Studies on the Motion of Viscous Flows— V 



Plane Axisymmetric Flows without Radial Velocity —In such flows, ui = 

 -qx2, U2 = qxj, U3 = 0; where q =q(r,x3,t), and r2 = Xj2 + Xj^. Such flows 

 clearly satisfy Eqs. (2). 



Let us list the known exact solutions of the Navier- Stokes equations which 

 belong to the above two families: 



Examples of Rectilinear Flows — 



1. Steady flow between parallel plates 



2. Steady flow in a circular pipe (more generally, of arbitrary section) 



3. Flow in Stokes' first problem 



4. Flow in Stokes' second problem 



5. Pipe flow starting from rest ... ' ^ , : 



6. Flow between plates starting from rest. 



Plane Axisymmetric Flow — .: • -;■' "• ^ 



1. Rigid body rotation 



2. Steady flow between concentric cylinders 



3. Potential vortex 



4. Vortex of Hamel and Oseen [7] 



5. Vortex of Taylor [8] 



6. Vortex of Rouse and Hsu [9]. ■ > 



Since the above examples refer to plane motion, a doubt lingers as to 

 whether any three-dimensional motion satisfying Eqs. (2) is dynamically possible. 

 To remove this doubt, we give a three-dimensional solution. 



Flow through a Rotating Pipe — 



Uj = -Ax^, U2 = Ax^, U3 = B (^1 -^j , 



where A, B, and R are constants, and r^ = y.^ + x^. K we put A equal to zero, 

 the flow reduces to Poiseuille flow. If we take pressure P as 1/2 [A2(Xj2 + x^^)] - 

 4(B/r2)vx3, and substitute in Eq. (3), we have 



"i,t + ^1^1,1 + "2^2,2 + "3"i,3 = Ax,(-A) ; 



687 



