J. Crank, The Mathematics of Diffusion, Oxford University Press, London and New York ().Google Scholar. 2. H. S. Carslaw and J. C. Jaeger, Conduction of. Crank J.-the Mathematics of Diffusion-Elsevier() - Ebook download as PDF File .pdf), Text File .txt) or read book online. Crank J.-the Mathematics of. Crank's Mathematics of diffusion is a comprehensive summary of solutions to several diffusion related problems. The insights offered are clear and logical.
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The book contains a collection of mathematical solutions of the differential equations of diffusion and methods of obtaining them. They are discussed against a. British Library Cataloguing in Publication Data. Crank, John. The mathematics of diffusion. nd ed. 1. Diffusion-Mathematical models. 1. Title. ' OC of the differential equation for diffusion. Within the last ten years. Dr. Crank and his The mathematical theory of diffusion is similar to that of heat conduction.
Here, indices i and j are related to the various components and not to the space coordinates. The Chapman—Enskog formulae for diffusion in gases include exactly the same terms. Earlier, such terms were introduced in the Maxwell—Stefan diffusion equation. Applications[ edit ] Equations based on Fick's law have been commonly used to model transport processes in foods, neurons , biopolymers , pharmaceuticals , porous soils , population dynamics , nuclear materials, plasma physics , and semiconductor doping processes. Theory of all voltammetric methods is based on solutions of Fick's equation.
Little mention is made of molecular mechanisms, and I have made only fleeting excursions into the realms of irreversible thermodynamics. These I hope are self-explanatory. A number of general accounts of the subject are already available, but very few mathematical solutions of the equations of non-equilibrium thermodynamics have been obtained for practical systems. During the last years the widespread occurrence of concentrationdependent diffusion has stimulated the development of new analytical and numerical solutions.
The time-lag method of measuring diffusion coefficients has also been intensively investigated and extended. Similarly, a lot of attention has been devoted to moving-boundary problems since the first edition was published. These and other matters have now been included by extensive revision of several chapters.
Also, the chapter dealing with the numerical solution of the diffusion equations has been completely rewritten and brought up to date. It seems unbelievable now that most of the calculations in the first edition were carried out on desk calculating machines.
Two entirely new chapters have been added. In one are assembled some of the mathematical models of non-Fickian or anomalous diffusion occurring mainly in solvent-polymer systems in the glassy state.
The other attempts a systematic review of diffusion in heterogeneous media, both laminates and particulates. A succession of improved solutions are described to the problem of diffusion in a medium in which are embedded discrete particles with different diffusion properties.
I have resisted the temptation to lengthen appreciably the earlier chapters. The enlarged edition of Carslaw and Jaeger's book Conduction of heat in solids contains a wealth of solutions of the heat-flow equations for constant heat parameters. Many of them are directly applicable to diffusion problems, though it seems that some non-mathematicians have difficulty in makitfg the necessary conversions.
For them I have included a brief 'translator's guide'. A few new solutions have been added, however, some of them in the context in which they arose, that is the measurement of diffusion coefficients. I am deeply grateful to my academic colleagues who shared my administrative responsibilities and particularly to Professor Peter Macdonald who so willingly and effectively assumed the role of Acting Head of the School of Mathematical Studies.
I am most grateful to Mrs.
Joyce Smith for all the help she gave me, not least by typing the manuscript and checking the proofs. Alan Moyse kept me well supplied with the seemingly innumerable books, journals, and photostat copies which I requested. I owe a great deal to friendly readers who have pointed out mistakes in the first edition and made helpful suggestions for the second. In particular I have benefited from discussions with my friend and former colleague, Dr.
Geoffrey Park. I had an invaluable introduction to the literature on which Chapter 12 is based from Mr. Woodcock, who came to me for help but, in fact, gave far more than he received.
Finally, I have appreciated the understanding help and guidance afforded me by members of staff of the Clarendon Press. Uxbridge October J.
Faraday Soc; Fig. ScL; Fig. It deals with the description of diffusion processes in terms of solutions of the differential equation for diffusion. Little mention is made of the alternative, but less well developed, description in terms of what is commonly called 'the random walk', nor are theories of the mechanism of diffusion in particular systems included.
The mathematical theory of diffusion is founded on that of heat conduction and correspondingly the early part of this book has developed from 'Conduction of heat in solids' by Carslaw and Jaeger.
Mark, and H. MacDonald, eds. Nicholson and I. Shain, Anal. Wagner, J. Nicholson, Anal.
Feldberg, in: Feldberg in: Prater, in: Feldberg and C. Averbach, Anal. Feldberg, J.
Hawley and S. Strojek, T. Kuwana, and S.
Blount and H. Herman, J. Booman and D. Pence, Anal. Sandifer and R. Buck, J. Joslin and D. Pletcher, J. Forsythe and W. Crank and P. Here, indices i and j are related to the various components and not to the space coordinates. The Chapman—Enskog formulae for diffusion in gases include exactly the same terms. Earlier, such terms were introduced in the Maxwell—Stefan diffusion equation.
Applications[ edit ] Equations based on Fick's law have been commonly used to model transport processes in foods, neurons , biopolymers , pharmaceuticals , porous soils , population dynamics , nuclear materials, plasma physics , and semiconductor doping processes.
Theory of all voltammetric methods is based on solutions of Fick's equation. Much experimental research in polymer science and food science has shown that a more general approach is required to describe transport of components in materials undergoing glass transition. In the vicinity of glass transition the flow behavior becomes "non-Fickian".