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We remark on the use of regularized Stokeslets in the slender body theory (SBT) approximation of Stokes flow about a thin fiber of radius ϵ>0. Denoting the regularization parameter by δ, we consider regularized SBT based on the most common regularized Stokeslet plus a regularized doublet correction. Given sufficiently smooth force data along the filament, we derive L∞ bounds for the difference between regularized SBT and its classical counterpart in terms of δ, ϵ, and the force data. We show that the regularized and classical expressions for the velocity of the filament itself differ by a term proportional to log(δ/ϵ); in particular, δ=ϵ is necessary to avoid an O(1) discrepancy between the theories. However, the flow at the surface of the fiber differs by an expression proportional to log(1+δ2/ϵ2), and any choice of δ∝ϵ will result in an O(1) discrepancy as ϵ→0. Consequently, the flow around a slender fiber due to regularized SBT does not converge to the solution of the well-posed slender body PDE which classical SBT is known to approximate. Numerics verify this O(1) discrepancy but also indicate that the difference may have little impact in practice.
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An integral model based on slender body theory, with applications to curved rigid fibers
We propose a novel integral model describing the motion of both flexible and rigid slender fibers in viscous flow and develop a numerical method for simulating dynamics of curved rigid fibers. The model is derived from nonlocal slender body theory (SBT), which approximates flow near the fiber using singular solutions of the Stokes equations integrated along the fiber centerline. In contrast to other models based on (singular) SBT, our model yields a smooth integral kernel which incorporates the (possibly varying) fiber radius naturally. The integral operator is provably negative definite in a nonphysical idealized geometry, as expected from the partial differential equation theory. This is numerically verified in physically relevant geometries. We discuss the convergence and stability of a numerical method for solving the integral equation. The accuracy of the model and method is verified against known models for ellipsoids. Finally, we develop an algorithm for computing dynamics of rigid fibers with complex geometries in the case where the fiber density is much greater than that of the fluid, for example, in turbulent gas-fiber suspensions.
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- Award ID(s):
- 2001959
- PAR ID:
- 10539694
- Publisher / Repository:
- AIP
- Date Published:
- Journal Name:
- Physics of Fluids
- Volume:
- 33
- Issue:
- 4
- ISSN:
- 1070-6631
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract We consider the mapping properties of the integral operator arising in nonlocal slender body theory (SBT) for the model geometry of a straight, periodic filament. It is well known that the classical singular SBT integral operator suffers from high wavenumber instabilities, making it unsuitable for approximating theslender body inverse problem, where the fiber velocity is prescribed and the integral operator must be inverted to find the force density along the fiber. Regularizations of the integral operator must therefore be used instead. Here, we consider two regularization methods: spectral truncation and the‐regularization of Tornberg and Shelley (2004). We compare the mapping properties of these approximations to the underlying partial differential equation (PDE) solution, which for the inverse problem is simply the Stokes Dirichlet problem with data constrained to be constant on cross sections. For the straight‐but‐periodic fiber with constant radius, we explicitly calculate the spectrum of the operator mapping fiber velocity to force for both the PDE and the approximations. We prove that the spectrum of the original SBT operator agrees closely with the PDE operator at low wavenumbers but differs at high frequencies, allowing us to define a truncated approximation with a wavenumber cutoff. For both the truncated and‐regularized approximations, we obtain rigorous‐based convergence to the PDE solution as: A fiber velocity withregularity givesconvergence, while a fiber velocity with at leastregularity yieldsconvergence. Moreover, we determine the dependence of the‐regularized error estimate on the regularization parameter.more » « less
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0041521
