Analytical solution of the Darcy equation in parabolic, bi-polar cylindrical and parabolic cylindrical coordinates

Abstract

The analytical solution (stream function) of the Darcy equation in specific orthogonal curvilinear coordinates, including parabolic coordinates, parabolic cylindrical coordinates, and bi-polar cylindrical coordinates, is the subject of the current research. Stream function solution of (????ଶ − ????ଶ)????(????, ????) = 0, are also obtained analytically in parabolic co-ordinates and parabolic cylindrical co-ordinates. The resulting analytical expressions of the stream function solution are a suitable combination of trigonometric, exponential, Modified Bessel, Parabolic Cylinder, Whittaker, and Laguerre functions.