Interface Problems and Methods in Biological and Physical Flows

The following sections are included:

• CONTENTS

• FOREWORD

• PREFACE

This survey paper is based on the lecture notes for the Institute for Mathematical Sciences (IMS) program on Moving Interface Problems and Applications in Fluid Dynamics, held on January 8–March 31, 2007, at IMS, National University of Singapore, and given by Robert Dillon (Washington State University, US) and Zhilin Li (North Carolina State University, USA). This paper attempts to give a brief survey or tutorial on the immersed boundary (IB) method and the immersed interface method (IIM). The immersed boundary method was originally introduced by Peskin for studying flow patterns around heart valves and for studying blood flow in the heart [118] and has since been applied to many other problems, particularly, in biophysics. The IB method is a mathematical modeling framework as well as a numerical method. The original motivation of the immersed interface method [76, 80] was to improve the accuracy of the IB method, at least from first-order accuracy to second; and deal with discontinuous coefficients in the governing equations. The IIM is a sharp interface method based on Cartesian grids or triangulation. The IIM makes use of jump conditions across interfaces so that the finite difference/element discretization can be accurate. Some new developments in the immersed interface method include the augmented immersed interface method; immersed finite element method; and singularity removal techniques. The immersed boundary and the interface methods have been coupled with different evolution schemes for free boundary and moving interface problems.

Cancer is a result of the uncontrolled growth of abnormal cells. Solid tumors possess two distinct phases of growth — the avascular and the vascular phases. In the former, the tumor nodules do not contain a vascular network and thus receive nutrients and growth factors via diffusion through the host tissue. As a result, these nodules typically remain small and grow at most to several mm in diameter. The transition to metastasis and malignancy typically starts with angiogenesis. Angiogenesis is the development of a tumor-induced neovascular network that grows from the main circulatory system to the tumor in response to an imbalance of proangiogenic growth factors that are released by hypoxic tumor cells. This provides the tumor with a direct supply of nutrients and growthpromoting factors. The vascularized tumor typically invades the host microenvironment and has significant metastatic potential. By the time a tumor reaches a clinically detectable size, it is usually in the vascular growth phase. In these lecture notes, we provide a brief overview of the multiphase mathematical modeling of solid tumor growth that has been developed to describe all phases of tumor progression. We demonstrate that mathematical modeling and numerical simulation can provide important insight into the root causes of solid tumor invasion and metastasis, and can provide and aid in assessing effective treatment strategies.

A review of numerical modeling of detonation processes in a tube that is pertinent to the development of pulse detonation engines is given. Topics of interest are listed. A technique for handling numerical stiffness is described where it is suitable for simulating arc ignition. One- and two-dimensional examples of Euler solutions are also provided.

Direct numerical simulations (DNS) have recently emerged as a viable tool to understand finite Reynolds number multiphase flows. The approach parallels DNS of turbulence, but the unsteady motion of a deforming phase boundary adds considerable complexity. Here, we briefly describe one particular method for DNS, front-tracking, and then review results for bubbly flows. Several simulations have shown that bubble deformation plays a very significant role in both determining the microstructure of the flow (the most probable bubble distribution) as well as the large-scale behavior of the system. Results for homogeneous bubbly flows and drag reduction due to bubble injection into a turbulent channel are discussed briefly and an investigation of the flow of nearly spherical bubbles in vertical channels is described in more detail. In all cases, DNS has lead to a fundamental new understanding of the flow.