The Bladeless Propeller 



uniform in the direction of the velocity V' of f relative to F^ Indeed, since the 

 pressure is constant at any point fixed in F ^, at points fixed in F the pressure 

 varies with time at the rate Bp/Bt = - V^-Vp • Therefore, except when Vp is zero 

 or normal to V , the flow in F is nonsteady — more precisely, cryptosteady — and 

 the energy level of its particles undergoes nondissipative changes that are ab- 

 sent in F . Similarly, whereas two contiguous streams will exchange energy 

 only by irreversible transport processes in a frame of reference f ^ in which 

 they are both steady, they will exchange energy also by pressure exchange in 

 every other frame of reference. This additional transfer of mechanical energy 

 is essentially nondissipative. It is, as in all forms of pressure exchange, equal 

 to the work done by the pressure forces which the interacting flows exert on one 

 another at their interfaces. This work is zero in f^, where the interfaces are 

 stationary. In every other frame of reference the interfaces move and energy is 

 transferred from one flow to the other by pressure exchange. The special merit 

 of cryptosteady interactions is that they can be generated, controlled, and ana- 

 lyzed as steady-flow processes in f^, while retaining the potential advantages of 

 nonsteady interactions in the frame of reference in which they are utilized. 



In the following discussion, the frame of reference f ^ will be referred to 

 as the "relative" frame, and the velocities in it as "relative velocities," whereas 

 the frame of reference f , in which the sought energy transfer is to be effected 

 and utilized, will be referred to as the "absolute" frame, and the velocities in it 

 will be called "absolute velocities." 



Consider, in the absolute frame, the interaction schematically described in 

 Fig. 3, between two flows at different energy levels —that of a "primary" fluid I 

 and that of a "secondary" fluid II, both inviscid. Upstream of the interaction 

 region, the two flows are separated from one another by an infinitely thin con- 

 taining wall, extending to infinity in the + y direction. At infinity upstream, the 

 velocity is uniform and constant in each flow and both velocities are parallel to 

 the X axis. 



The primary fluid enters the interaction space through an orifice in the 

 containing wall. This wall moves in the + y direction at the constant velocity v , 

 but since the wall is infinitely thin and the fluid is inviscid, no momentum in the 

 y direction is imparted by the moving orifice on the primary fluid: the area of 

 emergence of this fluid moves at the velocity v, but the primary fluid particles 

 themselves emerge through it at a velocity p^ ^ which is parallel to the x axis. 

 Because of the stipulated absence of viscosity, no work is required to sustain 

 the motion of the containing wall; nor is any energy otherwise exchanged be- 

 tween the double-flow system considered and its surroundings. 



In the frame of reference fixed to the orifice —the relative frame — the two 

 flows are. both steady. Therefore, the flow system is cryptosteady in the abso- 

 lute frame. 



In contrast to the absolute velocities a of the two flows, their relative ve- 

 locities c upstream of their merger station are not parallel. On the other hand, 

 in the relative frame the streamlines of the two flows must be parallel to one 

 another at their interfaces, because the latter are stationary. Thus, as they 

 come in contact, the interacting flows must deflect each other, in the relative 



1355 



