

A 



dV 





h dE 



^x 



a 



d X 





a dy 





h 



dV 



h 



BE 



V 



a 



Ty^ 



a 



Ti 



f(x. y) = V{x, y) + i E{x. y) [7] 



The components of current are 



[8] 



Since the stream function and velocity potential satisfy identical equations for two- 

 dimensional flow, the role of ^ and ^ may be interchanged. Thus, in setting up an electrical 

 analogy, the streamlines ifj may be represented either by F or £" and the equipotential lines ^ 

 by the other function. Either one of two analogies may be used in an electrolytic tank. 



Analogy K: tfy = V ijj = E [9] 



A dielectric model representing the hydrodynamic body is placed in a semiconducting medium 

 in such a way that equipotential lines correspond to lines of constant electric potential and 

 streamlines correspond to lines of constant electric flux. 



Analogy B: (j> = E ip = V [10] 



A conducting model representing the hydrodynamic body is placed in a dielectric medium in 

 such a way that the equipotential lines of flow correspond to lines of constant electric flux 

 and streamlines correspond to lines of constant electric potential. 



When the flow is three-dimensional with axial symmetry, the physical interpretation of 

 the stream function is somewhat different from that in two-dimensional flow. Since in three 

 dimensions the stream functions define surfaces across which there is no flow, the stream 

 function at any point in the medium is proportional to the flux of flow through a surface gener- 

 ated by the revolution about the symmetry axis of any curve joining this point with the axis. 

 The velocity components parallel and normal to the symmetry axis are respectively:^ 



a<p 1 dip 



* ,9a; y dy 



[11] 

 dcfj 1 dip 



^ dy y dx 



Since the stream function here is a measure of the flux of flow through a surface rather than 

 the flow across a line in two-dimensional flow, the dimensions of \p are no longer the same as 

 those of (p and the form of the equations defining the velocity components [11] is different 

 from that of [6]. Although (p still satisfies the Laplace equation, xp does not. By use of the 

 continuity equation and the condition of irrotationality of the motion, the differential equations 



