Let x represent the length BM, 7 the whole length of the filament; e,, the inclinations of AB, CB, to the horizon; k the height of a column of the fluid which would exert a pressure equal to that of the atmosphere. Then the motion of the fluid will be defined by the differential equation d2x 7 + g {k - l sin ( + (sin ɛ + sin () x} = 0, dt2 which is easily integrated. The arbitrary constants introduced by the integration are to be determined by knowing the position and velocity of the filament at any instant of time. Euler: Comment. Acad. Petrop. tom. xv. p. 254. CHAPTER III. RESISTANCES. THE earliest problems of the resistance of a fluid, on the surface of a solid moving through it, were solved by Newton,* who ascertained the magnitude of the resistance on a globe and on a cylinder moving in the direction of its axis, and enunciated the fundamental property of the Solid of Least Resistance. The theory of resistances laid down by Newton received afterwards extensive applications in the hands of James Bernoulli, who has given the results of his calculations for various forms of surfaces in the Acta Eruditorum for 1693. John Bernoulli has also treated on this subject in his Nouvelle Theorie de la Manoeuvre, and Herman has devoted to it the twelfth chapter of the second book of his Phoronomia. If A be the area of a plane directly opposed to a stream of fluid, of which the density is p, moving with the velocity v; then, according to the ordinary theory, the resistance on the plane will be equal to pv2. This formula, which forms the basis of the ordinary theory of resistances, was first deduced from the equations of fluid motion by John Bernoulli,† the law of resistance which it expresses having been already adopted by Newton, James Bernoulli, and others. An interesting application of the theory of Resistances has been made by Euler, in a memoir on the maximum efficiency of Wind Mills, in the Commentarii Academiæ Petropolitana, tom. iv. p. 41, entitled De Constructione Aptissima Molarum Alatarum. In the following problems, wherever the contrary is not stated, the resistance will be supposed to take place according to the law of the square of the velocity. * Principia: lib. II. Sec. 7. + Comment. Acad. Petrop. 1737, p. 37. 1. An isosceles triangular lamina being exposed to the action of a stream, the direction of which is parallel to the perpendicular distance CD (fig. 51) of the vertex C of one face of the lamina from the base AB of this face: to compare the resultant pressure of the fluid on the lamina as its vertex or its base is opposed to the stream. Letv be the velocity of the stream, 2a the angle at the vertex of either face of the lamina, 7 the length of AC or BC, s the distance of any point P in AC from C, and the thickness of the lamina. Let P denote the pressure on the lamina, when C, and Q, when AB meets the current. Then, under the supposition that C meets the stream, the component of the velocity at P, at right angles to AC, is v sin a, and therefore the normal pressure, on an elemental rectangle rds at P, will be equal to prds. (v sin a)3; and the component of this pressure, parallel to CD, will be equal prds. (v sin a)2. sin a = Prv2 sin3 a ds. to Hence P = 2 'S" bpro" sin2 ads = prlo2 sin2 a. 0 When AB meets the stream, each of its elements being acted on perpendicularly by the fluid with a velocity v, we shall have James Bernoulli: Acta Eruditorum, Lips. 1693, p. 252. Bossut Traité d' Hydrodynamique, tom. I. p. 431. 2. A lamina in the form of a semi-ellipse, bounded by the minor axis, moves through a fluid, first with its vertex and next with its base foremost, in the direction of its axis: to compare the resultant resistances in the two cases. Letv be the velocity of the stream, a the length of the semiaxis major AC (fig. 52), 26 the length of the axis minor BB', of either face of the lamina. Let CAx, CBy, be taken as the axes of coordinates, and let CM = x, PM = y. Let Pp = ds, p being any point of the curve AB indefinitely near to P. Then, denoting the thickness of the lamina and p the density of the fluid, the normal resistance on the portion Pp of the lamina, the vertex A being supposed to move foremost, will be equal to dy2 έρτε ds, ds2 and the component of this, parallel to AC, will be equal to Hence, P denoting the resultant resistance on the lamina, hence, integrating from -y to + y, we have But, Q denoting the resistance when the motion of the lamina takes place in the opposite direction, James Bernoulli: Acta Eruditorum, Lips. 1693, Herman: Phoronomia, p. 246. p. 253, Art. 5. 3. A semicircular lamina is placed in a stream flowing in a direction parallel to the axis of the lamina: to compare the resultant stress experienced by the lamina, when its vertex meets the stream, with that which it experiences when its base meets the stream. = a, Let C(fig. 53) be the middle point of the rectilinear boundary of either face of the lamina, P any point in the curvilinear circumference of this face, and CA its axis. Let AC LACP = 0, arc AP = 8, T = the thickness of the lamina. P, Q, denote the resultant pressures of the fluid on the lamina, accordingly as A or C meets the stream. Let Then, v cos being the normal component of the velocity with which the fluid impinges upon the lamina at P, the component of the pressure upon an elemental rectangle rds of the edge of the lamina, parallel to AC, will be equal to 1⁄2 p. τds. (v cos 0)2. cos 0 = 1⁄2 pтav2 Cos3 0 d0. Hence, P being equal to twice the component of the pressure upon the quadrant AB, parallel to AC, we have The pressure Q upon the base of the lamina will be equal to Bossut Traité d'Hydrodynamique, tom. I. p. 433-437. |