Abstract
The well-known Stokes’ problems are reexamined by applying the Adomian decomposition method (ADM) associated with other mathematical techniques in this paper. Both the finite-depth (bounded) and infinite-depth (unbounded) cases are analyzed. The present paper raises and deals with two major concerns. The first one is that, for Stokes’ problems, it lacks one boundary condition at the expansion point to fully determine all coefficients of the ADM solution in which an unknown function appears. This unknown function which is dependent on the transformed variable will be determined by the boundary condition at the far end. The second concern is that the derived solution begins to deviate from the exact solution as the spatial variable grows for the unbounded problems. This can be greatly improved by introducing the Padé approximant to satisfy the boundary condition at the far end. For the second problems, the derived ADM solution can be easily separated into the steady-state and the transient parts for a deeper comprehension of the flow. The present result shows an excellent agreement with the exact solution. The ADM is therefore verified to be a reliable mathematical method to analyze Stokes’ problems of finite and infinite depths.
Highlights
The Adomian decomposition method (ADM) has been extensively applied to pursue approximate solutions of mathematical as well as practical problems in many disciplines [1]
The lowest term of the ADM solution is determined by the imposed initial or boundary condition of the problem, and other higher terms can be calculated by applying the integral operator of recursion form with the help of lower terms
Even if the Padeapproximant is successful in extending the valid range of the ADM solution, an unbounded problem still increases the difficulty to keep the accuracy in the whole domain
Summary
The Adomian decomposition method (ADM) has been extensively applied to pursue approximate solutions of mathematical as well as practical problems in many disciplines [1]. To improve the knowledge and verify the accuracy and applicability of the ADM for Stokes’ problems, the exact solution, which contains the steady-state and transient parts, will be examined by applying the ADM and other mathematical techniques for bounded and unbounded cases. Different from many of past studies, two boundary conditions which are given at different spatial positions, one at y = 0 and another at y = h (for bounded problems) or y → ∞ (for unbounded problems), result in extra mathematical efforts to determine all coefficients of the ADM solution as two conditions are required at y = 0 for solving the present second-order PDEs. an unknown function appeared in some coefficients of the ADM solution which has to be determined by the boundary condition at y = h or y → ∞.
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