**Plenary Lecture
Heat Transfer Processes with Boiling in Extended Surfaces – Approximate and Exact Solutions**

**Professor Andris Buikis**

Institute of Mathematics and Computer Science

University of Latvia

LATVIA

E-mail: buikis@latnet.lv

**Abstract: ** Systems with extended surfaces (with fins, spines and/or nanostructures) have very broad field of applications: from space apparatus, engines, conditioners, fridges etc. till cooling systems for microchips of PC. Modern servers use the fluid boiling heat transfer. From praxis point of view the mathematical description (mathematical models) must be formulated as conjugated problem. We have developed some new models for boiling performance for micro-nano hybrid structure via silicon nanowire arrays.

Intensive quenching (IQ) processes are one of important branches of to day’s metallurgical technologies. In ecologically clean steel quenching processes important aspect is the heat exchange with the surrounding cold water. But the type of heat exchange is mostly defined by initial heat flux densities. If the initial heat flux density is less, then on the sample’s surface the nucleate boiling establishes. In 2005 we proposed the hyperbolic heat equation as mathematical model for intensive steel quenching process. In several papers during 2005-2011 we have developed some approaches (Green function method, original conservative averaging method) for the solving of the direct and time reverse problems for the hyperbolic heat equation.

In this talk we present original (based on the Green function generalization method and conservative averaging method) approach for the determination of the exact and approximated solutions in the systems with extended surfaces of quite complicated geometrical and thermal structure. By modeling practically interesting processes, very often we need to consider the situation, when the medium has a layered structure. Speaking mathematically, such situation can be described by PDE (or its system) with piecewise constant coefficients. In this lecture I’ll show how conjugations conditions, non-ideal contact conditions etc. and their generalizations can be obtained by our original method of conservative averaging (CAM). The usage of CAM for separate relatively thin sub-domain or/and for sub-domain with large medium characteristic, leads to reduction of domain, in which the solution must be found.

**Brief Biography of the Speaker: ** Andris Buikis received the M.S. in numerical mathematics from University of Latvia (Faculty of Physics and Mathematics) in 1963 and Dr. math., University of Latvia, 1970. He was Junior Researcher, Senior Researcher, Computing Centre, University of Latvia, 1962 – 1972. Assistant Professor and Head of Chair of Applied Mathematics, 1972 – 1976 and Head of Chair of Differential Equations and Numerical Methods, Faculty of Physics and Mathematics, University of Latvia, 1976 – 1984. Dr. habil. math. (Doctor of Science in former USSR), University of Kasan, Russia, 1988. Professor, University of Latvia, 1991. Director, Institute of Mathematics, Latvian Academy of Sciences and Latvian University, 1991 - 1996; 2003 – 2006 and Director, Science and Dialogue Centre of Latvia, 1993 -2007. Head of Laboratory of Mathematical Technologies, Institute of Mathematics and Computer Science, University of Latvia 2006-2010.

In 2010 he was elected as Member of Saeima (Parliament of Latvia). Full Member, Latvian Academy of Sciences, 1997. Member, Senate of the Latvian Academy of Sciences, 1994 –2011. The first, who received Piers Bohl Prize established by Latvian Academy of Sciences for a cycle of papers “Method of Conservative Averaging, Theory and Applications”, 2005. Member of Editorial Advisory Board, Journal Mathematical Modelling and Analysis (The Baltic Journal on Mathematical Applications, Numerical Analysis and Differential Equations), Lithuania 1999- and Associate Editor of WSEAS Transactions in Heat and Mass Transfer 2007-, Associate Editor of Journal of Vortex Science and Technology, 2011-.