Abstract: In this lecture we first review the stability analyses that have been carried out in the past for systems of nonlinear partial differential equations (PDEs) with important applications in fluid mechanics. Next, we consider a relevant nonlinear stability system of three-dimensional and time dependent nonlinear PDEs and their associated boundary conditions with applications for convective flows. Under certain assumptions, scaling and conditions, we will derive a mathematical model for the nonlinear stability system. We then determine some class of nonlinear steady and unsteady solutions of the resulting system. We explain the mathematical and nonlinear properties of these solutions. We also analyze the stability of these solutions and discuss the corresponding results for the stable solutions and their applications.
 

Brief Biography of the Speaker: Daniel N. Riahi joined Dept of Theoretical and Applied Mechanics (TAM) of The University of Illinois at Urbana-Champaign (UIUC) in 1980 and later affiliated with Dept of Mechanical and Industrial Eng (MIE) at UIUC. He served as Full Professor at UIUC from 1995 to 2005 and as Professor Emeritus at UIUC since 2005 with the Home Dept of Mechanical Science and Eng (MechSE) after joining MIE & TAM as a combined MechSE Dept at UIUC in 2006. Professor Riahi also was appointed as Full Professor in the Dept of Math at University of Texas-Pan American since 2006. Dr. Riahi was a Cambridge Univ. (U.K.)-Visiting Scholar in 1986. Earlier than 1980, Dr. Riahi worked at UCLA, Winthrop Univ. and a three-year Post-Doctoral position at the Florida State Univ. (FSU). His academic degrees are Ph.D. in Applied Math (Fluid Mech.) from FSU in1974, M.S. in Math from FSU in 1970 and B.S. in Math from Tehran Univ. in 1966.

Dr. Riahi’s research work & interest in the last four decades include studies in convection, flow instabilities & turbulence, flow during solidification & crystal growth, applied partial differential equations, stability & asymptotic analyses, and math modeling and theoretical developments with applications to eng and physical sciences. His research accomplishments include new theories, such as those for flow in mushy layers, shear flow over wavy walls, rough turbulence and convective flow in the presence of imperfections, uncovering new types of flow patterns for simple- or mixed-modes and multi-modal cases, and a number of discoveries in fundamental areas of convective and shear flows, some of which were already confirmed by the experimental studies. These include, in particular, flow structure during alloy solidification, roughness roles in turbulent shear flow, flow patterns in layers with finite conducting boundaries and non-monotonic dependence of the heat flux with respect to the rotation rate. Professor Riahi received UIUC-MechSE &UIUC-TAM Service Appreciation Letters in 2006, a UIUC Service Recognition Certificate in 2006, a UIUC Honorific Title Award in 2005 and a UIUC-TAM Recognition Award in 2005. He was included in a UIUC List of Teachers Rank as Excellent by their Students. He is member of over seven professional societies and a Fellow of Wessex Institute of Great Britain. He is author of Chapters in a book on Centrifugal Processing that won the Best Basic Science Book-Award by International Academy of Aeronautics in 1997. Dr. Riahi also received a UIUC-COE Research Award in 1994 and an Outstanding UIUC Service Recognition Certificate in 1987. He is author of over 320 publications mostly published in rigorously refereed journals, including books, invited articles and chapters of books. Dr. Riahi’s Professional Activities include Chairman of Applied Math at Winthrop Univ. (1977-78), and UIUC Eng Mech Coordinator and Chief Advisor (1985-86). He was awarded NSF Grants and supervised NASA Sponsored Res. Projects. He also received several UIUC-RB & UTPA-FRC Research Grants and NCSA Awards. He is ABI’s Research Board Advisor, Member of the Program Committee of the 4th Int. Workshop on Materials Processing in High Gravity, Member of the Int. Scientific Committees of the 5th and 6th Int. Conferences on Advances in Fluid Mechanics and Member of Int. Scientific Advisory Board of Advances in Fluid Mech. He is Editor & Editorial Board Member of over 20 Technical Journals and Book Series.