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Exploration of Local Force Calculations Using the Methods of Regularized Stokeslets

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We analyze the performance of the Method of Regularized Stokeslets (MRS) and the Method of Auxiliary Regularized Stokeslets (MARS) in computing the forces necessary to translate a sphere with unit velocity in Stokes flow. In particular, we explore the dependence of local and global force calculations on various parameters associated with each method. The parameters we varied include the regularization parameter, the discretization of the sphere, and the spread and placement of the auxiliary Stokeslets (MARS only). One challenge when using the MRS is that there is no systematic way to choose the regularization parameter, and the error is sensitive to this choice. In the literature it is stated that, compared to the MRS, the MARS weakens the error dependence on the choice of regularization parameter. We found this to be true in some cases, but not in others. Specifically, the dependence is weakened when comparing the 1-norm of the global force error and 2-norm of the local force error. This behavior is not seen with the max-norm of the local force error. In addition, with the MARS, there is a strong dependence of error on the normalized patch length which controls the spread of the auxiliary Stokeslets. We find the conditions needed to optimize the MARS method to outperform the MRS on the translating sphere problem.

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