Rutherford: Fluid Dynamics
D E Rutherford
D.Sc. (St And), Dr.Math. (Amsterdam)
Reader in Applied Mathematics
at St Andrews University
Edinburgh and London
This book is based upon courses of lectures given over a number of years to Honours students in the University of St Andrews. These lectures are constantly being modified in the light of experience and I have not hesitated to profit from the improvements introduced by my colleagues Dr A G Mackie and Dr A R Mitchell who have also delivered these courses from time to time. I am much indebted to both these gentlemen for many helpful suggestions made while reading the proofs of the book. I am also grateful to Mr R R Burnside and to Mr D G Weir for checking the exercises. A special word of thanks is due to the National Physical Laboratory for supplying the excellent photographs which have been used in the production of the plates. The publishers, Messrs. Oliver and Boyd Ltd., have shown the utmost co-operation in the production of this book which is the first to be printed by them on the new "4-line mathematical" machines recently introduced by the Monotype Corporation. While this system undoubtedly represents an advance in mathematical printing, its novelty must have presented the compositors with some unfamiliar difficulties and I should like to express my gratitude to all who were concerned in the printing of this volume.
D. E. R.
§ 1. Real fluids. A portion of a real fluid is composed of a very large number of molecules each of which has its own mass and velocity. At any instant the several molecules within a given closed surface have a great variety of velocities, since the velocities of the molecules vary both in magnitude and direction from molecule to molecule. If the closed surface has a small but finite volume V it is possible to consider the average mass per unit volume and the average vector velocity within the surface. These quantities might be regarded as the density p and the velocity q of the fluid at some point within V, though it must be remembered that their values depend upon the size of the small volume considered. In fact, if the volume be too small it may contain only one, or two, particles or even none at all, and the quantities then evaluated could hardly be regarded as the density and velocity of the fluid. On the other hand, if the volume chosen be too large p and q can only be regarded as average values and will not give a meaning to density or velocity at a point in the fluid.
The truth of the matter is that the concepts of density and velocity at a point in the fluid pertain only to the idealised notion of a continuous fluid and are not strictly applicable to a real fluid. The mathematical difficulties indicated above arise from the fact that a real fluid is a discrete assemblage of molecules and is not a continuous fluid.
JOC/EFR July 2008
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