38. Moving Axes in Hydrodynamics.—In many problems, such as the motion of a solid in liquid, it is convenient to take coordinate axes fixed to the solid and moving with it as the movable trihedron frame of reference. The components of velocity of the moving origin are denoted by U, V, W, and the components of angular velocity of the frame of reference by P, Q, R; and then if u, v, w denote the components of fluid velocity in space, and u′, v′, w′ the components relative to the axes at a point (x, y, z) fixed to the frame of reference, we have

u = U + u′ − yR + zQ, v = V + v′- zP + xR, w = W + w′ − xQ + yP.

(1)

Now if k denotes the component of absolute velocity in a direction fixed in space whose direction cosines are l, m, n,

k = lu + mv + nw;

(2)

and in the infinitesimal element of time dt, the coordinates of the fluid particle at (x, y, z) will have changed by (u′, v′, w′)dt; so that

Dk= dlu + dmv + dnw
dt dtdt dt
+ l ( du+ u′ du+ v′ du+ w′ du)
dt dxdy dz
+ m ( dv+ u′ dv+ v′ dv+ w′ dv)
dt dxdy dz