SynopsisWe study the time-dependent compressible flow of a Newtonian fluid in slits using an arbitrary nonlinear slip law relating the shear stress to the velocity at the wall. This slip law exhibits a maximum and a minimum and so does the flow curve. According to one-dimensional stability analyses, the steady-state solutions are unstable if the slope of the flow curve is negative. The two-dimensional flow problem is solved using finite elements for the space discretization and a standard fully implicit scheme for the time discretization. When compressibility is taken into account and the volumetric flow rate at the inlet is in the unstable regime, we obtain self-sustained oscillations of the pressure drop and of the mass flow rate at the exit, similar to those observed with the stick-slip instability. The effects of compressibility and of the length of the slit on the amplitude and the frequency of the oscillations are also examined.
In this work, the use of the Method of Fundamental Solutions (MFS) for solving elliptic partial differential equations is investigated, and the performance of various least squares routines used for the solution of the resulting minimization problem is studied. Two modi®ed versions of the MFS for harmonic and biharmonic problems with boundary singularities, which are based on the direct subtraction of the leading terms of the singular local solution from the original mathematical problem, are also examined. Both modi®ed methods give more accurate results than the standard MFS and also yield the values of the leading singular coef®cients. Moreover, one of them predicts the form of the leading singular term.
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