On the comparison of the methods used for the solutions of the governing equation for unsteady unidirectional flows of second grade fluids

In order to solve the governing equation for unsteady unidirectional flows of second grade fluids, the use of the Laplace and the Fourier transform methods are discussed. Three characteristic examples which are unsteady flow between two parallel plates, unsteady pipe flow and unsteady flow over a plane wall are considered. It is found that the solution for unsteady flow in bounded regions obtained by the Laplace and the Fourier transform methods are exactly the same as the case of the unsteady flow of a Newtonian fluid. It is shown that the Laplace transform method for small values of time is useful for flows of Newtonian fluids but it is not convenient for flows in unbounded regions of second grade fluids. Furthermore, it is explained that for some unsteady flows of second grade fluids, the solution obtained by using the Laplace transform does not satisfy the initial condition and therefore the Fourier transform method is used.