Equivalent Circuits of Compressible and Incompressible Fluid Flow Fields
Gabriel Kron · Journal of the aeronautical sciences. [REQUEST TITLE] · 1945
(1) Div pv — Q; curl v — V b6 b6 (2) Div a grad = & = fc-^ + b2 -^ + h + h bt bt b\fr bil/ (3) Curl c curl ip = d = di--^ + d t ~ + dz-ip + d* bt bt under transient, sinusoidal, or steady-state conditions, thereby allowing their solution either by a network analyzer or by numerical and analytic circuit methods. In fluid dynamics these equations represent the steady potential flow of a compressible fluid when expressed in the hodograph plane or the general nonviscous flow of an incompressible fluid when expressed in the physical plane or space. (In representing the flow of a compressible fluid in the physical space, the networks have variable units, since the equations are nonlinear.) The same equations and networks may be considered to represent static electromagnetic fields in current-carrying regions or in regions with stationary charges or poles. The second equation may also represent the conduction of heat (or diffusion) in a nonisotropic medium, or it may represent the general wave equation; All networks are expressed in curvilinear orthogonal coordinates to simplify the networks in case of curvilinear boundaries and to express three-dimensional problems with axial or other symmetry by means of a two-dimensional network. The coefficients p, a, and c are not scalars but elements of diagonal matrices, allowing special types of nonisotropic properties in the fields. All coefficients may be constants or functions of the independent space variables, thereby allowing nonhomogeneous fields. The fields may contain multiply connected regions, and a method showing how to represent Riemann-surfaces by equivalent circuits is given. Most transient and steady-state equivalent circuits require only resistances, inductances, and capacitors, while some transient networks also require ideal transformers. The blocks into which space is divided may have arbitrary lengths in different directions. General nonisotropic fields and nonorthogonal reference frames require an extension of the present equivalent circuits. Numerical methods of solving the equivalent circuits are given in another publication. In general the circuits may be used to check the consistency and accuracy of the results arrived at by other methods, approximate or exact. The unbalanced currents at the junctions (easily calculated) give a quantitative measure of the deviations from the correct answer.