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In this article, an incompressible, two-dimensional (2D), time-dependent Newtonian fluid flow in a square cavity is simulated using finite difference method and alternating direction implicit technique. Navier-Stokes equations are solved numerically in stream function-vorticity formulation. In order to verify the numerical solver, the one-sided lid-driven cavity is studied and the results are compared with relevant data in the literature. They were in a very good agreement. Furthermore, two distinguished unexplored cases of the three-sided lid-driven cavity have been investigated. In case (1), the upper and lower walls are translated to the right, while the left wall is translated upward and the right wall remains stationary. Moreover, in case (2), the upper wall is translated to the right but the lower wall is translated to the left, while the left wall is translated downward and the right wall remains stationary. However, stream function and vorticity values in addition to the location of primary and secondary vortices' centers inside the square cavity have been revealed at low and intermediate Reynolds numbers (Re), typically (Re = 100 to 2000). Moreover, the higher the Re, the more main primary vortex approaches the cavity center and the bigger the secondary vortices. K E Y W O R D Sfinite difference method, incompressible flow, lid-driven cavity, Navier-Stokes equations, stream function-vorticity formulation 1 This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

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The applications of flexible manipulators are increasing and due to the high demand on fuel consumption there is a need to optimize the energy consumption for stable and durable operation of the flexible manipulators. In the present work the Genetic Algorithm (GA) is employed to optimize the total torque and the torque of the first link of a two-link flexible manipulator with a fourth order polynomial trajectory. The mathematical model of the manipulator is obtained using the extended Hamilton's Principle where the flexible links are treated as Euler- Bernoulli's beam theory. A fifth order polynomial trajectory undergoes a rest-to rest maneuvering is proposed as a bench mark for validation.