eliminate them and so make the solution determinate. 



4. BOUNDARY CONDITIONS 



At the boundaries of the fluid the continuity equation is replaced by special surface 

 conditions. For example, at a fixed boundary it is necessary that the fluid velocity have no 

 component normal to the surface. If I, m, n are the direction cosines of the normal to the 

 surface, this condition requires that 



lu + mv + nw = [4a] 



at every point on the surface; compare Equation [lb]. 



If the boundary is in motion, the normal component of the fluid velocity at the surface, 

 or q^ = lu + mv + nw, must equal the velocity of the surface normal to itself. This is equiva- 

 lent to saying that the velocity of the fluid relative to the surface is wholly tangential, or that 

 a particle on the surface remains on the surface. A method of finding I, m, n when the equation 

 of the surface is given is derived in Sec, 135. 



5. ROTATIONAL MOTION; THE CIRCULATION 



In the kinematics of rigid bodies, a distinction is made between translational and 

 rotational motion. In rotational motion, all particles not on the axis go round the axis in 

 circles. An analogous but more general conception of rotational motion in a fluid can be 

 developed as follows. 



Consider any closed curve C drawn in the fluid, and choose a positive direction of 

 motion around the curve, as in Figure 5a. At each point of the curve, divide the particle 

 velocity into a component perpendicular to the curve and a component q^ in the direction of 

 the tangent to the curve; let q^ be taken positive when it has the same direction as the chosen 

 positive direction of motion around the curve and negative when it has the opposite direction. 

 In Figure 5a, q is positive at P but negative at Q. Multiply each element of length ds along 



Figure a Figure b 



Figure 5 — Illustrating the definition of the circulation. 



6 



