Orr — stability or Instahility of Motions of a Viscous Liquid. 135 



Aet, 35. A circular Pipe ; the more General Investigation. 



In discussing the most general disturbance in this case, we may either 

 transform to cylindrical coordinates the equation (5), and obtain in those 

 coordinates the equations giving a stationary /.t, or else obtain in Cartesian 

 coordinates the equations which would now replace (7), and then transform 

 them. Adopting the latter procedure, the equations are 



2^V-Ux - pivclW/clx - clp/ch 

 2uVhiy - ptvdJV/chj - cljj/d-i/, 

 2fxVho - p (tc^clW/clx + Uyd Wjcly) = clpjclz, (77) 



where Ujc, v.y denote the velocity-components in the x, y directions transverse 

 to that of flow. Confining ourselves to the symmetrical case, which there is 

 little doubt will give the lowest critical velocity, we write 



Ux = xu/r, Uy = yulr, 



when the two former equations become 



2fx(V'u-vr-^)- pi'jclW/(lr = cl2//dr, ' ' ' ,78) 



and the latter is 



~ - 2,:iVhv - pudW/dr = djy/dz. - (79) 



Noting that 



d/dr.V = [Y- -r-')dldr, (80) 



and writing W = C [cC- - r^), the elimination cA p between these gives 



2^ (V^ - r-'){duldz - dwjdr) + W pWidwjch - du\clr) - u] = 0, (81) 



Introducing the stream-function -^ defined by the equations 



ru = dxp/dz, riv = - chp/dr, - -- - - - - 



this becomes 



^(V^ - r-') {r-\cr-^pidr: + d^/dz') - r-cmdr] - 2C'pd^/drdz = ; 

 or, ,xr-^ { d'ldr^ - r'' d/dr + cP/dz'- 'r4^ - 2 CpcPxp/drdz = 0. (82) 



[On multiplying by r, differentiating with respect to /■, and dividing by r^ 



this might be written - 



nVhv - '2C'pr-\r- {r'-vS)ldrdz = 0, (83) 



an equation which might be obtained more easily directly from the equations 

 which replace (7). In the subsequent investigation, iv might equally well be 

 taken as the unknown function, instead of it.] 



