422 Mr Wilson, On Convection of Heat. 



If instead of a slab we take a circular cylinder sliding in 

 a jacket the differential equation is in cylindrical coordinates, 

 supposing everything symmetrical about the axis of the cylinder, 



dM d?d ld0 _ dd _ 

 dx 2 dr 2 r dr ^ dec 



A solution of this of the form 9 = *ZA n e anX J (ii n r) can be 

 obtained where a n = p ± V /j, n 2 + p 2 and the /jus are constants. If 

 the radius of the cylinder is a, then, when for example the surface 

 of the cylinder is maintained at temperature zero, the fis have 

 to be the roots of the equation J (fia) = 0. This solution therefore 

 enables problems for the cylinder to be solved similar to those 

 considered in the case of the slab. It is not proposed to consider 

 any problems on this case in detail. 



A problem of some interest from the practical point of view 

 is the distribution of temperature in a liquid flowing through a 

 pipe ; the temperature of the pipe being supposed known. 



We shall take the axis of the tube as coinciding with the axis 

 of x, and denote the radius of the tube by a. Let the mean 

 velocity of the liquid through the tube be V, then the velocity at 

 any point in the tube is given by the equation 



2V , 



u=-(a 2 -r 2 ). 



Taking everything to be symmetrical about the axis of the 

 tube, the appropriate form of the differential equation is there- 

 fore 



dM \<L0 dM)_ 2s P V dd 



dr 2 r dr dx 2 ka 2 dx 



A solution of this equation for the case when the temperature 



of the pipe is represented by the equation d = cx + d can be easily 



dO 

 obtained. In this case it is evident that -=- = c everywhere 



inside the pipe, so that the differential equation becomes 



d 2 6 1 dd 2soV , 2 2X 



t-s + - -7 rV ( a - r 2 )c = 0. 



dr 2 r dr ka 2 



A solution of this is 



= A+cpr 2 -^, 



where p = -£r- and A is a constant. 



1 2k 



