arXiv subjects

Laurel Ohm

Publications and source records attributed to Laurel Ohm.

22 records · Page 2Linked to original sources

arXiv2020

An error bound for the slender body approximation of a thin, rigid fiber sedimenting in Stokes flow

We investigate the motion of a thin rigid body in Stokes flow and the corresponding slender body approximation used to model sedimenting fibers. In particular, we derive a rigorous error bound comparing the rigid slender body approximation to the classical PDE for rigid motion in the case of a closed loop with constant radius. Our main tool is the slender body PDE framework established by the authors and D. Spirn in [18,19], which we adapt to the rigid setting.

math.AP↗

arXiv2019

A slender body model for thin rigid fibers: validation and comparisons

In this paper we consider a computational model for the motion of thin, rigid fibers in viscous flows based on slender body theory. Slender body theory approximates the fluid velocity field about the fiber as the flow due to a distribution of singular solutions to the Stokes equations along the fiber centerline. The velocity of the fiber itself is often approximated by an asymptotic limit of this expression. Here we investigate the efficacy of simply evaluating the slender body velocity expression on a curve along the surface of the actual 3D fiber, rather than limiting to the fiber centerline. Doing so may yield an expression better suited for numerical simulation. We validate this model for two simple geometries, namely, thin ellipsoids and thin rings, and we compare the model to results in the literature for constant and shear flow. In the case of a fiber with straight centerline, the model coincides with the prolate spheroid model of Jeffery. For the thin torus, the computed force agrees with the asymptotically accurate values of Johnson and Wu and gives qualitatively similar dynamics to oblate spheroids of similar size and inertia

physics.flu-dyn↗

arXiv2019

Theoretical justification and error analysis for slender body theory

Slender body theory facilitates computational simulations of thin fibers immersed in a viscous fluid by approximating each fiber using only the geometry of the fiber centerline curve and the line force density along it. However, it has been unclear how well slender body theory actually approximates Stokes flow about a thin but truly three-dimensional fiber, in part due to the fact that simply prescribing data along a one-dimensional curve does not result in a well-posed boundary value problem for the Stokes equations in $\mathbb{R}^3$. Here, we introduce a PDE problem to which slender body theory (SBT) provides an approximation, thereby placing SBT on firm theoretical footing. The slender body PDE is a new type of boundary value problem for Stokes flow where partial Dirichlet and partial Neumann conditions are specified everywhere along the fiber surface. Given only a 1D force density along a closed fiber, we show that the flow field exterior to the thin fiber is uniquely determined by imposing a fiber integrity condition: the surface velocity field on the fiber must be constant along cross sections orthogonal to the fiber centerline. Furthermore, a careful estimation of the residual, together with stability estimates provided by the PDE well-posedness framework, allow us to establish error estimates between the slender body approximation and the exact solution to the above problem. The error is bounded by an expression proportional to the fiber radius (up to logarithmic corrections) under mild regularity assumptions on the 1D force density and fiber centerline geometry.

math.AP↗

arXiv2019

Theoretical justification and error analysis for slender body theory with free ends

Slender body theory is a commonly used approximation in computational models of thin fibers in viscous fluids, especially in simulating the motion of cilia or flagella in swimming microorganisms. In [23], we developed a PDE framework for analyzing the error introduced by the slender body approximation for closed-loop fibers with constant radius $ε$, and showed that the difference between our closed-loop PDE solution and the slender body approximation is bounded by an expression proportional to $ε|\logε|$. Here we extend the slender body PDE framework to the free endpoint setting, which is more physically relevant from a modeling standpoint but more technically demanding than the closed loop analysis. The main new difficulties arising in the free endpoint setting are defining the endpoint geometry, identifying the extent of the 1D slender body force density, and determining how the well-posedness constants depend on the non-constant fiber radius. Given a slender fiber satisfying certain geometric constraints at the filament endpoints and a one-dimensional force density satisfying an endpoint decay condition, we show a bound for the difference between the solution to the slender body PDE and the slender body approximation in the free endpoint setting. The bound is a sum of the same $ε|\logε|$ term appearing in the closed loop setting and an endpoint term proportional to $ε$, where $ε$ is now the maximum fiber radius.

math.AP↗