﻿464 Two-Dimensional Steady Motion of a Viscous Fluid. 



pressure head created by the pump when it rotates with a 

 given velocity and discharges a given volume of fluid per 

 unit time. 



Flow under pressure along a circular canal. — Finally, we 

 will consider solution V. 



i/r = Ar 2 log r -f Br 2 -f log r. 



If u } v are the radial and transverse components of velocity 



r q6 



» = |*=2Arloffr+(A + 2B)r+-. 

 Or r 



If A = 0, this is the well-known solution for rotating con- 

 centric cylinders. Using the equations of motion in polar 

 coordinates we can without difficulty find the value of the 

 mean pressure p. 



p=iv P A8+pf(r), 



where 



f(r) = 2 A (log rf(Ar 2 + C) + 2 log r (ABr 2 + AC + 2BC) 



+ l(A 2 + 4B 2 )7*-^ 2 \ 



The constants B, C may be chosen so that the velocity is 

 zero for any two values of r, and we have the solution for 

 the flow of a viscous fluid round a canal bounded by two 

 concentric circular arcs. The pressure will not be constant 

 across a radial cross- section, but will vary in a way which 

 is represented by f{r). It will be noted, however, that f(r) 

 contains only the squares of the coefficients A, B, C, and is 

 therefore of the order of the square of the velocity. 



Suppose the canal is bounded by circular arcs of radii 

 a, b, which subtend an angle a. at their common centre, and 

 let P be the pressure difference between corresponding points 

 on the two bounding cross-sections. Then 



P = 4/xA«, 



where \i is the coefficient of viscosity and is therefore equal 

 to vp. The condition that the velocity shall vanish when 

 r = a. b gives the following equations to determine the 

 constants B, C 



A(rlogr 2 + r) + 2Br+ - =0, (r=a, b), 



