3 t'2 Prof. Heimlich/, on Discontinuous Movements of Fluids. 



It is easy to see that the conditions of the state of rest are 

 satisfied by such an infinitely long cylindrical current which, 

 issuing out of a tube of the same diameter, enters into a still 

 external fluid, and contains throughout fluid which moves pa- 

 rallel to its axis with uniform velocity. 



I shall here merely give a sketch of the mathematical treat- 

 ment of a case of the inverse kind, where the current passes out 

 of a wide space into a narrow channel, in order at the same time 

 to give an example of a method by which certain problems can 

 be solved in the doctrine of potential functions which have 

 hitherto presented difficulty, 



I confine myself to the case where the motion is "stationary," 

 and depends alone upon two rectangular coordinates r, //, and 

 where, further, in the frictionless fluid there are no rotating par- 

 tieles, and consequently none such can arise. For the particle 

 of fluid at the point (x } y) t let us denote by u the component 

 velocity parallel to a?, and by v that parallel to y. Then two 

 functions of a? and y may, as is well known, be found such that 



dd> dslr 

 dx dy 



dd> dy 



"~ dy ~ dx 



By means of these equations the condition is at once fulfilled 

 in the interior of the fluid, that the mass remains constant for 

 every unit of volume, namely 



* 4. * - *+ 4. «*** - d ** 4-^-0 f] a) 



d.v + dy ~ dv~ + dy 2 ~ dx* + dif ~ U ' * ' [U) 



With the constant density //, and when the power of the external 

 force is represented by V, the pressure in the interior of the fluid 

 is given by the equation 



(i) 



The curves 





(lb) 



yjr= const. 



are the current lines of the fluid, and the curves 



(f>= const, 

 are orthogonal to them. The latter are the curves of equal power 

 in the case of electricity, or of equal temperature in the case of 

 heat, flowing in a stationary current in conductors of constant 

 conductivity. 



