446 HYDROMECHANICS [HYDRODYNAMICS. accelerations reversed, combined with the impressed forces per unit of mass, will form a system in equilibrium according to D Alem- bert s principle. To determine the component accelerations, suppose F any function of x, y, z, t, and determine the rate of change of F per unit of time "DTP for a moving particle ; denoting this change by , we have (tt DF_, F(x-}-u5t, ~ z + u-5t, t + St) - F(x,y,z,t) ~~ dF -jj at dF ~j~ ax so that D d d -ji = j7 + u j~ at at ax dF v ^r + ay d v j~ ay dF ~r > dz d w T-- dz - is called particle differentiation, because we follow the rate of dt change of the particle as it leaves the point xyz ; but , , dj- ^? are the rates of change of F at the time t at the point dy dz xyz, fixed in space. Consequently the component accelerations parallel to the axes of coordinates of a particle of fluid are du du du du
- H U-, h V-, h W-r- ,
dt dx dy dz dv dv dv dv -j-. + U-j- + V-r- dt dx dy dw dw -jr + u, dt dx V Iz dw dw V-T-+ W-j- dy dz leading to the equations of motion last established. If F(x,y,z,t) = Q be the equation of a surface containing always the same particles of fluid, it follows from the preceding that DF dF dt dF dx dF v dy dF dz This is called the differential equation of the bounding surface, as particles of fluid once in the bounding surface always remain in it. To integrate the equations of motion (4), (5), and (6), suppose the impressed forces due to a potential V, such that the force in any direction is the rate of diminution of V in that direction, then X = - and putting dx Y-- dz dz the equations may be written du dv ~dt~ dw ~dt dE, (8), (9) (10), where R = and g 2 = tt 2 + v s + w; 2 (Lamb, Motion of Fluids, Appendix D; also Proc. London Math. Society, vol. ix.). A stream line is defined to be the actual path of a particle, and a line of flow to be a line such that the tangent at every point is in the direction of the velocity at the point ; the stream lines and lines of flow are coincident only when the motion is steady ; and when the motion is irrotational, the lines of flow are orthogonal to the surfaces obtained by equating the velocity function to a constant. A vortex line is defined to be a line whose tangent at any point is in the direction of the resultant w of the component angular velocities |, 17, C at that point ; and u is called the spin (Clifford, Kinematic). ,TJ, are called the components of molecular rotation (or spin) at xyz, for a reason to be explained afterwards ; and when they vanish the motion is said to be irrotational, and a function <f> exists, called the velocity function, such that dd> dd> dtt> u = -~- v = r w = j dx dy dz and, generally, the velocity in any direction is then the space varia tion of <. When the motion is irrotational, equations (8), (9), and (10) become dxdt dx ~ dydt + dy~ dzdi + ~dz = and therefore dt a constant throughout the fluid, which may, however, be a function of the time. If, however, the motion be steady, that is, if the velocity at any point of space does not change with the time, then
- i=o
dt U and the equations become dx -o -o ~ U di " dz ,dR dx dE -j- dy dE dE . r- = 5 so that .. , , , , az _ _ dE . dx dy dz and therefore the surface R, = constant, contains both stream lines and vortex lines ; and therefore
- +Y +
! = constant (11) along a stream line, and along a vortex line ; and if the motion is irrotational, the constant is the same for all the space filled with the fluid ; for then dE dE . dE A dx dy dz Taking the axis of x for an instant in the direction of the normal to the surface R = constant, then u = Q and | = 0, and (8), (9), and (10), if the motion is steady, reduce to -5 2t - 2wr) = 2q<n> sin 0, where is the angle between the stream and the vortex line. It is sometimes convenient to use moving exes of coordinates in Hydrodynamics, and the equations of motion then become du . - .du . . du , , du _ 1 dp with two similar equations ; w : , 2 , o> 3 denoting the component angular velocities of the moving axes, and u, v, w the components of the velocity of the fluid in space at the point xyz at the time t parallel to the axes. For if q denote the component velocity of the particle xyz at the time t in a direction fixed in space whose direction-cosines are I, in, n, then q = lu + mv + nw ; and in the infinitesimal element of time dt the coordinates of the particle will have become x + (u + ?/a> 3 - zo> 2 )dt , y + (v + zw-i - xca 3 )dt , z + (w + xu 2 - ywjdt ; Do dl dm dn . , fi , m I - dt dt * dt dt du dw dv dx dw dw dw) But, since I, m, n are the direction-cosines of a line fixed in space, dl _ dm _ _, ^ = 7 dt a>3 u 2 dt l 3 dt 2 .du 2 dx du dt + m <!t .dv (u + yo> 3 -zu. 2 )-^ .dv .dw .dw
- )^ + (* + 2 *i -*);*;
(v ! d P + m I Y ~
p dyj P
for all values of I, m, n, leading to the equations of motion. ll Y : dp
- *| A *~ ~" ~T~
p dx
