Lines of constant <p, known as the velocity potential, are everywhere normal to the 

 direction of flow, and the velocity components are obtained from the gradient of this function. 



d (h d <p d (p 



"^ dx y dy ' dz ^'^^ 



Equations [1] and [2] are valid for both two- and three-dimensional flows. 



The Laplace equation with similar boundary conditions also occurs in other fields of 

 physics, particularly in electrostatics where the electric potential V satisfies the Laplace 

 equation 



d'^V d^V d^V 



AF = ^ + X + ^ =0 rql 



dx^ dy^ dz^ LSI 



The components of electric current are obtained from the gradient of V: 



h dV h dV h dV 



a o X ^ a o y adz 



where cris the specific resistance of the medium and A the depth of the medium. Since the 

 forms of the equations are identical, it is possible to determine the hydrodynamic flow from a 

 study of the electrical counterpart in which the hydrodynamic velocity potential <p corresponds 

 to the electric potential 7. Such analogies are possible both in two- and three-dimensional 

 flows whether the flow is axisymmetrical ornot. 



In two-dimensional flow and in three-dimensional flow in which there is an axis of sym- 

 metry, there exists a system of stream functions i/f which are constant along lines of flow and 

 are everywhere orthogonal to equipotential lines. In two-dimensional flow, i^i as well as ^ is 

 a potential function and satisfies the Laplace equation. Hence, ^ and ^A may be represented 

 as the real and imaginary parts, respectively, of a complex potential function: 



w{x, y) = 4>{x, y) + iiPix, y) [5] 



The velocity components in terms of <j4 and i/» are given by the Cauchy-Riemann equations: 



[6] 



In the two-dimensional electric field analogy, a system of lines of constant electrical 

 flux E exists which forms an orthogonal network with the lines of constant electrical potential, 

 where E also satisfies the Laplace equation. Thus, V and E may also be represented as the 

 real and imaginary parts of a complex potential function: 



'""' dx ~ 



dip 





dip 



dx 



