Download Vectors, Tensors and the Basic Equations of Fluid Mechanics (Dover Books on Mathematics) ePub

Note: I am halfway through the book, about to go into the chapter on tensors, though I am already familiar with them, having already gone through Pavel Grinfeld's excellent "Introduction to Tensor Analysis and the Calculus of Moving Surfaces".

THE NEGATIVE

As mentioned in other reviews, this book would best be described as a thorough introduction to the mathematics of fluid mechanics, along with a concise refresher on the necessary foundations from vector calculus, linear algebra, and tensors ( some of which is found in the appendix ). Note: the basic equations of FM are initially derived in Cartesian coordinates in the first half of the book, and later reformulated using a coordinates-free approach in the second half of the book, following the chapter on tensors ( chapter 7 ). As such, the book either skims or skips over core concepts from basic physics, specifically from rigid-body dynamics and thermodynamics. For example, the "moment of linear momentum" ( that's the "angular momentum" caused by body forces and normal stresses ), is never properly introduced as a physical concept. Neither is force, nor body torque ( also referred to as the "moment of the external couple" ), nor the concept of energy and energy conservation from thermodynamics, etc. The book uses these various physical concepts and laws, however, in order to lay out basic equations as a starting point, from which the author then derives equations relevant to fluid mechanics. Also absent is a proper, gradual introduction to the various fluid types and what their properties mean from a physical standpoint ( ex: what is a non-elastic fluid? Is it the same as a incompressible fluid? what is viscosity? pressure?, ... ). In other words, besides basic physics, the author assumes that the reader is already familiar with some aspects of fluids and FM! Additionally, the book occasionally gets a bit ahead of itself, by mentioning terms and ideas BEFORE formally introducing them. For example: "Newtonian fluid", "constitutive equation", "equation of state", or the "functional relationship" between the stress tensor and the deformation tensor, leaving readers previously unfamiliar with FM temporarily puzzled. In a few instances, some relations or conditions are implied in the derivations which are not explicitly stated beforehand. For example: stress tensor T-sub(ji,j) = div(T) is replaced in some places by T-sub(ij,j), which is only valid if T is symmetric ( and therefore we are referring to a non-polar fluid: either Stokesian or Newtonian ), or, when deriving the rate of change of kinetic energy for a non-polar fluid ( section 6.14, p.117 ), the reason the velocity gradient tensor in the derivation is replaced at one point by the deformation tensor ( E-sub(ij) ) is not as it would seem because the antisymmetric part ( named Omega, rf. p.89 ) of the velocity gradient tensor is null ( which would imply that the vorticity is 0 and therefore that the fluid is irrotational ), but because the double dot product ( aka. dyadic product ) of two second order tensors, one symmetric ( T in this case ) and the other anti-symmetric ( Omega ) is always zero -- far from obvious.

The book is dense -- mostly in a good way, but occasionally not so good --, so expect to spend a few months on it, reading and re-reading. Two or three passages were too terse in my opinion ( ex: the explanation leading to the constitutive equation for Stokesian fluids -- note: check out "The Derivation of Stress-Deformation Relations for a Stokesian Fluid", an 11 pages paper by James Serrin for help ). I occasionally had to go to other sources to refresh on a few math and physics concepts. For example: eigenvectors from linear algebra ( although given in the Appendix, I preferred another reference ), the operator for the directional derivative of a vector filed -- v dot Del -- ( rf. Borisenko's "Vector and Tensor Analysis", p.164 ), angular momentum, torque, work, kinetic energy... Although my having a superficial knowledge of physics was not an impediment to following the derivations, on the math front however, I would say that previous experience with vector calculus and linear algebra -- even tensors -- is definitely recommended, and probably a must.

THE POSITIVE

Where the book excels, however, is in thoroughly exposing the reader to the mathematical underpinnings of fluid mechanics. For instance, the explanation of the Eulerian vs Lagrangian viewpoints, giving rise to the material derivative, is exemplary in its clarity. Another great thing the book does is to provide the equations for the general case ( meaning for viscous + compressible "Stokesian" fluids ) early on, and derive the various special cases from them. One exception is the elasticity property, which is treated later in the book, in chapter 8. This is different from many other FM books which start by giving equations for the simplest case ( typically for incompressible, non-viscous + non-polar fluids, i.e. "perfect" or "ideal" fluids ), and then add complexity in later chapters, thus revisiting previously given equations. As seductive as this approach might be, a significant drawback is that one needs to remember which specific conditions match which version of the equations ! I believe that a much better approach is to start by introducing the equations for the broadest case whenever possible, and then par them down as the properties of the fluid get specified ( e.g. Is the fluid viscous? What about density: divergence-free or not? Is the stress tensor symmetric ? etc... ). For example, some books initially give Cauchy's equation of motion with the gradient of pressure as the stress term ( a sub-case pertaining to incompressible + non-viscous flow ), whereas Aris uses the divergence of the stress tensor from the start -- which has the advantage of being always applicable --, with the stress tensor defined in terms of both pressure and the viscosity tensor ( aka. the viscosity "stress" tensor ) rather than pressure alone. Other examples: the equation of conservation of angular momentum is promptly given for polar fluids ( of which non-polar fluids is a special case, i.e. when the stress tensor is symmetric ). The constitutive equation, which governs fluid behavior ( i.e. stress tensor in terms of the deformation tensor ), is given for Stokesian fluids first, which is the broader "general case", and then pared down for Newtonian fluids ( a special case of the former, in which the eigenvalues of the viscosity tensor are related to those of the deformation tensor via a linear equation rather than via a quadratic ). To that end, tensor ( aka. "index", or "component" ) notation often helps to elucidate steps in the derivation of the equations as well as -- sometimes -- abbreviate results, which may otherwise seem either opaque or verbose in either matrix, differential or vector calculus notation -- in my opinion! Besides the notation, the author leverages the principal advantage of tensors, which is to be able to write equations that are valid in any coordinate system ( such as curvilinear coords ), in the second half of the book following the chapter on tensors.

The pay off... Once you are through, you may become aware how confusing, cluttered and/or incomplete, other texts on fluid mechanics sometimes are. If you are a graphics software engineer, looking to dive into the computational aspect of FM for simulation, I believe this book to be a good jumping off point, though any other type of engineer ( e.g. nautical, civil, aeronautical ) will likely require a more concrete and extensive physics-based introduction. The math presented here should be invaluable regardless, though.

ADDENDUM

The first half of the book, which is aimed at introducing the basic equations of fluid mechanics in Cartesian coordinates, appears to be a partially abridged retelling -- with a preference for tensor ( i.e. index, or component ) notation over matrix and vector calculus notation -- of James Serrin's seminal 138 page review paper, "Mathematical Principles of Classical Fluid Mechanics" ( a paper influenced heavily by the works of his teacher, Clifford Truesdell, as well as of R. S. Rivlin and J. L. Ericksen ). This paper was originally published in a 1959 book titled "Encyclopedia of Physics / Handbuch der Physik , Band VIII/1", edited by Truesdell ( p.125 to 263 ), and was republished in a 2012 Springer book titled "Encyclopedia of Physics - Fluid Dynamics I", edited by S. Flugge. I was fortunately able to find the full paper online... Overall, I found Serrin's exposition to be more gradual and in many ways more complete ( ex: he includes a discussion of gases and shock waves ), helping me clear some misunderstandings I had following Aris' exposition. While Aris does a better job at introducing the prerequisite math foundations, with an emphasis on tensors and tensor ( index ) notation -- as tensors are used for expressing the basic FM equations in a coordinates-free ( tensorial ) form, in the second half of the book --, Serrin's exposition seems more physics oriented, and does a much better job at introducing fluid types and their properties. I think the ideal would be to read both Aris' and Serrin's versions in parallel to gain a fuller understanding.

I have taken a liking to thinking of Rutherford Aris as the Yogi Berra of science. I’ve quoted him often, and in many ways, he was the great American philosopher of his field. However, do not let that fool you. Dr. Aris was an exceptional mathematician and physical scientist, and it shows in his treatment of fluid mechanics for this text.

Aris begins this book with the standard introduction to the algebra of vectors and tensors, and he advances this notion early on, with his explanations involving indicial notation and a Cartesian geometry. Many will find his notation to be a bit off the beaten path, but a disciplined reader will gain great insight in mathematical fluid mechanics, provided that they are willing to pay close attention to his definitions. After his algebraic treatment, Aris moves into the calculus of vectors and tensors, with a bit more thorough treatment than what is typically seen in a modern college course. It is only after laying this strong mathematical foundation, that Aris proceeds to the advanced descriptions involving fluid mechanics.

Dr. Aris gets to the real meat and potatoes, so to speak, in chapters 4 and 5, where he develops the kinematics and stress for fluids. In chapter 6 he derives the equations of motion and energy, while chapter 7 is dedicated solely to a more advanced discussion on tensors. Chapter 8 is where the reader will finally get their proper explanation pertaining to the Navier-Stokes Equations, and it is arguably the standout of this text. From there, Aris moves into the more geometrical concepts of curved surfaces, and in chapter 10 he describes the equations of surface flow. He then finishes with a chapter on reacting fluids, and of central importance to everyone is section 11.31, where Aris derives the Law of Conservation of Energy for fluids.

In addition to an excellent appendix, this book also includes exercises that can help one understand problem solving techniques for fluids, but admittedly, I found it more useful to keep my scratch-pad handy while reading the text, and to think long and hard about the descriptions. Perhaps, the only glaring omissions, are advanced theoretical descriptions pertaining to turbulence. Nonetheless, considering when it was written, I am willing to forgive such imperfections. I thoroughly enjoyed this book, and I would highly recommend it for anyone with an interest in vectors and tensors, as well as those who are seeking a deeper mathematical understanding of fluid mechanics.

Good Book.

This would make a good introduction to tensors for physics students (e.g. for General Relativity), though the approach is a completely classical, using index notation; you won't find anything on manifolds or differential forms here. An interesting feature is an extensive chapter on local surface theory (e.g. Gaussian curvature, but only after introducing the full Riemann tensor), which is good for building intuition about curvature in higher dimensions. While the applications are all in n <= 3 dimensions, the mathematics is done in a way that easily generalizes to higher dimensions.

Quick delivery and item as promised.

great price

I got this book for my research. This book is amongst the more readable introductions to tensors in complicated coordinate systems. Also has a very good chapter on free surface problems that is very well written and easy to understand.

This is one of those pithy classics that one needs to tread through very very carefully. The book is an uphill task, wherein rests its utility and fun. Fluid Mechanics in its simplified diluted version is served in many OTHER textbooks. This one serves it in its raw mathematical glory, and as the author says in the preface of the book, any serious engineer and fluid mechanician must realize that math must be befriended and mastered. The author makes very terse, and powerful introduction to the field. I would recommend a physicist to supplement it with a text by Faber, for Chemical Engineer to study it after the famous BSL (Transport Phenomenon), for a mathematician to pick it on any day, and for everyone in general to approach it with determination required to climb a rocky terrain. (Ah! I guess I would write an even better review if and when I finish reading it:)