126 



Mb. stokes, on SOME CASES OF FLUID MOTION. 



A similar result may be proved to be true for any solid symmetrical with respect to two 

 planes at right angles to each other, and moving in the direction of the line of their intersection 

 in an infinitely extended fluid, the solid and fluid having been at first at rest. Let the planes 

 of symmetry be taken for the planes of xy and xx, the origin being fixed in the body : then it 

 is evident that the resultant of the pressure on the solid due to the motion will be in the direction 

 of the axis of x, and that there will be no resultant couple. Let C be the velocity of the solid 

 at any time; then the value of (h at that time will be of the form C^{x, y, z), where C alone 

 contains t, (Art. 4), and the velocity of the particle whose co-ordinates are x, y, x, being pro- 

 portional to C, the vis vhui of the solid and fluid together will be proportional to C'. Now if no 

 forces act on the fluid and solid, except the pressure of the fluid, this vis viva must be constant*; 

 therefore C must be constant; therefore the resultant of the fluid pressure on the solid must be 

 zero. If now C be a function of t we shall have 



p = -p^\^{x,y, ~)-J^ +P^ 



J) being the pressure when C is constant. Since therefore the resultant of the fluid pressure 



dC 

 varies for the same solid and fluid as the effective force, and for different fluids varies as p, 



the effect of the inertia of the fluid will be, to increase the mass of the solid by n times that of 

 the fluid displaced, n depending only on the particular solid considered. 



Let us consider two such solids, similar to each other, and having the co-ordinates planes 

 similarly situated, and moving with the same velocities. Let the linear dimensions of the second 

 be greater than those of the first in the ratio of m to 1. Let ?<, v, w, be the velocities, parallel 

 to the axes, of the particle (x, y, z) in the fluid about the first ; then shall the corresponding 

 velocities at the point {mx, my, mz) in the fluid about the second be also u, v, w. For 

 udmx + vdmy + wdnix = m{udx + vdy + wdz) (24), 



and is therefore an exact differential, since udx + vdy + wdz is one: also the normal at the 

 point (.r, y, z) in the first surface will be inclined to the axes at the same angles as the normal 

 at the point (ma?, my, mz) of the second surface is inclined to its axes, and therefore the normal 

 velocities of the two surfaces at these points are the same ; and the velocities of the fluid at these 

 two points parallel to the axes being also the same, it follows that the normal velocity of each point 

 of the second surface is equal to that of the fluid in contact with it. Lastly, the motion about 

 the first solid being supposed to vanish at an infinite distance from it, that about tlie second will 

 vanish also. Hence the supposition made with respect to the motion of the fluid about the second 

 surface is correct. Now putting <p for f(jidx + vdy + wdz) for the fluid in the first case, the 

 corresponding integral for the fluid in the second case will be mcp, if the constant be properly 

 chosen, as follows from equation (24.). Consequently the value of that part of the expression for 

 the pressure, on which the resistance depends, will be m times as great for any point in the 



• If an incompressible fluid which is homogeneous or hetero- 

 geneous, and contains in it any number of rigid bodies, be in 

 motion, the rigid bodies being also in motion, if the rigid bodies 

 are perfectly smooth, and no contacts are formed or broken among 

 them, and if no forces act except the pressure of the fluid, tlie 

 principle of vis viva gives 



— r— = 'iffp.vdS «)> 



at 



where v is the whole velocity of tlie mass w, and the sign 2 ex- 

 tends over the whole fluid and the rigid bodies spoken of, and 

 where dS is an clement of the surface which bounds the whole, 

 71, the pressure about the element dS^ and v the normal velocity of 



the particles in that element, reckoned positive when tending into 

 the fluid, and where the sign jf^extends to all points of the bound- 

 ing surface. To apply equation («) to the case of motion at 

 present considered, let us flrst confine ourselves to a spherical 

 portion of the fluid, whose radius is r, and whose centre is near 

 the solid, so that dS refers to the surface of this portion. Let us 

 now suppose r to become inflnite : then the second side of (a) will 

 vanish, provided p, remain finite, and v decrease in a higher ratio 

 than — . Both of these will be true, (Art. It. J; for v will vary 



ultimately as — , since there is no alteration of volume. Hence 

 if the sign S extend to inflnity, we shall have Smu- constant. 



