same element forms an oblique parallelepiped. The corner corresponding to (a, b, c) has for its co-ordinates x, y, z; and the co-ordinates relative to this point of the other extremities of the dx three edges meeting in it are respectively da, dy da, dz da da • da da; dc. The volume of the par dc Hence, since the mass of the element is unchanged, we have In the case of an incompressible fluid p = P。, so that (23) becomes d(x, y, z) Weber's Transformation. 18. If the forces X, Y, Z have a potential, i.e. if they can be expressed as the partial differential coefficients with respect to x, y, z of a single function which we denote by V (so that Vis the potential energy, due to those forces, of unit mass placed in the position (x, y, z)), the equations (22) may be written * Salmon, Geometry of Three Dimensions. Let us integrate these equations with respect to t between the limits 0 and t. We remark that where u is the initial value of the x-component of velocity of the Ио particle (a, b, c). we have Hence if we write dx dx dy dy, dz dz + + dt da dt da dt da and the equation of continuity, are the partial differential equations to be satisfied by the five unknown quantities x, y, z, P, X; p being supposed already eliminated by means of one of the relations of Art. 7. In the case of a liquid, p occurs in (27) only, so that (26) and (24) may be employed to find x, y, z, and x, while p may be found afterwards from (27). The initial conditions to be satisfied are x = a, y=b, z=c, x=0. The boundary conditions vary with the particular problem under investigation. 19. The equations (26) and (27) may be applied to find the equations of impulsive motion of a liquid. Let the impulse act from t=0 to t=7, where T is infinitely small, and let т be the * H. Weber, Crelle, t. 68. upper limit of integration in (25). We find x=- V'-' is the impulsive pressure and V' the potential of the external impulsive forces at the point (a, b, c). Since xa, y = b, z = c, &c., &c., 20. In the method of Art. 16 the quantities a, b, c need not be restricted to mean the initial co-ordinates of a particle; they may be considered to be any three quantities which serve to identify a particle, and which vary continuously from one particle to another. If we thus generalize the meanings of a, b, c, the form of equations (22) is not altered; to find the form which (23) assumes, let 。, y, z now denote the initial co-ordinates of the particle to which a, b, c refer. The initial volume of the parallelepiped, three of whose edges are drawn from the particle (a, b, c) to the particles (a + da, b, c), (a, b + db, c), (a, b, c + dc), respectively, is 21. If we compare the two forms of the fundamental equations to which we have been led, we notice that the Eulerian equations of motion are linear and of the first order, whilst the Lagrangian equations are of the second order, and also contain products of differential coefficients. In Weber's transformation the latter are replaced by a system of equations of the first order, and of the second degree. The Eulerian equation of continuity is also much simpler than the Lagrangian, especially in the case of liquids. In these respects, therefore, the Eulerian forms of the equations possess great advantages over the Lagrangian. Again, the form in which the solution of the Eulerian equations appears corresponds, in many cases, more nearly to what we wish to know as to the motion of a fluid, our object being, in general, to gain a knowledge of the state of motion of the fluid mass at any instant, rather than to trace the career of individual particles. * On the other hand, whenever the fluid is bounded by a moving surface, the Lagrangian method possesses certain theoretical advantages. In the Eulerian method the functions u, v, w . have no existence beyond this surface, and hence the range of values of x, y, z for which these functions exist varies in consequence of the motion which we have to investigate. In the other method, on the contrary, the range of values of the independent variables a, b, c is given once for all by the initial conditions. The difficulty, however, of integrating the Lagrangian equations has hitherto prevented their application except in certain very special cases. Accordingly in this treatise we deal almost exclusively with the Eulerian equations. The integration and simplification of these in certain cases form the subject of the following chapter. * H. Weber, Crelle, t. 68. CHAPTER II. INTEGRATION OF THE EQUATIONS IN SPECIAL CASES. 22. IN most cases of interest the external impressed forces have a potential; viz. we have In a large and important class of cases the component velocities u, v, w can be similarly expressed as the partial differential coefficients of a function 4, so that Such a function is called a 'velocity-potential,' from its analogy to the potential function which occurs in the theories of Attractions, Electrostatics, &c. The general theory of the velocitypotential is reserved for the next chapter; but we give at once a proof of the following important theorem : 23. If a velocity-potential exist, at any one instant, for any finite portion of a perfect fluid in motion under the action of forces which have a potential, then, provided the density of the fluid be either constant or a function of the pressure only, a velocity-potential exists for the same portion of the fluid at all subsequent instants. In the equations of Art. 18, let the instant at which the velocity-potential, exists be taken as the origin of time; we have then uda+v.db+w,dc=do, throughout the portion of the mass in question. Multiplying the |
