E. Bruce Pitman

Bio: E. Bruce Pitman is an academic researcher from New Jersey Institute of Technology. The author has contributed to research in topics: Tubuloglomerular feedback & Hopf bifurcation. The author has an hindex of 3, co-authored 5 publications receiving 65 citations.

Ill‐posedness in three‐dimensional plastic flow

Abstract: This paper examines partial differential equations for frictional materials flowing via plastic yield, including the equations given by the Critical State Theory of Soil Mechanics. In particular, the material density is considered as a dependent variable. In previous work we demonstrated that two-dimensional plastic flow may be ill posed due to an instability along two rays in Fourier transform space. In this paper, we show that in three dimensions the equations are linearly well posed provided all three strain rates are nonzero.

Tubuloglomerular feedback in a dynamic nephron

Abstract: A dynamic model for a short-looped mammalian nephron is developed to study tubuloglomerular feedback (TGF). Evolution equations for salt and urea concentrations and for fluid flux in the nephron are derived and coupled to a resistance network that serves as a schematic model of the glomerulus and associated structures. The evolution equations, which are semi-linear hyperbolic partial differential equations, are solved by the method of flux-corrected transport. The implementation and testing of this method is described and numerical results are presented. This investigation suggests that: (i) the concentrating nephron exhibits high gain, i.e., a small increase in single nephron glomerular filtration rate produces a large increase in the salt concentration of tubular fluid in the cortical thick ascending limb at the macula densa; (ii) the nephron, as a concentrating system, acts as a low-pass filter, i.e., high frequency pressure oscillations (1 Hz) of a prescribed amplitude at the proximal tubule produce relatively low amplitude oscillations in tubular concentrations, while low frequency oscillations (1/30 Hz) produce relatively high amplitude oscillations in tubular concentrations; and (iii) as a consequence of long time delay in TGF, some perturbations in afferent arteriolar blood pressure induce sustained periodic oscillations similar to those observed in recent experiments.

A reduced model for nephron flow dynamics mediated by tubuloglomerular feedback

Abstract: Previously we developed a “minimal” mathematical model of the tubuloglomerular feedback (TGF) system in a short-looped nephron of the mammalian kidney. In that model, a hyperbolic partial differential equation (PDE) represented the advection and transepithelial transport of chloride in the thick ascending limb (TAL). The feedback response was represented by an empirical relationship that determined the glomerular filtration rate as a function of time-delayed TAL luminal chloride concentration alongside the macula densa. This PDE model system with feedback and a time delay presents analytical and computational challenges. In this report, we derive a reduced model that is based on the minimal model. The reduced model, which is formulated as an integral equation in time, is easier to study than the PDE model. As in the case of the minimal model, analysis of the reduced model suggests that sustained oscillations in nephron fluid flow arise from a Hopf bifurcation, with delay time and system gain as bifurcation parameters. Both analysis and numerical calculations indicate that the principal bifurcation locus predicted by the reduced model coincides with the analogous locus obtained from the minimal model. Near the principal bifurcation locus, numerical solutions of the two models nearly coincide. For bifurcation parameters that differ sufficiently from the principal bifurcation locus, the numerical solutions to the two models differ somewhat. The reduced TGF model has the potential to facilitate simulation and analysis of interactions among TGF systems in multiple nephrons.

MFIX documentation theory guide

Abstract: This report describes the MFIX (Multiphase Flow with Interphase exchanges) computer model. MFIX is a general-purpose hydrodynamic model that describes chemical reactions and heat transfer in dense or dilute fluid-solids flows, flows typically occurring in energy conversion and chemical processing reactors. MFIX calculations give detailed information on pressure, temperature, composition, and velocity distributions in the reactors. With such information, the engineer can visualize the conditions in the reactor, conduct parametric studies and what-if experiments, and, thereby, assist in the design process. The MFIX model, developed at the Morgantown Energy Technology Center (METC), has the following capabilities: mass and momentum balance equations for gas and multiple solids phases; a gas phase and two solids phase energy equations; an arbitrary number of species balance equations for each of the phases; granular stress equations based on kinetic theory and frictional flow theory; a user-defined chemistry subroutine; three-dimensional Cartesian or cylindrical coordinate systems; nonuniform mesh size; impermeable and semi-permeable internal surfaces; user-friendly input data file; multiple, single-precision, binary, direct-access, output files that minimize disk storage and accelerate data retrieval; and extensive error reporting. This report, which is Volume 1 of the code documentation, describes the hydrodynamic theory used in the model: the conservation equations,more » constitutive relations, and the initial and boundary conditions. The literature on the hydrodynamic theory is briefly surveyed, and the bases for the different parts of the model are highlighted.« less

Physics of the Granular State

Abstract: Granular materials display a variety of behaviors that are in many ways different from those of other substances. They cannot be easily classified as either solids or liquids. This has prompted the generation of analogies between the physics found in a simple sandpile and that found in complicated microscopic systems, such as flux motion in superconductors or spin glasses. Recently, the unusual behavior of granular systems has led to a number of new theories and to a new era of experimentation on granular systems.

Computing granular avalanches and landslides

Abstract: rock fragments that might range from centimeters to meters in size, are typically O(10 m) deep, and can run out over distances of tens of kilometers. This vast range of scales, the rheology of the geological material under consideration, and the presence of interstitial fluid in the moving mass, all make for a complicated modeling and computing problem. Although we lack a full understanding of how mass flows are initiated, there is a growing body of computational and modeling research whose goal is to understand the flow processes, once the motion of a geologic mass of material is initiated. This paper describes one effort to develop a tool set for simulations of geophysical mass flows. We present a computing environment that incorporates topographical data in order to generate a numerical grid on which a parallel, adaptive mesh Godunov solver can simulate model systems of equations that contain no interstitial fluid. The computational solver is flexible, and can be changed to allow for more complex material models, as warranted. © 2003 American Institute of Physics. @DOI: 10.1063/1.1614253#

Well-posed and ill-posed behaviour of the -rheology for granular flow

Abstract: In light of the successes of the Navier–Stokes equations in the study of fluid flows, similar continuum treatment of granular materials is a long-standing ambition. This is due to their wide-ranging applications in the pharmaceutical and engineering industries as well as to geophysical phenomena such as avalanches and landslides. Historically this has been attempted through modification of the dissipation terms in the momentum balance equations, effectively introducing pressure and strain-rate dependence into the viscosity. Originally, a popular model for this granular viscosity, the Coulomb rheology, proposed rate-independent plastic behaviour scaled by a constant friction coefficient . Unfortunately, the resultant equations are always ill-posed. Mathematically ill-posed problems suffer from unbounded growth of short-wavelength perturbations, which necessarily leads to grid-dependent numerical results that do not converge as the spatial resolution is enhanced. This is unrealistic as all physical systems are subject to noise and do not blow up catastrophically. It is therefore vital to seek well-posed equations to make realistic predictions. The recent -rheology is a major step forward, which allows granular flows in chutes and shear cells to be predicted. This is achieved by introducing a dependence on the non-dimensional inertial number in the friction coefficient . In this paper it is shown that the -rheology is well-posed for intermediate values of , but that it is ill-posed for both high and low inertial numbers. This result is not obvious from casual inspection of the equations, and suggests that additional physics, such as enduring force chains and binary collisions, becomes important in these limits. The theoretical results are validated numerically using two implicit schemes for non-Newtonian flows. In particular, it is shown explicitly that at a given resolution a standard numerical scheme used to compute steady-uniform Bagnold flow is stable in the well-posed region of parameter space, but is unstable to small perturbations, which grow exponentially quickly, in the ill-posed domain.