New integral equation formulations for steady and unsteady flow problems of an incompressible viscous fluid are presented. The so-called direct approach in which the velocity vector and the pressure are inclued as unknowns is employed in this paper. The nonlinear boundary value, and the initial-boundary value problems described with the Navier-Stokes equations are transformed into integral equations by the method of weighted residuals. Fundamental solutions of the Stokes approximate equations are used as the weight function. The fundamental solution tensors are presented for the steady-state and unsteady-state problems. For the unsteady-state problem, we derive not only the time-dependent fundamental solution tensor but also the one using the finite difference approximation for the time derivative. A numerical example of the two-dimensional driven cavity flow is given to show the validity and effectiveness of the method.
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