Applications of Mathematical Heat Transfer and Fluid Flow Models in Engineering and Medicine

Albert Einstein

Preface

This textbook for advanced graduate and post-graduate courses presents the applications of the modern heat transfer and fluid flow mathematical models in engineering, biology, and medicine. By writing this work, the author continues the introduction of brand-new efficient methods in fluid flow and heat transfer that have been developed and widely used during the last 50 years after computers became common. While his previous two monographs presented these contemporary methods on an academic level in heat transfer only [119] or in both areas heat transfer and fluid flow on the preliminary level [121], this manual introduces the modern approaches in studying corresponding mathematical models-a core of each research means determining its efficiency and applicability. Two types of new mathematical models are considered: the conjugate models in heat transfer and in fluid flow and models of direct numerical simulation of turbulence. The current situation of applications of these models is presented in two parts: applications of conjugate heat transfer in engineering (Part I) and applications of conjugate fluid flow (peristaltic flow) in medicine and biology and applications in engineering of direct numerical simulation (Part II). These parts contain theory, analysis of mathematical models, and methods of problem solution introduced via 134 detailed and 231 shortly reviewed examples selected from a list of 448 cited original papers adopted from 152 scientific general mathematic, computing, and different specific oriented journals, 42 proceedings, reports, theses, and 37 books. This list of 448 references comprehends the whole period of the new methods existing from the 1950s to present time, including more than 100 results published during the last 5 years among which more than half of the studies were issued in 2014 to 2016.

The term conjugate, or coupled problem, was coined in the 1960s to designate the heat transfer strict investigation that requires matching temperature fields of bodies flowing around or inside the fluids. Later on, it became clear that these terms and procedures are important to many other natural and technology processes, consisting of interactions between elements and/or substances. In particular, the peristaltic flow is an inherent conjugate phenomenon because such flow occurs due to interaction between the elastic channel walls and the fluid inside channel. The conjugate formulation reflects the basic features of a studied phenomenon. Due to that, the models of this type are reliable and significantly improve the correctness and physical understanding of the results. Conjugate methods constitute a powerful tool for solving contemporary problems, substituting the previous approximate approaches. At the same time, it is important to know when the common more simple approaches may be used with comparable exactness, instead of more complex conjugate procedures. The textbook answers this question, as well as significant other questions governing the applications of conjugate methods.

The other group of new methods considered in this text is based on direct numerical solution of exact unsteady (without averaging) Navier-Stokes equations. Because the unsteady Navier-Stokes equations describe the complete space- and time-dependent field of turbulent flow, the results of direct numerical simulation are considered as an experimental data gained computationally. Such results provide highly accurate instantaneous turbulence characteristics giving further insight into physics of turbulence, opening new possibilities, fresh ideas and improving applications.

The discussion goes along with 239 exercises and 136 comments. Whereas the former allowed the reader to improve his or her skills and experience, the latter are used to clear up specific terms and to note some instructive historical facts. The majority of exercises are used by the author to divide the derivation of particular expressions or formulae with a reader. To realize such an offer, the way of solution and the result are given in the text. However, the mathematical procedure is left for the reader as an exercise. Such a type of exercise gives a person a choice to be satisfied only by results, or use the suggested drill to improve their own expertise. For convenience, it is pointed out in the text when each exercise should be performed, and to find a specific one, the reader may apply the contents where the locations of exercises are indicated. Also, for the reader orientation, the more sophisticated exercises and examples (and hence, corresponding publications) are marked by an asterisk .

As mentioned above, comments provide significant information required for understanding. Such valuable subjects as, for example, special means, like tridiagonal matrix algorithm (TDMA), or alternating direction implicit method (ADI), or scientific terms, such as order of the value of magnitude, function singularities, tensor or factor of nonisothermicity, are explained via comments incorporated in the text. Meaningful historical notes are also introduced through comments at the relevant manual points. Thus, after discussing the benefits of the boundary layer theory, it is noticed that boundary-layer methods was not utilized for the first 25 years until Prandtl's lecture at the Royal Aeronautical Society meeting in 1927. The other examples of historical notes given by comments are explanations of the name BBO of differential equation, the Saffman slip boundary condition, the Paul Erlich role in monoclonal antibodies, and the Smagorinsky contribution in the direct numerical simulation of turbulence.

In view of the intended audience, special attention is given to the balance between strictness and comprehensibility of the writing. Such a compromise is realized using a strict formulation of the problems on one side and the detailed explanation of definitions, special terms and procedures on the other. For example, it is justified that both problems-heat transfer of flow past a body and peristaltic flow in a flexible channel-are similar, and both are inherently conjugate. At the same time, it is explained in detail why a nonlinear model of peristaltic flow differs in essence from a linear heat transfer pattern.

In contrast to exciting college courses on heat transfer presenting basically simple empirical approaches based on the heat transfer coefficients, the conjugate methods are grounded on contemporary fluid flow and heat transfer models. Therefore, to help the reader to understand the conjugate principles and procedures, the third part of this textbook offers fundamental laws of laminar and turbulent fluid flows and applied mathematic methods frequently used in engineering (Part III). Setting subsidiary chapters behind the body text, it is assumed that the reader takes a relatively small part of the information required to understand only a specific thesis or topic, rather than studying the whole subject in advance. In addition, the references given in the text in the form: Chap. (Chapter), S. (Section), Exam. (Example), Exer. (Exercise), and Com. (Comment) help the reader to find directly the desired portion of knowledge. Such a book structure permits the reader to get explanation step by step during studying. At the same time, an experienced person may read the text ignoring those citations.

As a whole, the textbook is written so as to be usable to senior and post-graduate students and engineers with the prerequisites of calculus, fluid mechanics, and heat transfer college engineering courses.

The textbook begins with Part I presenting applications in heat transfer, which starts with an introduction containing two pieces. The first writing, "When and why conjugate procedure is essential" explains in detail where the term conjugate came from, what it means, and in which cases conjugation procedure is important. The second piece entitled "A core of conjugation" presents the qualitative analysis of a simple problem of heat transfer from a plate heated from one end. This assay clarifies a physical meaning of the conjugation principle by showing the contrasting distributions of the heat transfer characteristics on the interface in two flow directions, from heated and unheated ends.

This part consists of four chapters, incorporating the theory of conjugate heat transfer based on universal functions (Chapter 1) and three chapters of applications: universal function applications (Chapter 2), conjugate problem applications in flows around bodies and inside channels (Chapter 3), and special application of conjugate heat transfer models in industrial and technological processes (Chapter 4).

The first chapter begins from the formulation of conjugate heat transfer problems specifying two sets of equations, the initial and boundary conditions governing the conjugate problem for a body and fluid. Each equation, such as Navier-Stokes or Laplace equation, is followed by references to chapter or section from the third part, presenting appropriate explanation. The initial conditions, the three kinds of common boundary conditions, and the Dirichlet and Neumann problem formulation for elliptic differential equations are considered in detail. The conjugate conditions on fluid/body interface (fourth kind boundary condition) and specific methods for performing the conjugate procedure are discussed also.

The next section introduces the universal functions that are widely used in this text. It explains what universal functions are, and shows that these...