The present formulation is here mathematically analysed based on Scheurer's theorem, . Stability conditions and error estimates are established when the Sisko relation is considered. Numerical examples are presented to confirm the stability analysis.
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It can be seen that from continuous classical Galerkin formulation associated with problem 1 , we can obtain. To generate the stabilized finite element method proposed here, the following definitions will be used. Remark 3. In this work, to obtain velocity and pressure approximations to problem 1 , we define the following variational form, constructed by adding the least squares of the linear momentum and of the continuity equations to the Galekin formulation, generating the following problem with homogeneous boundary condition considered without lost of generalities:. Problem PG hd.
The finite element analysis is developed here by considering solutions in Hilbert Spaces, as in . We start the analysis by rewriting the discontinuous pressure approximation p h as.
The discontinuous pressure allows the satisfaction of the incompressibility constraint at element level in contrast to the continuous approximations, which satisfies the constraint only in global sense. Lemma 4. Assuming the same considerations of Lemma 4. From the triangular inequality, the Lemma 4.
The lemma is obtained as a consequence of. Definition 4. In other hand, from the definition of U h , from the inverse estimate 12 and from the inverse estimate for the pressure, see ,. Using the classical inequality,. With the above results we can establish the following result that will be needed later to generate the estimates in Theorem 4. Theorem 4. By the consistency of the problem and from 4 we have.
From the continuity of the two first terms in the right hand side above, , and using the inequality 14 we have. By using Lemma 4. From Lemma 4. The inequality 16 gives a sufficient condition, providing a useful relation to be used to choose the stabilizing parameters.
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With the above results, we can establish the following error approximation estimates. From Theorem 4. In order to obtain an estimate to - h 0 , we note that from PG hd problem, we have. In order to obtain numerical results for the finite element method presented here to nonlinear problem, the following numerical algorithm will be used. There are many methods to solve nonlinear equations. In this case, we lag nonlinear terms in the system of equations and start with an initial guess generating a sequence of functions that is expected to converge for the solution.
For the convergence of the algorithm, we imposed a tolerence of 10 We note magnitude decreasing in the velocity and in the pressure fields and also the flatteness of the pressure next to the two corners 0, 1 and 1, 1 due to the pseudoplastic effect, as expected. Formulations that use the continuous pressure interpolations may present lack of accuracy in those regions, where critical boundary conditions exist.
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Unlike, discontinuous pressure interpolations, as is the case here, recover the accuracy at those regions, due to the satisfaction of the incompressibility constraint locally. It can be seen that the greater is the non-Newtonian effect, the larger is the number of iteractions required to achieve convergence, as expected for a fixed mesh. In this work, a consistent stabilized mixed Petrov-Galerkin-like finite element formulation in primitive variables, with continuous velocity and discontinuous pressure interpolations, has been mathematically analyzed for flows governed by the nonlinear Sisko relation.
To generate the mathematical stability conditions, it was possible to split the discontinuous pressure. Only the constant by part pressure resulted as responsible to fulfill the LBB condition.
The other part, the null mean pressure part, contributed to achieve the required ellipticity in the Scheurer's theorem sense together with the stabilizing terms. For this formulation ellipticity was the key for the stability, since the constant part of the pressure fulfills in standard ways the LBB. It was possible from the ellipticity to provide a sufficient condition to choose the stabilizing parameters not only as a function of the quasi-newtonian constitutive parameter but considering both constitutive constants coming from the Sisko relation.
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