Stony Brook AMS - Downloadable Preprints - 2004

We present results on the pore space characterization including distributions for pore volume, pore surface area, throat surface area, and principal direction diameters for pores and throats. We present results on oil and water distribution in the pore space at residual oil and water conditions. We also consider the effects on residual fluid distribution due to the injection and gelation of a water-based gel.

In extensive studies of Berea cores it has been observed that introducing water-based gels in the displacement process (for example, in water shut-off treatment of production wells) reduces permeability to water more than to oil. A number of micro-scale mechanisms have been proposed for this disproportional permeability reduction. Our results provide supporting evidence for the involvement of gel dehydration and oil trapping

To appear in Cerebral Cortex
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SUNYSB-AMS-04-03	Errors in Numerical Solutions of Spherically Symmetric Shock Physics Problems	J. Glimm, J.W. Grove, Y. Kang, T. Lee, D.H. Sharp, Y. Yu, K. Ye and M. Zhao

We seek robust and understandable error models for shock physics simulations. The purpose of this paper is to explore complications introduced by spherical flow in the analysis of errors in the numerical solution of shock interaction problems. In contrast to the case of planar waves, the spherical waves are not constant in strength between interactions and the solution is not piecewise constant between waves. Nevertheless simple power laws predict the dependence of the solution on the radius. We find that the same power laws predict the evolution of the error, as the error, once formed, propagates according to the same laws as govern the solution structures ({\it i.e.} the waves) themselves. We analyze errors in composite wave interaction problems based on the analysis of single interactions and a multi-path scattering formula to combine the effects of errors propagating through the individual interactions.
We refine the wave filters we have previously introduced for the identification and analysis of wave strength and position in planar (1D) shock physics simulations. The filter now must be applicable to the case of non-constant states between waves. The numerical solutions, in contrast to the physical solutions, are approximately constant in a narrow region immediately adjacent to the numerical waves. For this reason, the planar one dimensional wave filters provide sufficient accuracy and are used without change. However, as we contemplate the solution of the same problem in a two dimensional cylindrical geometry ($r, z$) or three dimensional rectangular geometry ($x,y,z$) and also the solutions of perturbed spherical problems, such as the spherical Richtmyer-Meshkov instability problem, there will be a need for higher dimensional wave filters. We offer a solution to this pattern recognition problem.

Submitted to Contemporary Mathematics.
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SUNYSB-AMS-04-04	Studies of the microstructure and properties of dense ceramic coatings produced by high-velocity oxygen-fuel combustion spraying	A. Kulkarni, J. Gutleber, S. Sampeth, A. Goland, W.B. Lindquist, H. Hermann, J. Allen and B. Dowd.

High-velocity oxygen-fuel (HVOF) spraying stands out among the various processes to improve metal and ceramic coating density and surface characteristics. This paper explores microstructure development, coating characterization and properties of HVOF sprayed alumina coatings and compares these with those produced using the conventional air plasma spray process. We report on the characterization of these coatings using small-angle neutron scattering (SANS) and X-ray computed microtomography (XMT) to explain the behavior observed for the two coating systems. Microstructure information on porosity, void orientation distribution, void mean opening dimensions and internal surface areas have been obtained using SANS. XMT (X-ray synchrotron microtomography) has been used to nondestructively image the microstructural features in 3D at a 2.7-um spatial resolution over a 2-3 mm field of view. 3D medial axis analysis has been used for the quantitative analysis of the coarse void space in order to obtain information on the porosity, specific surface area, pore connectivity and size distribution of the larger voids in the coatings. The results reveal different pore morphologies for the two spray processes. While only globular pores are imaged in the plasma sprayed coatings due to the spatial resolution limit, highly layered porosity is imaged in the HVOF coating. When the quantitative SANS and XMT information are combined, the different thermal and mechanical properties of the two different coating types can be explained in terms of their distinctly different void microstructures.

Appears in J. Mat. Sci. Engin. A, 369 (2004) 124-137.
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SUNYSB-AMS-04-05	Self similarity of Rayleigh-Taylor mixing rates	E. George and J. Glimm.

We establish a renormalized self similar scaling law for fluid mixing in the deeply compressible regime. Compressibility introduces a new length scale into the mixing, but our time dependent analysis of the density contrast largely removes the effects of this length scale, so that self similarity is maintained. Dynamically induced density changes lead to a dynamic Atwood number to measure density contrasts. We improve previous density renormalizations to allow a unified treatment of mass diffusion and compressible density stratification in a range of weakly to strongly compressible three dimensional multimode \RT simulations. Some of these simulations use front tracking to prevent numerical interfacial mass diffusion, while the others are untracked and diffusive. Using the dynamic Atwood number to define growth rate constants, approximate universality of the mixing rate constant $\alpha_b$ is obtained at low compressibility. Furthermore, earlier results giving consistent simulation, experiment and theory for nearly incompressible mixing, are now extended to show renormalized self similar scaling with moderate changes in the mixing rates for simulations of highly compressible mixing. The renormalized (i.e., dynamic Atwood number corrected) mixing rates of the diffusive and non-diffusive simulations agree, and show very similar compressibility dependence, with self similar mixing rates tripling as compressibility becomes strong.

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SUNYSB-AMS-04-06	Rayleigh-Taylor mixing rates for compressible flow	H. Jin, X.F. Liu, T. Lu, B. Cheng, J. Glimm and D.H. Sharp.

We study Rayleigh-Taylor instability in both the moderately compressible and weakly compressible regimes. For the two dimensional single mode case, we find that the dimensionless terminal velocities (and associated Froude numbers) are nearly constant over most of this region of parameter space, as the thermodynamic parameters describing the equation of state are varied. The phenomenological drag coefficient which occurs in the single mode buoyancy-drag equation is directly related to the terminal velocities and has a similar behavior. Pressure differences and interface shape, however, display significant dependence on the EOS parameters even for the weakly compressible flows. For three dimensional multimode mixing, we expect accordingly that density stratification rather than drag will provide the leading compressibility effect. We develop an analytical model to account for density stratification effects in multimode self-similar mixing. Our theory is consistent with and extends numerically based conclusions developed earlier which also identify density stratification as the dominant compressibility effect for multimode three dimensional mixing.

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SUNYSB-AMS-04-07	Radiation Coupled Front Tracking Simulations for Laser Driven Shock Experiments	Y. Zhang, R. P. Drake, J. Glimm, J. W. Grove, and D. H. Sharp

The purpose of this paper is to develop a numerical algorithm to track the preheat interface motion driven by radiation transfer in high-intensity laser experiments. Our front tracking algorithm is coupled to a radiation process through an inter-package coupling by connecting the output from a radhydro code hyades TO THE INPUt of FronTier code of front tracking. Our coupled algorithm is validated by comparing simulation results from both codes in both low and high radiation cases. Significant interface motion and deformation of the harmonic perturbation due to radiation preheat are observed in high radiation heat case.

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SUNYSB-AMS-04-08	Error Analysis of Composite Shock Interaction Problems	T. Lee, Y. Yu, M. Zhao, J. Glimm, X. Li and K. Ye

We propose statistical models of uncertainty and error in numerical solutions. To represent errors in shock physics simulations we propose a composition law. The law allows us to estimate errors in the solutions of composite problems in terms of the errors from simpler ones as discussed in a previous paper. In this paper, we conduct a detailed analysis of the errors. One of our goals is to understand the relative magnitude of the input uncertainty and the solution errors. In more detail, we wish to understand the contribution of each interaction to the errors at the end of the simulation.

To be presented at the Ninth ASCE Joint Specialy Conference on Probabilistic Mechanics & Structural Reliability, Albuquerque, New Mexico, July 2004
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SUNYSB-AMS-04-09	Shock Wave Interactions in Spherical and Perturbed Spherical Geometries	S. Dutta, E. George, J. Glimm, J. Grove, H. Jin, T. Lee, X. Li, D. H. Sharp, K. Ye, Y. Yu, Y. Zhang and M. Zhao

The interaction of shock waves with spherical and perturbed spherical layers provides an interesting and important class of problems. We present recent results for these and related planar geometry problems. We address the following issues: 1. The accuracy of different numerical solution methods, 2. The magnitude of numerical solution errors and their causes, 3. The development of the instabilities, 4. Reduced descriptions for chaotic flows.

To appear in special issue of Nonlinear Analysis
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SUNYSB-AMS-04-10	3D image-based study of fluid displacement in a Berea core	M. Prodanovic, W.B. Lindquist, and R.S. Seright

We report on three dimensional (3D) pore scale medium characterization, permeability computations for throat structures, and residual fluid distribution in a Berea core. X-ray computed microtomography combined with X-ray attenuating dopants is used to obtain 3D images of the pore network and to resolve phase distributions in the pore space.

We present results on pore characterization, including distributions for pore volume, pore surface area, throat surface area, and principal direction diameters for pores and throats. Lattice Boltzmann computations are used to predict permeabilities for individual throats reconstructed from the images. We present results on oil and water distribution in the pore space at residual conditions. We also consider the effects on residual fluid distribution due to the injection and gelation of a water-based gel. In extensive studies of Berea cores it has been observed that introducing water-based gels in the displacement process reduces permeability to water more than to oil. more than to oil. Our results provide supporting evidence for the involvement of gel compaction (dehydration) and oil trapping, while discounting gel blockage in throats, as mechanisms contributing to this effect.

Morphogen gradients contribute to pattern formation by determining positional information within morphogenetic fields. Interpretation of positional information is thought to rely on direct, concentration threshold dependent mechanisms for establishing multiple differential domains of target gene expression. In Drosophila, maternal gradients establish the initial position of boundaries for zygotic gap gene expression, which in turn convey positional information to pair-rule and segment-polarity genes, the latter forming a segmental prepattern by the onset of gastrulation. Based on quantitative gene expression data, we report substantial anterior shifts in the position of gap domains after their initial establishment. Using a data-driven mathematical modelling approach, we show that these shifts are based on a regulatory mechanism which relies on asymmetric gap-gap cross-repression and does not require diffusion of gap proteins. Our analysis implies that threshold dependent interpretation of maternal morphogen concentration is not sufficient to determine shifting gap domain boundary positions, and suggests that establishment and interpretation of positional information are not independent processes in the Drosophila blastoderm.

Genetic studies have revealed that segment determination in Drosophila melanogaster is based on hierarchical regulatory interactions among maternal coordinate and zygotic segmentation genes. The gap gene system constitutes the most upstream zygotic layer of this regulatory hierarchy, responsible for the initial interpretation of positional information encoded by maternal gradients. We present a detailed analysis of regulatory interactions involved in gap gene regulation based on gap gene circuits, which are mathematical gene network models used to infer regulatory interactions from quantitative gene expression data. Our models reproduce gap gene expression at high accuracy and temporal resolution. Regulatory interactions found in gap gene circuits provide consistent and sufficient mechanisms for gap gene expression, which largely agree with mechanisms previously inferred from qualitative studies of mutant gene expression patterns. Our models predict activation of Kr by Cad, and clarify several other regulatory interactions. Our analysis suggests a central role for repressive feedback loops between complementary gap genes. We observe that repressive interactions among overlapping gap genes show anteroposterior asymmetry with posterior dominance. Lastly, our models suggest a correlation between timing of gap domain boundary formation and regulatory contributions from the terminal maternal system.

We study the relationship between pattern formation in Drosophila segment determination and nuclear structure by replacing the nuclei by a homogeneous continuum. Because this replacement cannot be performed experimentally, mathematical simulation is applied by transforming a previously published model of the segmentation system formulated in terms of explicit nuclear structure into partial differential equations. This transformation has changed the mathematical type of model equations and is therefore interpreted as a structural perturbation of the model. Parameter values are found for three continuum models by means of a new optimal steepest descent algorithm. Each of these models contains a different mathematical representation of nuclear divisions (mitoses). We obtained correct pattern dynamics from all of them, as well as from the model with explicit nuclear structure. This leads us to conclude that nuclear divisions are not coupled to pattern formation and serve only to populate the blastoderm with nuclei. We also investigate whether the calculated patterns in the developmental period modelled resemble their attractors, and find that they fail to do so. The implications of our results for models of biological pattern formation based on partial differential equations are discussed.