The subsidence of the motion in a cup of tea which has been stirred may be mentioned as a familiar instance of friction, or, which is the same, of a deviation from the law of normal pressure; and the absolute regularity of the surface when it comes to rest, whatever may have been the nature of the previous disturbance, may be considered as a proof that all tangential force vanishes when the motion ceases. It does not fall in with the object of this Note to enter into the theory of the friction of fluids in motion*, and accordingly the hypothesis of normal pressure will be adopted. The usual notation will be employed, as in the preceding Notes. Consider the elementary parallelepiped of fluid comprised between planes parallel to the coordinate planes and passing through the points whose coordinates are x, y, z, and x + dx, y+dy, z+dz. Let X, Y, Z be the accelerating forces acting on the fluid at the point (x, y, z); then, p and X being ultimately constant throughout the element, the moving force parallel to x arising from the accelerating forces which act on the element will be ultimately pX dx dy dz. The difference between the pressures, referred to a unit of surface, at opposite points of the faces dy dz is ultimately dp/dx.dx, acting in the direction of a negative, and therefore the difference of the total pressures on these faces is ultimately dp/dx. dx dy dz; and the pressures on the other faces act in a direction perpendicular to the axis of x. The effective moving force parallel to a is ultimately p. D2x/Dt. dx dy dz, where, in order to prevent confusion, D is used to denote differentiation when the independent variables are supposed to be t, and three parameters which distinguish one particle of the fluid from another, as for instance the initial coordinates of the particle, while d is reserved to denote differentiation when the independent variables are x, y, z, t. We have therefore, ultimately, D2x = (px_dp (pX - dp) dx dy dz, dx * The reader who feels an interest in the subject may consult a memoir by Navier, Mémoires de l'Académie, tom. vi. p. 389; another by Poisson, Journal de l'École Polytechnique, Cahier xx. p. 139; an abstract of a memoir by M. de SaintVenant, Comptes Rendus, tom. xvII. (Nov. 1843) p. 1240; and a paper in the Cambridge Philosophical Transactions, Vol. VIII. p. 287. [Ante, Vol. 1. p. 75.] with similar equations for y and z. Dividing by p da dy dz, transposing, and taking the limit, we get These are the dynamical equations which must be satisfied at every point in the interior of the fluid mass; but they are not at present in a convenient shape, inasmuch as they contain differential coefficients taken on two different suppositions. It will be convenient to express them in terms of differential coefficients. taken on the second supposition, that is, that x, y, z, t are the independent variables. Now Da/Dt=u, and on the second supposition u is a function of t, x, y, z, each of which is a function of t on the first supposition. We have, therefore, by Differential Calculus, The equations (1) or (2), which are physically considered the same, determine completely, so far as Dynamics alone are concerned, the motion of each particle of the fluid. Hence any other purely dynamical equation which we might set down would be identically satisfied by (1) or (2). Thus, if we were to consider the fluid which at the time t is contained within a closed surface S, and set down the last three equations of equilibrium of a rigid body between the pressures exerted on S, the moving forces due to the accelerating forces acting on the contained fluid, and the effective moving forces reversed, we should not thereby obtain any new equation. The surface S may be either finite or infinitesimal, as, for example, the surface of the elementary parallelepiped with which we started. Thus we should fall into error if we were to set down these three equations for the parallelepiped, and think that we had thereby obtained three new independent equations. If the fluid considered be homogeneous and incompressible, p is a constant. If it be heterogeneous and incompressible, p is a function of x, y, z, t, and we have the additional equation Dp/Dt=0, or dp dt dp + u + v +w = (3), which expresses the fact of the incompressibility. If the fluid be elastic and homogeneous, and at the same temperature @ throughout, and if moreover the change of temperature due to condensation and rarefaction be neglected, we shall have where k is a given constant, depending on the nature of the gas, and a a known constant which is the same for all gases [nearly]. The numerical value of a, as determined by experiment, is 00366, being supposed to refer to the centigrade thermometer. If the condensations and rarefactions of the fluid be rapid, we may without inconsistency take account of the increase of temperature produced by compression, while we neglect the communication of heat from one part of the mass to another. The only important problem coming under this class is that of sound. If we suppose the changes in pressure and density small, and neglect the squares of small quantities, we have, putting p, p, for the values of p, p in equilibrium, K being a constant which, as is well known, expresses the ratio of the specific heat of the gas considered under a constant pressure to its specific heat when the volume is constant. We are not, however, obliged to consider specific heat at all; but we may if we please regard K merely as the value of d log p/d log p for p = P12 p being that function of p which it is in the case of a mass of air suddenly compressed or dilated. In whichever point of view we regard K, the observation of the velocity of sound forms the best mode of determining its numerical value. It will be observed that in the proof given of equations (1) it has been supposed that the pressure exerted by the fluid outside the parallelepiped was exerted wholly on the fluid forming the parallelepiped, and not partly on this portion of fluid and partly on the fluid at the other side of the parallelepiped. Now, the pressure arising directly from molecular forces, this imposes a restriction on the diminution of the parallelepiped, namely that its edges shall not become less than the radius of the sphere of activity of the molecular forces. Consequently we cannot, mathematically speaking, suppose the parallelepiped to be indefinitely diminished. It is known, however, that the molecular forces are insensible at sensible distances, so that we may suppose the parallelepiped to become so small that the values of the forces, &c., for any point of it, do not sensibly differ from their values for one of the corners, and that all summations with respect to such elements may be replaced without sensible error by integrations; so that the values of the several unknown quantities obtained from our equations by differentiation, integration, &c. are sensibly correct, so far as this cause of error is concerned; and that is all that we can ever attain to in the mathematical expression of physical laws. The same remarks apply as to the bearing on our reasoning of the supposition of the existence of ultimate molecules, a question into which we are not in the least called upon to enter. There remains yet to be considered what may be called the dynamical equation of the bounding surface. Consider, first, the case of a fluid in contact with the surface of a solid, which may be either at rest or in motion. Let P be a point in the surface, about which the curvature is not infinitely great, o an element of the surface about P, PN a normal at P, directed into the fluid, and let PN=h. Through N draw a plane A perpendicular to PN, and project w on this plane by a circumscribing cylindrical surface. Suppose h greater than the radius r of the sphere of activity of the molecular forces, and likewise large enough to allow the plane A not to cut the perimeter of w. For the reason already mentioned r will be neglected, and therefore no restriction imposed on h on the first account. Let II be the pressure sustained by the solid, referred to a unit of surface, II having the value belonging to the point P, and let p' be the pressure of the fluid at N. Consider the element of fluid comprised between w, its projection on the plane A, and the projecting cylindrical surface. The forces acting on this element are, first, the pressure of the fluid on the base, which acts in the direction NP, and is ultimately equal to p'w; secondly, the pressure of the solid, which ultimately acts along PN and is equal to IIw; thirdly, the pressure of the fluid on the cylindrical surface, which acts everywhere in a direction perpendicular to PN; and, lastly, the moving forces due to the accelerating forces acting on the fluid; and this whole system of forces is in equilibrium with forces equal and opposite to the effective moving forces. Now the moving forces due to the accelerating forces acting on the fluid, and the effective moving forces, are both of the order wh, and therefore, whatever may be their directions, vanish in the limit compared with the force p'w, if we suppose, as we may, that h vanishes in the limit. Hence we get from the equation of the forces parallel to PN, passing to the limit, p being the limiting value of p', or the result obtained by substituting in the general expression for the pressure the coordinates of the point P for x, y, z. It should be observed that, in proving this equation, the forces on which capillary phenomena depend have not been taken into account. And in fact it is only when such forces are neglected that equation (6) is true. In the case of a liquid with a free surface, or more generally in the case of two fluids in contact, it may be proved, just as before, that equation (6) holds good at any point in the surface, p, II being the results obtained on substituting the coordinates of the point considered for the general coordinates in the general expressions for the pressure in the two fluids respectively. In this case, as before, capillary attraction is supposed to be neglected. |