The science of Hydrodynamics may be divided into two separate branches, viz. the motion of liquids and the motion of gases. The chief interest arising from the latter branch of the subject is due to the fact that air is the vehicle by means of which sound is transmitted, and consequently the discussion of special problems relating to the motion of gases belongs to the theory of sound rather than to hydrodynamics; it must also be recollected that in order to deal satisfactorily with many problems connected with the motion of gases, it is necessary to take into account changes of temperature and other matters which properly belong to the science of thermodynamics. In the earlier chapters of the present treatise the general theory of the motion of fluids is discussed, including those peculiarities of motion which are alike common to liquids and gases; but the subsequent chapters are limited almost entirely to the consideration of special problems relating to the motion of liquids.
In ancient times very little advance in hydrodynamics appears to have been made. In modern times the earliest pioneers were Torricelli and Bernoulli, whose investigations were due to the hydraulic requirements of Italian ornamental landscape gardening; but the first great step was taken by D'Alembert and Euler, who in the last century successfully applied dynamical principles to the subject, and thereby discovered the general equations of motion of a perfect fluid, and placed the subject on a satisfactory basis. The discovery of the general equations of motion was followed up by the investigations of the great French mathematicians Laplace, Lagrange and Poisson, the first of whom has left us a splendid memorial of his genius in his celebrated Theory of the Tides.
The next advance was made by Poisson and Green, the former of whom in 1831 discovered the velocity potential due to the motion of a sphere in an unlimited liquid, and the latter of whom in 1833, without a knowledge of Poisson's work, discovered the velocity potential due to the motion of translation of an ellipsoid in an unlimited liquid. Green's investigation was completed for the case of rotation by Clebsch in 1856.
The velocity potential due to the motion of a variety of cylindrical surfaces has also been discovered during the last fifteen years; but a similar advance has not been made as regards the motion of two or more solids. The kinetic energy of a liquid due to the motion of two cylinders whose cross sections are circular, has been obtained by Hicks and Greenhill. The former has also written several valuable papers on the motion of two spheres, which have placed this problem in a perfectly satisfactory condition. A complete discussion of the motion of two oblate or prolate spheroids whose excentricities are nearly equal to zero or unity, would be an attractive subject for investigation, and would throw light on the motion of two ships sailing alongside one another.
In 1845 Professor Stokes published his well-known theory of the motion of a viscous liquid, in which he endeavoured to account for the frictional action which exists in all known liquids, and which causes the motion to gradually subside by converting the kinetic energy into heat. This paper was followed up in 1850 by another, in which he solved various problems relating to the motion of spheres and cylinders in a viscous liquid. Previously to this paper no problem relating to the motion of a solid body in a liquid had ever been solved, in which the viscosity bad been taken into account.
Since the time of Lagrange the essential difference between the motion of a fluid when a velocity potential exists and when it does not exist had been recognised; and an opinion very generally prevailed that if at any particular instant some particular portion of the fluid were moving in such a manner that a velocity potential existed, the subsequent motion of this same portion of fluid would always be such that the component velocities of its elements would be derivable from a velocity potential. The first rigorous proof of this important proposition was given by Cauchy, and a different one was subsequently given by Stokes, but until the year 1858 no complete investigation respecting the peculiarities of rotational motion had ever been made. This was effected by Helmholtz in his celebrated memoir on Vortex Motion, which may perhaps be considered the most important step in hydrodynamics which has been made during the present century. The same subject was subsequently taken up by Sir W Thomson and the theory of polycyclic velocity potentials fully investigated. During the last six years important additional investigations on the theory of vortex rings have been made by Hicks and J J Thomson.
The last twenty years have witnessed a great advance in hydrodynamics, and numerous important papers have been written by many eminent mathematicians both British and foreign, which will be considered in detail in the present work.