[PDF] Fluid and Thermodynamics Volume 2

Fluid and Thermodynamics Volume 2: Superior Fluid Mechanics and Thermodynamic Fundamentals by Kolumban Hutter and Yongqi Wang | PDF Free Download.

Fluid and Thermodynamics Contents

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• Creeping Movement Round Spheres at Relaxation in a Newtonian Fluid
• Three-Dimensional Creeping Stream—Systematic Derivation of the Shallow Stream Approximations
• Shallow Fast Granular Avalanches 
• Uniqueness and Stability 
• Turbulent Modeling 
• Turbulent Mixing Size Fashions and Their Purposes to Elementary Stream Configurations
• Thermodynamics—Subject Formulation 
• Gasoline Dynamics 
• Dimensional Evaluation, Similitude and Bodily Experiments

Preface to Fluid and Thermodynamics PDF

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Fluid and thermodynamics (FTD) are usually taught at technical universities as separate topics and this separation might be justified just by causes of the assigned time; the weather of every topic might be launched inside a semester of *15 weeks.

Almost certainly, these outer academic boundaries might even have nicely furthered this separation. Intellectually

The 2 topics, nevertheless, belong collectively, particularly since for all however ultimate fluids the second legislation of thermodynamics imposes constraint circumstances on the parameters of the governing equations (usually partial differential equations) which might be then used within the fluid dynamic a part of the joint effort to assemble options to bodily motivated preliminary boundary worth issues that train us essential info of the conduct of the movement of the fluid underneath sure circumstances.

One of many authors (Ok.H.) discovered this mixture of fluid and thermodynamics as an assigned one-semester course, when he began in 1987 within the Division of Mechanics at Technische Universität Darmstadt (at the moment ‘Technische Hochschule’) because the successor of the late Prof. Dr. rer.nat. ERNST BECKER (1929–1984).

With Ok.H’s emphasised curiosity in continuum mechanics and thermodynamics, this twin understanding of the mathematical description of the fluid matter was ultimate and the task to show it was a welcome problem, which was declared as a ‘credo’ to the working atmosphere in each instructing and analysis in his group.

The course notes of FTD taught to upper-class electrical engineers for 18 years have been shortly labored out into the ebook ‘Fluid und Thermodynamik Eine Einführung’ and revealed by Springer Verlag, Berlin, and so on., (ISBN 3-540-59235-0, second version).

All of the chapters of this ebook—some barely prolonged—have been translated (by Ok.H.) into the English language and are interwoven on this treatise with chapters, which, as an entire, ought to present a reasonably detailed understanding of FTD.

All topics of this treatise of FTD have been taught in a single or one other type as lectures in programs to college students at Technische Universität Darmstadt, Swiss Federal Institute of Expertise in Zürich (ETHZ), and in visitor lectures in superior programs at different universities and analysis establishments worldwide.

The viewers in these programs consisted of scholars, doctoral candidates and postdoctoral assistants of engineering (civil, mechanical, chemical, mechanics), pure sciences (meteorologists, oceanographers, geophysicists), arithmetic, and physics. Among the matters included are as follows: 

• Fluid mechanics
• Continuum mechanics and thermodynamics
• Mechanics of environmentally associated techniques (glacier, ice-sheet mechanics, bodily oceanography, lake physics, soil movement, avalanches, particles, and mudflows)
• Vorticity and angular momentum,
• Turbulence modeling (of zeroth, first and second-order)
• Common and singular perturbations
• Continuum mechanics and thermodynamics of mixtures
• Continuum mechanics and thermodynamics of COSSERAT continua and COSSERAT mixtures,
• Theoretical glaciology
• Shallow creeping flows of landslides, glaciers, and ice sheets

And others. It’s hoped that we have been profitable in designing a coherent image of the supposed textual content FTD. Writing the ebook chapters additionally profited from books that have been written earlier by us and co-authors [1–6].

Fluid and Thermodynamics Volume 1: Primary Fluid Mechanics

This quantity consists of 10 chapters and begins in an introductory Chap. 1 with some historic info, the definition of the topic subject and lists a very powerful properties of liquids.

This descriptive account is then adopted in Chap. 2 by the easy mathematical description of the basic hydrostatic equation and its use in analyses of equilibrium of fluid techniques and stability of floating our bodies, the derivation of the ARCHIMEDEAN precept and willpower of the strain distribution within the environment.

Chapter 3 offers with the hydrodynamics of ultimate incompressible (density preserving) fluids. Streamlines, trajectories, and streaklines are outlined. A cautious derivation of the balances of mass and linear momentum is given and it’s proven how the BERNOULLI equation is derived from the stability legislation of momentum and how it’s utilized in purposes.

In a single-dimensional easy circulation issues, the momentum and BERNOULLI equations are equal.

For discontinuous processes with jumps, this isn’t so. However, the BERNOULLI equation is a really helpful equation in lots of engineering purposes. This chapter ends with the stability legislation of second of momentum and its utility for EULER’s turbine equation.

The conservation legislation of angular momentum, offered in Chap. 4, gives the event to outline circulation and vorticity and the vorticity theorems, amongst them these of HELMHOLTZ and ERTEL. The objective of this chapter is to construct a elementary understanding of vorticity.

In Chap. 5 a set of straightforward circulation issues in ultimate fluids is offered. It’s proven how vector analytical strategies are used to show the differential geometric properties of vortex-free circulation fields and to judge the motion-induced drive on a physique in a possible subject.

The idea of digital mass is outlined and two-dimensional fluid potential circulation is printed. This almanac of flows of ultimate fluids is complemented in Chap. 6 by the presentation of the answer methods of two-dimensional potential circulation by complex-valued operate theoretical strategies utilizing conformal mappings.

Potential flows round two-dimensional air foils, laminar free jets, and the SCHWARZ– CHRYSTOFFEL transformations are employed to assemble the mathematical descriptions of such flows by a slit or a number of slits, round air wings, free jets, and in ducts bounding a super fluid.

The mathematical bodily research of viscous flows begins in Chap. 7 with the derivation of the overall stress–pressure price relation of viscous fluids, specifically NAVIER–STOKES fluids and extra usually, non-NEWTONian fluids.

Software of those equations to viscometric flows, liquid movies, POISEUILLE circulation, and the slide bearing idea resulting from REYNOLDS and SOMMERFELD show their use in an engineering context. Creeping circulation for a pseudo-plastic fluid with a free floor then reveals the appliance within the glaciological-geological context.

Chapter 8 continues with the research of the two-dimensional and three-dimensional easy circulation of the NAVIER–STOKES equations. HAGEN–POISEUILLE circulation and the EKMAN idea of the wall-near wall-parallel circulation on a rotating body (Earth) and its generalization are offered as options of the NAVIER–STOKES equations within the half-space above an oscillating wall and that of a stationary axisymmetric laminar jet.

This then results in the presentation of PRANDTL’s boundary layer idea with flows round wedges and the BLASIUS boundary layer and others.

In Chap. 9 two- and three-dimensional boundary layer flows within the neighborhood of a stagnation level are studied as circulation round wedges and alongside wedge sidewalls. The circulation, induced within the half aircraft above a rotating aircraft, can be decided.

The strategy of the boundary layer method is commenced with the BLASIUS circulation, however extra importantly, the boundary layer resolution method for the NAVIER–STOKES equations is defined by way of the tactic of matched asymptotic expansions.

Furthermore, the worldwide legal guidelines of the regular boundary layer idea are defined with the help of the HOLSTEIN–BOHLEN process.

The chapter ends with a short research of non-stationary boundary layers, during which an impulsive begin from relaxation, circulation within the neighborhood of a pulsating physique, oscillation induced drift present, and non-stationary plate boundary layers are studied.

In Chap. 10 pipe circulation is studied for laminar (HAGEN–POISEUILLE) in addition to for turbulent flows; this example culminates through a dimensional evaluation to the well-known MOODY diagram. The amount ends on this chapter with the aircraft boundary layer circulation alongside a wall resulting from PRANDTL and VON KÁRMÁN with the well-known logarithmic velocity profile.

This final downside is later reanalyzed because the controversies between energy and logarithmic velocity profile close to partitions are nonetheless ongoing analysis at this time.

Fluid and Thermodynamics Volume 2: Superior Fluid Mechanics and Thermodynamic Fundamentals

This quantity consists of 10 chapters and commences in Chap. 11 with the willpower of the creeping movement round spheres at relaxation in a NEWTONian fluid. It is a classical downside of singular perturbations within the type of matched asymptotic expansions.

For creeping circulation, the acceleration phrases in NEWTON’s legislation might be ignored to roughly calculate circulation across the sphere by this so-called STOKES approximation.

It seems that distant from the sphere the acceleration phrases grow to be bigger than these within the STOKES resolution in order that the latter resolution violates the boundary circumstances at infinity. This lowest order correction of the circulation across the sphere is because of OSEEN (1910).

In a scientific perturbation enlargement, the outer —OSEEN—collection and the interior—STOKES—collection with the small REYNOLDS quantity as perturbation parameter should be matched collectively to find out all boundary and transition circumstances of interior and outer expansions.

This process is relatively tough, i.e., not simple to know for newbies. This idea, initially resulting from KAPLUN and to LAGERSTRÖM has been prolonged, and the drag coefficient for the sphere, which additionally might be measured is expressible by way of a collection enlargement of powers of the REYNOLDS quantity.

Nonetheless, for REYNOLDS numbers bigger than unity, convergence to measured values is poor. About 20–30 years in the past a brand new mathematical method was designed the so-called Homotopy Evaluation Methodology

It’s primarily based on a completely completely different enlargement method, and outcomes for the drag coefficient lie a lot nearer to the experimental values than values obtained with the ‘classical’ matched asymptotic enlargement, as proven in Fig. 11.11. By the way, the laminar circulation of a viscous fluid round a cylinder can analogously be handled however will not be contained on this treatise.

Chapter 12 is dedicated to the approximate willpower of the rate subject in a shallow layer of ice or granular soil, handled as a non-Newtonian materials flowing underneath the motion of its personal weight and assuming its velocity to be so small that STOKES circulation might be assumed.

Two limiting instances might be analyzed: (i) Within the first, the flowing materials on a steep slope (which is the case for creeping landslides or snow on mountain topographies with inclination angles which might be massive).

(ii) Within the second case, the inclination angles are small. Scenario (ii) is apt to ice circulation in massive ice sheets comparable to Greenland and Antarctica, essential in local weather eventualities in a warming environment.

We derive perturbation schemes by way of a shallowness parameter within the two conditions and talk about purposes underneath real-world circumstances. In shallow speedy gravity pushed free floor flows the acceleration phrases in NEWTON’s legislation are not negligible.

Chapter 13 is dedicated to such granular flows in an try to introduce the reader to the difficult idea of the dynamical conduct of fluidized cohesionless granular supplies in avalanches of snow, particles, and mud, and so on.

The theoretical description of shifting layers of granular assemblies begins with the one-dimensional depth-integrated MOHR–COULOMB plastic layer flows down inclines the so-called SAVAGE–HUTTER idea

However then continues with the overall formulation of the mannequin equations referred to topography following curvilinear coordinates with all its peculiarities within the idea and using shock-capturing numerical integration methods.

Chapter 14 on uniqueness and stability gives the primary taste into the topic of laminar-turbulent transition. Two completely different theoretical ideas are in use and each assume that the laminar–turbulent transition is a query of lack of stability of the laminar movement.

With using the vitality technique one tries to seek out higher sure circumstances for the laminar circulation to be steady.

Extra profitable for pinpointing the laminar-turbulent transition has been the tactic of linear instability evaluation, during which a lowest sure is looked for, at which the onset of deviations from the laminar circulation is happening.

In Chap. 15, an in depth introduction to the modeling of turbulence is given. Filter operations are launched to separate the bodily stability legal guidelines into evolution equations for the averaged fields on the one hand, and into fluctuating or pulsating fields alternatively.

This process generates averages of merchandise of fluctuating portions, for which closure relations should be formulated.

Relying upon the complexity of those closure relations, so-called zeroth, first, and increased order turbulence fashions are obtained: easy algebraic gradient-type relations for the flux phrases, one or two equation fashions, e.g., k-ε; k-ω, during which evolution equations for the averaged correlation merchandise are formulated, and so on.

That is finished for density preserving fluids in addition to so-called BOUSSINESQ fluids and convection fluids on a rotating body (Earth), that are essential fashions to explain atmospheric and oceanic flows.

Chapter 16 goes again one step by scrutinizing the early zeroth-order closure relations as proposed by PRANDTL, VON KÁRMÁN, and collaborators.

The premise is BUSSINESQ’S (1872) ansatz for the shear stress in plane-parallel circulation, τ12, which is expressed to be proportional to the corresponding averaged shear price o v1=ox2 with a coefficient of proportionality ρε, the place ρ is the density and ε a kinematic turbulent viscosity or turbulent diffusivity [m2 s −1 ].

In turbulence idea the flux phrases of momentum, warmth, and suspended mass are all parameterized as gradient-type relations with turbulent diffusivities handled as constants.

PRANDTL realized from information collected in his institute that ε was not a relentless however relied on his mixing size squared and the magnitude of the shear price (PRANDTL 1925).

This proposal was later improved (PRANDTL 1942) to amend the unsatisfactory settlement at positions the place shear charges disappeared.

The 1942-law continues to be native, which implies that the REYNOLDS stress tensor at a spatial level will depend on spatial velocity derivatives in the identical place. PRANDTL within the second proposal of his 1942-paper steered that the turbulent diffusivity ought to depend upon the rate distinction on the factors the place the rate of the turbulent path assumes most and minimal values.

This proposal introduces some non-locality, yielded higher settlement with information, however PRANDTL left the gradient-type dependence so as to keep in conformity with BOUSSINESQ.

It does neither grow to be obvious nor clear that PRANDTL or the modelers at the moment would have realized that non-local results could be the trigger for the higher settlement of the theoretical formulations with information.

The proposal of full non-local conduct of the REYNOLDS stress parameterization got here in 1991 by P. EGOLF and subsequent analysis articles throughout *20 years, during which additionally the native pressure price (= native velocity gradient) is changed by a distinction quotient.

We encourage and clarify the proposed Distinction Quotient Turbulence Mannequin (DQTM) and show that for normal two-dimensional configurations analyzed on this chapter its efficiency is superior to different zeroth order fashions.

The subsequent two chapters are dedicated to thermodynamics; first, fundamentals are attacked and, second a subject formulation is offered and explored.

Class expertise has taught us that thermodynamic fundamentals (Chap. 17) are obscure for novel readers.

Utmost warning is subsequently exercised to exactly introduce terminology comparable to ‘states’, ‘processes’, ‘in depth’, ‘intensive’, and ‘molar state variables’ in addition to ideas like ‘adiabatic’, and ‘diathermal partitions’, ‘empirical’ and ‘absolute temperature’, ‘equations of state’, and ‘reversible’ and ‘irreversible processes’.

The core of this chapter is, nevertheless, the presentation of the First and Second Regulation of Thermodynamics.

The primary legislation balances the energies. It states that the time price of change of the kinetic plus inner energies is balanced by the mechanical energy of the stresses and the physique forces plus the thermal analogies, that are the flux of warmth by the boundary plus the precise radiation additionally known as vitality provide.

This conservation legislation then results in the definitions of the caloric equations of state and the definitions of particular heats.

The Second Regulation of Thermodynamics is probably going probably the most obscure and it’s launched right here as a stability legislation for the entropy and states that each one bodily processes are irreversible.

We encourage this legislation by going from simple and easy techniques to extra complicated techniques by a generalization and culminate on this tour with the Second Regulation because the assertion that the entropy manufacturing price can’t be unfavorable.

Examples illustrate the implications in easy bodily techniques and present the place the 2 variants of entropy ideas might result in completely different solutions. Chapter 18 extends and applies the above ideas to steady materials techniques.

The Second Regulation is written in a world type as a stability legislation of entropy with flux, provide and manufacturing portions, which might be written in a neighborhood type as a differential assertion.

The actual type of the Second Regulation then relies upon upon which postulates the person phrases within the entropy stability are subjected to.

When the entropy flux equals warmth flux divided by absolute temperature and the entropy manufacturing price density is requested to be non-negative, the entropy stability legislation seems because the CLAUSIUS–DUHEM inequality and its exploitation follows the axiomatic process of open techniques thermodynamics as launched by COLEMAN and NOLL.

When the entropy flux is left arbitrary however is of the identical operate class as the opposite constitutive relations and the entropy provide price density is identically zero, then the entropy inequality seems within the type of MÜLLER.

In each instances, the Second Regulation is expressed by the requirement that the entropy manufacturing price density should be non-negative, however particulars of the exploitation of the Second Regulation within the two instances are subtly completely different from each other.

For traditional media comparable to elastic and/or viscous fluids, the outcomes are the identical. Nonetheless, for complicated media, they might nicely differ from each other. Examples will illustrate the procedures and outcomes.

Chapter 19 on gasoline dynamics illustrates a technically essential instance of a fluid subject idea, the place the data deduced by the Second Regulation of Thermodynamics delivers essential properties, expressed by the thermal and caloric equations of state of, say, ultimate and actual gases.

We briefly contact issues of acoustics, regular isentropic circulation processes, and their stream filament idea.

The outline of the propagation of small perturbations in a gasoline serves in its one-dimensional type ideally as a mannequin for the propagation of sound, for e.g. in a flute or organ pipe, and it may be used to clarify the DOPPLER shift occurring when the sound supply is shifting relative to the receiver.

Furthermore, with the stream filament idea the sub- and supersonic circulation by a nozzle might be defined. In a remaining part, the three-dimensional idea of shocks is derived because the set of soar circumstances on surfaces for the stability legal guidelines of mass, momentum, vitality, and entropy.

Their exploitation is illustrated for regular surfaces for easy fluids underneath adiabatic circulation circumstances.

These issues are classics; gasoline dynamics, certainly kinds an essential superior technical subject that was developed within the twentieth century as a topic of aerodynamics and astronautics and essential specialties of mechanical engineering.

Chapter 20 is dedicated to the topics ‘Dimensional evaluation, similitude and bodily experimentation at laboratory scale’, matters usually not systematically taught at increased technical training. Nonetheless, no insider would deny their usefulness.

Books treating these topics individually and in enough element have appeared for the reason that mid-twentieth century.

We give an account of dimensional evaluation, outline dimensional homogeneity of features of mathematical physics, the properties of which culminate in BUCKINGHAM’s theorem (which is proved in an appendix to the chapter)

Its use is illustrated by a variety of issues from normal fluid dynamics, gasoline dynamics, and thermal sciences, e.g., propagation of a shock from some extent supply, rising gasoline bubbles, RAYLEIGH–BÉNARD instability, and so on.

The idea of bodily fashions develops guidelines, how you can down- or up-scale bodily processes from the dimensions of a prototype to the dimensions of the mannequin.

The idea reveals that usually, such scaling transformations are virtually by no means precisely potential in order that scale results enter in these instances, which distort the mannequin outcomes compared to these within the prototype.

In hydraulic purposes, this results in the so-called FROUDE and REYNOLDS fashions, during which both the FROUDE or REYNOLDS quantity, respectively, stays a mapping invariant however not the opposite.

Software on sediment transport in rivers, warmth switch in pressured convection, and so on., illustrate the difficulties.

The chapter ends with the characterization of dimensional homogeneity of the equations describing bodily processes by their governing differential equations. The NAVIER–STOKES–FOURIER–FICK fluid equations function an illustration.

The intention of this treatise is, aside from presenting its addressed topics, a transparent, detailed, and considerably rigorous mathematical presentation of FTD on the premise of restricted information as a prerequisite.

Calculus or evaluation of features of a single or a number of variables, linear algebra and the fundamentals of unusual and partial differential equations are assumed to be recognized, as is the idea of complicated features.

The latter will not be universally taught in engineering curricula of universities; we consider that readers not geared up with the idea of complicated features can simply familiarize themselves with its fundamentals in a couple of weeks’ studying efforts.

The second objective of this treatise is to border the person topics of their historic context by offering biographical sketches of the inventors of the actual ideas.

The science of fluid and thermodynamics started within the Western world greater than 2000 years in the past, e.g., by ARCHIMEDES in Syracuse.

First cautious observations on turbulence have been described by LEONARDO DA VINCI and on the movement of falling our bodies by GALILEO GALILEI. Mathematical description of the movement of bodily our bodies was begun by ISAAC NEWTON and DESCARTES.

EULER and father JOHANN and son DANIEL BERNOULLI launched, amongst others, the continual strategies for ultimate, i.e., reversible supplies.

Most of this analysis occurred within the seventeenth and eighteenth centuries and was perfected within the upcoming nineteenth and twentieth centuries.

The popularity of the vitality stability equation and the entropy imbalance assertion as bodily legal guidelines are achievements of the nineteenth and first a part of the 20th centuries and are related to scientists like SADI CARNOT, JULIUS MAYER, HERMANN HELMHOLTZ, RUDOLF CLAUSIUS, PIERRE MAURICE MARIE DUHEM, WILLIAM THOMSON (LORD KELVIN), WILLIARD GIBBS, and MAX PLANK, to call a couple of.

The options of the (preliminary) boundary worth issues which ensue from the rising equations have been solved by numerous follow-up scientists from the mid-nineteenth century to current, of whom a couple of stand out distinguishingly:

OSBORNE REYNOLDS, LORD RAYLEIGH, LUDWIG PRANDTL, THEODORE VON KÁRMÁN, G.I. TAYLOR, HERMANN SCHLICHTING, and many others.

The historical past, which advanced from the work of all these scientists, is fascinating. By itemizing quick biographical sketches of these scientists who contributed to the event of fluid and thermodynamics, we hope to information the reader to a coherent historic growth of the fascinating topic of fluid and thermodynamics.

We regard this twin method as a justified process, particularly for the reason that twenty-first-century college college students do not sufficiently respect the actual fact, on which shoulders of giants and predecessors we stand.

The books have been collectively drafted by us from notes that collected throughout years. As talked about earlier than, the Chaps. 1–3, 5, 7, 10, 17–20 are translated (and partly revised) from ‘Fluid- und Thermodynamik – eine Einführung’.

Most of the different chapters have been composed in handwriting and TEXed by Ok.H. and considerably improved and polished by Y.W. We share equal accountability for the content material and the errors that also stay.

Figures, that are taken from others, are reproduced and largely redrawn, however talked about within the acknowledgment and/or determine captions. However, a considerable variety of figures have been designed by us.

Nonetheless, we acquired assist for his or her digital manufacturing: Mr. ANDREAS ROHRER, from the Laboratory of Hydraulics, Hydrology, and Glaciology at ETH Zurich (VAW), drew figures for Chaps.

8 and 9 and the scholar assistants Mr. WALDEMAR SURNIN and Mr. JAN BATTRAM from the Institute of Fluid Dynamics at Technische Universität Darmstadt aided within the manufacturing of figures of a number of different chapters.

Mr. ANDREAS SCHLUMPF from VAW and Ms. ALEXANDRA PAUNICA and Prof. IOANA LUCA drew figures for Chap. 6 and a number of different chapters. It’s customized of most publishers to ask referees to overview ebook manuscripts shortly earlier than submission for printing by specialists of the topics handled within the forthcoming ebook.

It’s, nevertheless, additionally nearly consequential that reviewers for a two-volume treatise of greater than 1200 pages can hardly be discovered, merely due to the extreme labor that goes with such an task.

However, this burden was taken up by two emeriti, Dr.-Ing. PETER HAUPT, Professor of Mechanics on the College of Kassel, Germany, and Dr. rer. nat, Dr. h.c. HANS DIETER ALBER, Professor of Arithmetic, Technical College, Darmstadt, Germany.

We totally thank these colleagues for his or her in depth assist. Their criticisms and suggestions are gratefully integrated within the remaining model of the manuscript.

Ok.H. needs to specific his honest because of ETH Zurich and specifically to Prof. Dr. R. BOES for the allowance to share a desk as an emeritus professor from Darmstadt on the Laboratory of Hydraulics, Hydrology and Glaciology at ETH Zurich and he equally thanks, Profs.

Dr. MARTIN FUNK and Dr. WILLI H. HAGER, members of this laboratory, for his or her help. Y.W. want to specific his because of Prof.

Dr. MARTIN OBERLACK for the free and constructive collaboration in his fluid dynamic working unit at Technische Universität Darmstadt. This treatise was deliberate as a three-volume venture, and, certainly, two chapters of a potential quantity III have already been written.

We nonetheless maintain up this intention, however the superior age of one in all us doesn’t assure that we are going to achieve success on this endeavor. We will see …

Lastly, we thank Springer Verlag, and specifically Dr. Annett Buettner, for the curiosity in our FTD treatise and AGEM2, usually.

Fluid and Thermodynamics: Volume 2: Advanced Fluid Mechanics and Thermodynamic Fundamentals PDF

Author(s): Kolumban Hutter, Yongqi Wang (auth.)

Series: Advances in Geophysical and Environmental Mechanics and Mathematics

Publisher: Springer International Publishing, Year: 2016

ISBN: 978-3-319-33635-0

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