Instructors:
- Prof. Suncica Canic (Mathematics, UH)
- Dr. Rosenstrauch (UH, UT Medical Center Houston, Texas Heart Institute)
Office Hours: Tuesday and Thursday: 1-2:30pm (PGH 671)
E-mail: canic@math.uh.edu
Texts:
- Y.C. Fung: Biomechanics: Circulation, ISBN: 0387943846.
- Y.C. Fung: "Biomechanics: Mechanical properties of living tissues", ISBN 0-387-97947-6
- Alexandre J. Chorin, Jerrold E. Marsden: A Mathematical Introduction to Fluid Mechanics, # ISBN: 0387979182
- E.N. Marieb, Human Anatomy and Physiology. Sixth Edition. Pearson Benjamin Cummings, San Francisco 2004.
- McOwen: "Partial Differential Equations", ISBN 0-13-121880-8
- L.C. Evans: "Partial Differential Equations", ISBN 0-8218-0772-2,
- Randall J. Leveque: Numerical Methods for Conservation Laws, # ISBN: 0817627235,
- Walter A. Strauss: Partial Differential Equations: An Introduction, # ISBN: 0471548685
Course Description
Please note that this course is listed as a graduate course, however a motivated undergraduate student who has completed Calculus III, a course on differential equations and an introductory course on partial differential equations should be able to complete the course succesfully. Students with majors in Mathematics, Biology, Engineering/Bioengineering or Physics are welcome. In addition to the standard lectures held at the University of Houston by Prof. Canic, six lectures will be offered by Dr. Rosenstrauch at the Texas Heart Institute. Observation of Open Heart Surgery from the dome of one of the Operating Suites at Texas Heart Institute, and a visit to the Mock Circulatory Flow Loop facility are included in this course.
This course introduces students to the basic tools from mathematics, fluid dynamics, mechanics and human cardiovascular anatomy and physiology, necessary to study problems in cardiovascular fluid dynamics by covering the following topics.
Part I: Linear Partial Differential Equations (Overview)
- Introduction; Well Posedness; Linear v.s. Nonlinear PDEs
- First order linear PDEs; characteristics, transport equation
- Classification of second order linear PDEs
- The Wave Equation: causality, energy, characteristics, D'Alambert formula
- Diffusion: Maximum principle. Energy, Green's function
- Comparison between waves and diffusion
- Laplace's equation, Poisson formula, maximum principle, separation of variables
Part II: Nonlinear Partial Differential Equations: Conservation Laws
- Introduction; examples, characteristics, shock formation
- Weak formulation; Weak solutions; Examples.
- Basic Finite Difference Methods for Linear Conservation Laws
- Finite Difference Methods for Nonlinear Conservation Laws
Part III: Basic Fluid Mechanics
- Equations of Motion
- Transport Theorem, Conservation of Energy, Bernoulli's Theorem
- Navier-Stokes equations I
- Navier-Stokes equations II
Part IV: Basics of Human Cardiovascular Anatomy and Physiology
(lectures held by Dr. Rosenstrauch at the Texas Heart Institute)
- The Blood (composition and characteristics)
- The Heart (anatomy and physiology)
- The Cardiovascular System (Blood Vessels)
- Observing Cardiovascular Surgery at THI
Part V: Cardiovascular Fluid Mechanics
- Laminar flow in a Channel or Tube
- Composition and Properties of Blood Vessel Walls;
- Linear Elasticity; Modeling Vessel Walls ;
- Blood flow properties and modeling (large versus small vessels)
- Wave propagation in blood vessels
Part VI: Axi-Symmetric Models
- Derivation; Asymptotic analysis; Initial and Boundary Data;
- Analysis of the reduced equations (characteristics, hyperbolicity)
- Numerical Methods for the Reduced Equations
- Visiting the THI Mock Flow Loop Facility (Measurements)
- Research Projects
- Guest lectures
