128-131. Determination of motion in terms of expansion and rotation Boundary-conditions. 182-185. Examples: Flux through a straight tube Effect of viscosity on motion of sound. Uniform motion of a sphere. A. Establishment of the fundamental equations on the molecular hypothesis 231 B. Multiply-connected regions. Definitions self-consistent. Barriers . C. Formula for component momenta of a solid immersed in a liquid ON FLUID MOTION. CHAPTER I. THE EQUATIONS OF MOTION. 1. THE following investigations proceed on the assumption that the fluids with which we deal may be treated as practically continuous and homogeneous in structure; i.e. we assume that the properties of the smallest portions into which we can conceive them to be divided are the same as those of the substance in bulk. It is shewn in note (A), at the end of the book, that the fundamental equations arrived at on this supposition, with proper modifications of the meanings of the symbols, still hold when we take account of the heterogeneous or molecular structure which is most probably possessed by all ordinary matter. 2. The fundamental property of a fluid is that it cannot be in equilibrium in a state of stress such that the mutual action between two adjacent parts is oblique to the common surface. This property is the basis of Hydrostatics, and is verified by the complete agreement of the deductions of that science with experiment. Very slight observation is enough, however, to convince us that oblique stresses may exist in fluids in motion. Let us suppose for instance that a vessel in the form of a circular cylinder, containing water (or other liquid), is made to rotate about its axis, which is vertical. If the motion of the vessel be uniform, the fluid is soon found to be rotating with the vessel as one solid body. If the vessel be now brought to rest, the motion of the fluid continues for some time, but gradually subsides, and at length ceases altogether; and it is found that during this process the portions of fluid which are further from the axis lag behind those which are nearer, and have their motion more rapidly checked. These phenomena point to the existence of mutual actions between contiguous elements which are partly tangential to the common surface. For if the mutual action were everywhere wholly normal, it is obvious that the moment of momentum, about the axis of the vessel, of any portion of fluid bounded by a surface of revolution about this axis, would be constant. We infer, moreover, that these tangential stresses are not called into play so long as the fluid moves as a solid body, but only whilst a change of shape of some portion of the mass is going on, and that their tendency is to oppose this change of shape. 3. It is usual, however, in the first instance, to neglect the tangential stresses altogether. Their effect is in many practical cases small, but, independently of this, it is convenient to divide the not inconsiderable difficulties of our subject by investigating first the effects of purely normal stress. The further consideration of the laws of tangential stress is accordingly deferred till Chapter IX. If the stress exerted across any small plane area situated at a point P of the fluid be wholly normal, its intensity (per unit area) is the same for all aspects of the plane. The following proof of this theorem is given here for purposes of reference. Through P draw three straight lines PA, PB, PC mutually at right angles, and let a plane whose direction-cosines relatively to these lines are l, m, n, passing infinitely close to P, meet them in A, B, C. * Let p, P1, P2, P, denote the intensities of the stresses across the faces ABC, PBC, PCA, PAB, respectively, of the tetrahedron PABC. If A be the area of the first-mentioned face, the areas of the others in order are l▲, mA, nA. Hence if we form the equation of motion of the tetrahedron parallel to PA we have P1. lA = pl. A, where we have omitted the terms which express the rate of change of momentum, and the component of the external impressed forces, because they are ultimately proportional to the mass of the tetrahedron, and therefore of the third order of small quantities, whilst the terms retained in the equation of motion are of the second. We have then, ultimately, p = p1, and similarly P=P.-P., which proves the theorem. 4. The equations of motion of a fluid have been obtained in two different forms, corresponding to the two ways in which the problem of determining the motion of a fluid mass, acted on by given forces and subject to given conditions, may be viewed.. We may either regard as the object of our investigations a knowledge of the velocity, the pressure, and the density, at all points of space occupied by the fluid, for all instants; or we may seek to determine the history of each individual particle. The equations obtained on these two plans are conveniently designated, as by German mathematicians, the Eulerian' and the 'Lagrangian' forms of the hydrokinetic equations, although both forms are in reality due to Euler †. The Eulerian Forms of the Equations. 5. Let u, v, w be the components, parallel to the co-ordinate axes, of the velocity at the point (x, y, z) at the time t. These quantities are then functions of the independent variables x, y, z, t. For any particular value of t they express the motion at that * Reckoned positive when pressures, negative when tensions. Ordinary fluids are, however, incapable of supporting more than an exceedingly slight degree of tension, so that p is nearly always positive. + Principes généraux du mouvement des fluides. Hist. de l'Acad. de Berlin, 1755. De principiis motus fluidorum. Novi Comm. Acad. Petrop. t. 14, p. 1, 1759. Lagrange starts in the Mécanique Analytique with the second form of the equations, but transforms them at once to the 'Eulerian' form. |