For a body rotating about a fixed axis, similar considerations hold. The kinetic energy 

 of body and fluid can be written 



T ^ — (I + kl') w^ [146c] 



where / is the moment of inertia of the body about the axis of rotation, 



/' is the moment of inertia of the displaced fluid when rotating as if solid, and 

 tu is the angular velocity. 



The rate at which work is done then takes the form 



„ dT dco 



Geo = — ^(I + kl') (o -—, 

 dt ^ ' dt' 



where G is the torque acting on the body. 



Hence G = {I + kI')—. [146d] 



dt 



This equation shows that no torque is required for constant angular rotation about a 

 fixed axis in an ideal medium. The constant k is here the coefficient of inertia for rotation 

 about the given axis, and its value for rotation is usually different from that for translation. 



Two-dimensional flow, as described in Section 13, is an important special case of fluid 

 flow in which the motion occurs in a set of parallel planes, so that there is no component of 

 velocity or acceleration in the direction perpendicular to these planes. In two-dimensional 

 cases it will be understood that all quantities refer to the portion of the body and of the fluid 

 that is contained between two planes drawn parallel to the planes of motion and unit distance 

 apart, and fj, 4/j, M[, /j, /j, will be written as referring to this portion. The coefficient of 

 inertia, on the other hand, being merely a constant of proportionality, does not require a subscript. 



It has been assumed that the motion is irrotational and is therefore entirely determined 

 by the motion of the body. This assumption is essential. Furthermore, in defining the coef- 

 ficient of inertia, only one component of the force or torque was considered; and the discus- 

 sion was limited to certain special types of motion. It is of interest to consider how the coef- 

 ficient of inertia may be used in certain other cases; and certain other features of the force 

 action of fluids upon moving bodies may also be mentioned without proof. 



For a given body of finite dimensions, with its mass distributed in any given manner, 

 it can be shown that there is always at least one set of mutually perpendicular directions, 

 fixed relative to the body, in any one of which the body can move through frictionless fluid 

 without the action of any forces upon it and without exhibiting any tendency to rotate. These 



381 



