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Laurel Ohm

Publications and source records attributed to Laurel Ohm.

At least 19 recordsLinked to original sources
arXiv2026

Boundary layers and vanishing diffusivity in run-and-tumble models

A notable feature of confined active matter systems is the tendency for motile particles to accumulate near solid boundaries. In various linear models with no-flux boundary conditions, this accumulation is realized through the development of sharp boundary layers at small particle diffusivity $\kappa$. In this paper, we present the first rigorous investigation of nonlinear boundary layers in the context of confined active matter. Specifically, we consider a family of 1D run-and-tumble models with nonlinear advection and tumbling on the half-line $\mathbb{R}_+$. We rigorously prove the vanishing diffusivity limit with quantitative convergence rates. In the limiting system, the boundary mass enters as a new variable which solves a nonlinear ODE, coupled to the PDE through a dynamic boundary condition. Interestingly, the nonlinearity on the boundary at $\kappa = 0$ cannot be obtained without reference to the boundary layer analysis at $\kappa \ll 1$. Numerically, these models exhibit rich behavior, including phase transition and hysteresis in the boundary layer.

math.AP↗
arXiv2026

A nonlocal curve evolution for an immersed elastic filament: global existence and convergence to resistive force theory

We consider a nonlocal curve evolution belonging to a hierarchy of models for the dynamics of an inextensible elastic filament in a 3D Stokes fluid. This model captures the principal part of a full free boundary problem for an elastic filament in Stokes flow. The fluid effects on the filament evolution are encoded in a pseudodifferential force-to-velocity operator which may be regarded as an interpolation between resistive force theory at low wavenumbers and a Stokes boundary value problem at high wavenumbers. Here the curve is considered to be the centerline of a 3D filament with constant cross sectional radius $\epsilon>0$. We show global well-posedness for the curve evolution in the natural energy space. This loosely suggests that the full evolution may be globally well-posed if the large-scale geometry is controlled. Furthermore, we prove convergence to resistive force theory dynamics as $\epsilon\to 0$, which illustrates how resistive force theory emerges from more detailed models.

math.AP↗
arXiv2025

The slender body free boundary problem

We consider the slender body free boundary problem describing the evolution of an inextensible, closed elastic filament immersed in a Stokes fluid in $\mathbb{R}^3$. The filament elasticity is governed by Euler-Bernoulli beam theory, and the coupling between this 1D elasticity law and the surrounding 3D fluid is governed by the slender body Neumann-to-Dirichlet (NtD) map, which treats the filament as a 3D object with constant cross-sectional radius $0<\epsilon\ll1$. This map serves as a mathematical justification for slender body theories wherein such 3D-1D couplings play a central role. We develop a solution theory for the filament evolution under this coupling. Our analysis relies on two main ingredients: (1) an extraction of the principal symbol of the slender body NtD map, from the author's previous work, and (2) a detailed treatment of the tension determination problem for enforcing the inextensibility constraint. Our work provides a mathematical foundation for various computational models in which a slender filament evolves according to a 1D elasticity law in a 3D fluid. This forms a key development in our broader program to place slender body theories on firm theoretical footing.

math.AP↗
arXiv2025

A hierarchy of blood vessel models, Part I: 3D-1D to 1D

We propose and analyze a family of models describing blood perfusion through a tissue surrounding a thin blood vessel. Our goal is to rigorously establish convergence results among 3D-3D Darcy--Stokes, 3D-1D Darcy--Poiseuille, and 1D Green's function methods commonly used to model this process. In Part I, we propose a 3D-1D Darcy--Poiseuille system where the coupling across the permeable vessel surface involves an angle-averaged Neumann boundary condition coupled with a geometrically constrained Robin boundary condition. We show that this model is well-posed and moreover limits to a 1D Green's function model as the maximum vessel radius $\epsilon\to 0$. In the 1D model, the exterior blood pressure is given by an explicit Green's function expression involving the interior blood pressure. The interior pressure satisfies a novel 1D integrodifferential equation in which the integral term incorporates the effects of the exterior pressure and the vessel geometry. Much of this paper is devoted to analyzing this integrodifferential equation. Using the \emph{a priori} bounds obtained here, we show that the solution to the 1D model converges to the 3D-1D solution with a rate proportional to $\epsilon^{1/2}|\log\epsilon|$. In Part II [Ohm \& Strikwerda, arXiv preprint July 2025], we rely on the 1D estimates to show that both the 1D and 3D-1D models converge to a coupled 3D-3D Darcy-Stokes system as $\epsilon\to 0$, thereby establishing a convergence chain among all hierarchy levels.

math.AP↗
arXiv2025

A hierarchy of blood vessel models, Part II: 3D-3D to 3D-1D and 1D

We propose and analyze a hierarchy of three models of blood perfusion through a tissue surrounding a thin arteriole or venule. Our goal is to rigorously link 3D-3D Darcy--Stokes, 3D-1D Darcy--Poiseuille, and 1D Green's function methods commonly used to model this process. Here in Part II, we consider the most detailed level, a 3D-3D Darcy-Stokes system coupled across the permeable vessel surface by mass conservation and pressure/stress balance conditions. We derive a convergence result between the 3D-3D model and both the 3D-1D Darcy--Poiseuille model and 1D Green's function model proposed in Part I [Ohm \& Strikwerda, arXiv preprint July 2025] at a rate proportional to $\epsilon^{1/6}|\log\epsilon|$, where $\epsilon$ is the maximum vessel radius. The rate is limited by the inclusion of a degenerate endpoint where the vessel radius vanishes, i.e. becomes indistinguishable from a capillary. Key to our proof are \emph{a priori} estimates for the 1D integrodifferential model obtained in Part I.

math.AP↗
arXiv2025

Rods in flows: the PDE theory of immersed elastic filaments

We investigate a family of curve evolution equations approximating the motion of a Kirchhoff rod immersed in a low Reynolds number fluid. The rod is modeled as a framed curve whose energy consists of the bending energy of the curve and the twisting energy of the frame. The equations we consider may be realized as gradient flows of the rod energy under a certain anisotropic metric coming from resistive force theory. Ultimately, our goal is to provide a comprehensive treatment of the PDE theory of immersed rod dynamics. We begin by analyzing the problem without the frame, in which case the evolution is globally well-posed and solutions asymptotically converge to Euler elasticae. Next, in the planar setting, we demonstrate the existence of large time-periodic solutions forced by intrinsic curvature relevant to undulatory swimming. Finally, the majority of the paper is devoted to the evolution of an immersed Kirchhoff rod in 3D, which involves a strong coupling between the curve and frame. Under a physically reasonable assumption on the curvature, solutions exist globally-in-time and asymptotically converge to rod equilibria.

math.AP↗
arXiv2024

Arrested development and traveling waves of active suspensions in nematic liquid crystals

Active particles in anisotropic, viscoelastic fluids experience competing stresses which guide their trajectories. An aligned suspension of particles can trigger a hydrodynamic bend instability, but the elasticity of the fluid can drive particle orientations back towards alignment. To study these competing effects, we examine a dilute suspension of active particles in an Ericksen-Leslie model nematic liquid crystal. An anchoring strength linking the active and passive media tunes the system between active suspension theory in Newtonian fluids in one limit, and active nematic theory in another. For extensile active stresses, beyond a critical active Ericksen number or particle concentration, the suspension first comes into alignment, then buckles via a classical bend instability. Rather than entering the fully developed roiling state observed in isotropic fluids, the development is arrested into a steady, flowing state by fluid elasticity. Arrested states of higher wavenumber appear at yet larger extensile activity, and a phase transition is identified at finite anchoring strength. If the active particles are motile, the particles can surf along the bent environment of their own creation. More exotic states are also observed, including a oscillatory 'thrashing' mode. Moment equations are derived, compared to kinetic theory simulations, and analyzed in asymptotic limits which admit exact expressions for the traveling wave speed and both particle orientation and director fields in the first arrested state.

cond-mat.soft↗
arXiv2024

A free boundary problem for an immersed filament in 3D Stokes flow

We consider a simplified extensible version of a dynamic free boundary problem for a thin filament with radius $\epsilon>0$ immersed in 3D Stokes flow. The 3D fluid is coupled to the quasi-1D filament dynamics via a novel type of angle-averaged Neumann-to-Dirichlet operator for the Stokes equations, and much of the difficulty in the analysis lies in understanding this operator. Here we show that the main part of this angle-averaged NtD map about a closed, curved filament is the corresponding operator about a straight filament, for which we can derive an explicit symbol. Remainder terms due to curvature are lower order with respect to regularity or size in $\epsilon$. Using this operator decomposition, it is then possible to show that the simplified free boundary evolution is a third-order parabolic equation and is locally well posed. This establishes a more complete mathematical foundation for the myriad computational results based on slender body approximations for thin immersed elastic structures.

math.AP↗
arXiv2023

On an angle-averaged Neumann-to-Dirichlet map for thin filaments

We consider the Laplace equation in the exterior of a thin filament in $\mathbb{R}^3$ and perform a detailed decomposition of a notion of slender body Neumann-to-Dirichlet (NtD) and Dirichlet-to-Neumann (DtN) maps along the filament surface. The decomposition is motivated by a filament evolution equation in Stokes flow for which the Laplace setting serves as an important toy problem. Given a general curved, closed filament with constant radius $\epsilon>0$, we show that both the slender body DtN and NtD maps may be decomposed into the corresponding operator about a straight, periodic filament plus lower order remainders. For the straight filament, both the slender body NtD and DtN maps are given by explicit Fourier multipliers and it is straightforward to compute their mapping properties. The remainder terms are lower order in the sense that they are small with respect to $\epsilon$ or smoother. While the strategy here is meant to serve as a blueprint for the Stokes setting, the Laplace problem may be of independent interest.

math.AP↗
arXiv2022

Well-posedness of a viscoelastic resistive force theory and applications to swimming

We propose and analyze a simple model for the evolution of an immersed, inextensible filament which incorporates linear viscoelastic effects of the surrounding fluid. The model is a closed-form system of equations along the curve only which includes a `memory' term due to viscoelasticity. For a planar filament, given a forcing in the form of a preferred curvature, we prove well-posedness of the fiber evolution as well as the existence of a unique time-periodic solution in the case of time-periodic forcing. Moreover, we obtain an expression for the swimming speed of the filament in terms of the preferred curvature. The swimming speed depends in a complicated way on the viscoelastic parameters corresponding to the fluid relaxation time and additional polymeric viscosity. We study this expression in detail, accompanied by numerical simulations, and show that this simple model can capture complex effects of viscoelasticity on swimming. In particular, the viscoelastic swimmer is shown to be faster than its Newtonian counterpart in some situations and slower in others. Strikingly, we even find an example where viscoelastic effects may lead to a reversal in swimming direction from the Newtonian setting, although this occurs when the displacement for both the Newtonian and viscoelastic swimmers is practically negligible.

math.AP↗
arXiv2022

Well-posedness and applications of classical elastohydrodynamics for a swimming filament

We consider a classical elastohydrodynamic model of an inextensible filament undergoing planar motion in $\mathbb{R}^3$. The hydrodynamics are described by resistive force theory, and the fiber elasticity is governed by Euler-Bernoulli beam theory. Our aim is twofold: (1) Serve as a starting point for developing the mathematical analysis of filament elastohydrodynamics, particularly the analytical treatment of an inextensibility constraint, and (2) As an application, prove conditions on internal fiber forcing that allow a free-ended filament to swim. Our analysis of fiber swimming speed is supplemented with a numerical optimization of the internal fiber forcing, as well as a novel numerical method for simulating an inextensible swimmer.

math.AP↗
arXiv2022

On the stabilizing effect of swimming in an active suspension

We consider a kinetic model of an active suspension of rod-like microswimmers. In certain regimes, swimming has a stabilizing effect on the suspension. We quantify this effect near homogeneous isotropic equilibria $\overline{\psi} = \text{const}$. Notably, in the absence of particle (translational and orientational) diffusion, swimming is the only stabilizing mechanism. On the torus, in the non-diffusive regime, we demonstrate linear Landau damping up to the stability threshold predicted in the applied literature. With small diffusion, we demonstrate nonlinear stability of arbitrary equilibrium values for pullers (front-actuated swimmers) and enhanced dissipation for both pullers and pushers (rear-actuated swimmers) at small concentrations. On the whole space, we prove nonlinear stability of the vacuum equilibrium due to generalized Taylor dispersion.

math.AP↗
arXiv2022

Weakly nonlinear analysis of pattern formation in active suspensions

We consider the Saintillan--Shelley kinetic model of active rodlike particles in Stokes flow (Saintillan & Shelley 2008a,b), for which the uniform, isotropic suspension of pusher particles is known to be unstable in certain settings. Through weakly nonlinear analysis accompanied by numerical simulations, we determine exactly how the isotropic steady state loses stability in different parameter regimes. We study each of the various types of bifurcations admitted by the system, including both subcritical and supercritical Hopf and pitchfork bifurcations. Elucidating this system's behavior near these bifurcations provides a theoretical means of comparing this model with other physical systems which transition to turbulence, and makes predictions about the nature of bifurcations in active suspensions that can be explored experimentally.

physics.flu-dyn↗
arXiv2021

A single-layer based numerical method for the slender body boundary value problem

Fluid flows containing dilute or dense suspensions of thin fibers are widespread in biological and industrial processes. To describe the motion of a thin immersed fiber, or to describe the forces acting on it, it is convenient to work with one-dimensional fiber centerlines and force densities rather than two-dimensional surfaces and surface tractions. Slender body theories offer ways to model and simulate the motion of immersed fibers using only one-dimensional data. However, standard formulations can break down when the fiber surface comes close to intersecting itself or other fibers. In this paper we introduce a numerical method for a recently derived three-dimensional slender body boundary value problem that can be stated entirely in terms of a one-dimensional distribution of forces on the centerline. The method is based on a new completed single-layer potential formulation of fluid velocity which removes the nullspace associated with the unmodified single layer potential. We discretize the model and present numerical results demonstrating the good conditioning and improved performance of the method in the presence of near-intersections. To avoid the modeling and numerical choices involved with free ends, we consider closed fibers.

math.NA↗
arXiv2021

Remarks on regularized Stokeslets in slender body theory

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 $L^\infty$ 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 $\log(1+δ^2/ε^2)$, and any choice of $δ\proptoε$ will result in an $O(1)$ discrepancy as $ε\to 0$. Consequently, the flow around a slender fiber due to regularized SBT does not converge to the solution of the well-posed \emph{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.

physics.flu-dyn↗
arXiv2020

An integral model based on slender body theory, with applications to curved rigid fibers

We propose a novel integral model describing the motion of curved slender fibers in viscous flow, and develop a numerical method for simulating dynamics of 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 non-physical idealized geometry, as expected from PDE theory. This is numerically verified in physically relevant geometries. We propose a convergent numerical method for solving the integral equation and discuss its convergence and stability. The accuracy of the model and method is verified against known models for ellipsoids. Finally, a fast algorithm for computing dynamics of rigid fibers with complex geometries is developed.

physics.flu-dyn↗
arXiv2020

Accuracy of slender body theory in approximating force exerted by thin fiber on viscous fluid

We consider the accuracy of slender body theory in approximating the force exerted by a thin fiber on the surrounding viscous fluid when the fiber velocity is prescribed. We term this the slender body inverse problem, as it is known that slender body theory converges to a well-posed PDE solution when the force is prescribed and the fiber velocity is unknown. From a PDE perspective, the slender body inverse problem is simply the Dirichlet problem for the Stokes equations, but from an approximation perspective, nonlocal slender body theory exhibits high wavenumber instabilities which complicate analysis. Here we consider two methods for regularizing the slender body approximation: spectral truncation and the $δ$-regularization of Tornberg and Shelley (2004). For a straight, periodic fiber with constant radius $ε>0$, we explicitly calculate the spectrum of the operator mapping fiber velocity to force for both the PDE and the approximations. We show that the spectrum of the original slender body approximation agrees closely with the PDE solution at low wavenumbers but differs at high frequencies, allowing us to define a truncated approximation with a wavenumber cutoff $\sim1/ε$. For both the truncated and $δ$-regularized approximations, we obtain similar convergence results to the PDE solution as $ε\to0$: a fiber velocity with $H^1$ regularity gives $O(ε)$ convergence, while a fiber velocity with at least $H^2$ regularity yields $O(ε^2)$ convergence. Moreover, we determine the dependence of the $δ$-regularized error estimate on the regularization parameter $δ$.

math.AP↗