Stony Brook AMS - Downloadable Preprints, 1997

We analyse and improve a recently-proposed two-phase flow model for the statistical evolution of two-fluid mixing. A hyperbolic equation for the volume fraction, whose characteristic speed is the average interface velocity $v^*$, plays a central role. We propose a new model for $v^*$ in terms of the volume fraction and fluid velocities, which can be interpreted as a constitutive law for two-fluid mixing. In the incompressible limit, the two-phase equations admit a self-similar solution for an arbitrary scaling of lengths. We show that the constitutive law for $v^*$ can be expressed directly in terms of the volume fraction, and thus it is an experimentally measurable quantity. For incompressible Rayleigh-Taylor mixing, we examine the self-similar solution based on a simple zero-parameter model for $v^*$. It is shown that the present approach gives improved agreement with experimental data for the growth rate of a Rayleigh-Taylor mixing layer. Closure of the two-phase flow model requires boundary conditions for the surfaces that separate the two-phase and single-phase regions, \ie the edges of the mixing layer. We propose boundary conditions for Rayleigh-Taylor mixing based on the inertial, drag, and buoyant forces on the furthest penetrating structures which define these edges. Our analysis indicates that the compatibility of the boundary conditions with the two-phase flow model is an important consideration. The closure assumptions introduced here and their consequences in relation to experimental data are compared to the work of others.

In clinical trials comparing several treatments, it is often the case that the response variance varies from treatment to treatment. This paper discusses the optimal treatment allocation scheme in this situation. For the one-objective scenario, i.e$.$, pariwise comparison of all treatments, three allocation rules, namely, the optimal rule, the equal allocation rule and the naive rule, are given and compared. A rule of thumb is given consequently. For the multiple-objective scenario, several example are discussed and optimal allocation rules are derived.

One common problem in many randomized clinical trials is how to assign patients to several treatment protocols in an optimal way. The traditional optimal design is usually derived under a single criterion, although in reality, there are usually several objectives in a clinical trial. In this paper, optimal treatment allocation schemes are derived for multiple-objective clinical trials with binary outcome measurement. A graphical method for finding the optimal strategy is proposed and several illustrative examples are discussed.

The toxicity of a potential drug must be studied before it can go on to the clinical trial stage. Quantal dose-response experiments are conducted to relate the drug dose level to the probability of a response. The binary response denotes whether the drug is toxic or not at the given dose level. The results of quantal dose-response experiments are often summarized by estimates of dose levels at which the probabilities of being poisoned are 0.25, 0.50 and 0.75. The corresponding dose levels are the three quartiles in the logit scale. One of the research questions addressed here is to find the optimal dosage allocation scheme for the estimation of the second quartile, also called the ``median lethal dose'', subject to the efficiencies for estimating the other two quartiles are not too low. A searching strategy for such an optimal design is proposed and several illustrative examples are given.

We describe a new constitutive theory for two-phase flow models of chaotic mixing layers, which form as two incompressible fluids interpenetrate. This theory is compatible with arbitrary velocities of the edges of a mixing layer, and it gives analytic solutions for the distribution of fluid variables across the layer in terms of these velocities. Our results are in agreement with all available data from planar Rayleigh-Taylor instability experiments. The model that we discuss can be embedded in a larger system of two-phase flow equations in order to predict other important physical quantities, such as the fluid pressures and internal energies in compressible mixing.

This paper describes surface evolution formulated in terms of a Hamilton-Jacobi equation and a solution algorithm based on a three-dimensional front tracking algorithm. Our method achieves sharp resolution in the evolution of surface edges and corners.

This study is motivated by semiconductor chip evolution during deposition and resputtering processes. For this reason, we discuss here the effects of diffuse rescattering on surface features.

We illustrate some of the three-dimensional capabilities of the front tracking algorithm. We also present a validation study by display of two-dimensional cross sections of three-dimensional simulations of a finite length trench. The cross sections correspond to two-dimensional simulations of S. Hamaguchi and S. M. Rossnagel.

We investigate solutions of Riemann problems for systems of two conservation laws in one spatial dimension. Our approach is to organize Riemann solutions into strata of successively higher codimension. The codimension-zero stratum consists of Riemann solutions that are structurally stable: the number and types of waves in a solution are preserved under small perturbations of the flux function and initial data. Codimension-one Riemann solutions, which constitute most of the boundary of the codimension-zero stratum, violate structural stability in a minimal way. At the codimension-one stratum, either the qualitative structure of Riemann solutions changes or solutions fail to be parameterized smoothly by the flux function and the initial data.

In this paper, we give an overview of the phenomena associated with codimension-one Riemann solutions. We list the different kinds of codimension-one solutions, and we classify them according to their geometric properties, their roles in solving Riemann problems, and their relationships to wave curves.

In this paper, we present three new results. The first is a new closure theory for all of the interface averages that couple the individual phases, generalizing a previous result for the average interface velocity. The major physics assumption underlying this closure theory is clearly identified: an absence of length scales beyond those contained in the primitive variables which define the interface average. The second result is a closed form solution of the two-phase flow model in the incompressible limit, which is valid for arbitrary trajectories of the mixing layer boundaries. The third result is an exact balance of forces relation for each such boundary, relating its acceleration to buoyancy, drag, and other forces. By replacing certain terms in this relation with phenomenological laws, one can close the model and thus uniquely specify the two-phase flow.

A marginal model for the analysis of binomial data involving one or two random factors is presented. Two variance-covariance models are derived based on the multiplicative error formulation. The parameters of mean and variance components are estimated using the quasi-likelihood and method of moments, respectively. An application of the model is illustrated by an analysis of multivariate over-dispersed binomial data from a developmental toxicity experiment.

Key words: Extra-binomial variation; mixed effects model; nested design; random effects model; two-way layout.