and compared in the LF. Of course, the radiation pressures are not equivalent. It is expected that simplified physical models and numerical discretizations of the RH equations that do not preserve this first-order accuracy will not retain the correct equilibrium-diffusion solutions. As a practical example, we show that nonequilibrium-diffusion radiative-shock solutions devolve to equilibrium-diffusion solutions when the asymptotic parameter is small.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗