104. TWO COAXIAL CYLINDERS. 



In the flow described at the end of Section 67, at any given value of r such as ;• = 6 the 

 radial velocity varies with d in the same way as does the normal component of the velocity of 

 a moving circular cylinder. Let such a cylinder, of radius 6, be inserted with its axis at the 

 origin, and let it move perpendicularly to its axis with a velocity F toward (9 = where 



F = f; (-1 + — |. [104a] 



Then the surface of the cylinder and the fluid have the same radial component of velocity, so 

 that the necessary boundary condition is satisfied at the surface of the cylinder. 



Assume that a > b. Then the formulas of Section 67 will represent the flow between 

 two coaxial cylinders of which the outer, of radius a, is stationary. After substituting from 

 Equation [104a] for U in terms of V, Equations [67b,c,i,j] give for the motion of the fluid 

 between the cylinders 



b^Va'-\ b^V 



r + cos t?, ^ = \t .sin 6, [104b, c 



b^V I a^ \ b^V I c? 

 q, = 1 cos 61, qQ^ ( +1] sin 0. [104d,e] 



a^-b^ \r'^ j ^ a^-b^ 



In Equation [17d] for the kinetic energy, the contribution of the outer boundary at r=c 

 vanishes, since q^ = 0. Hence the kinetic energy of the fluid per unit length of the cylinders, 

 at the instant at which they are coaxial, is 



if I y 1 , a^ _^ ^2 „ 



^i^i; P h^n'^^-TP c/^q^bdd= - 77 pb^ V^, [I04f] 



a^ -b^ 



2 1" 2 ' 2 .2 ,2 



after substituting the values of (p and q^ with r = 6 and integrating. 



105. FLUID WITHIN A ROTATING SHELL OF ELLIPTIC OR OTHER SHAPE 



If Equation [99c] in Section 99 is to represent an ellipse, ip must be a quadratic function 

 of X and y. Consider, therefore, 



w = (^ + lip = lA z-^ = iA{x + iy)'^, (p = -2Axy, 4i = A{x'^ - y-^). 



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