Studies on the Motion of Viscous Flows— III 



Vj = v^(x,y) (l - — 



where Uo(x,y) and vo(x,y) are the velocity components of the two-dimensional 

 potential flow past a given body. The flow represented by the above equations 

 has the same streamlines as the potential flow about the body, and the stream- 

 lines for all parallel layers z = constant are congruent. The condition of no- 

 slip at the plates z = ±h is satisfied, but the same at the surface of the body is 

 not satisfied [19]. 



Using the above solution to calculate the nonlinear inertia terms, he obtains 

 a second approximation. For the case of a circular cylinder he gives the radial 

 and tangential velocity components as :^ . . 



= Reh2 



Reh2 



cos 2( 



15 h« 



and the component in the z direction is given as 



Reh^ 



4_ z_ 

 21 h 



44 z^ 

 105 h3 



15 hs 



105 h7 



Here Re = M'L/v , where U is the maximum velocity of the stream in the center 

 of the plate, i^ is the kinematic viscosity of the fluid, and L is a characteristic 

 dimension of the obstacle, which for a circular cylinder is taken as the radius. 



It can be seen that u^, Vj ^^^ ^2 satisfy the no-slip boundary condition at 

 the plate z = ±h, but they do not satisfy it at the cylinder wall. What is inter- 

 esting is that the normal component U2 is also not zero at the cylinder wall. 

 The total solution is given by the sums u^ + u^, v^ + V2, and W2. 



Since u^ and Vj depend on the number (Reh^) and W2 depends on the num- 

 ber (Reh^), h being the nondimensional thickness of the fluid layer between the 

 walls, Riegels introduces a characteristic Reynolds number for the flow config- 

 uration as 



Re 



"r2" 



Uh' 



vR 



Reh^ 



h' 



h = — 



R 



For 



Re 



h'' 



Uh'' 



^ Reh^ 



525 



