No abstract

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 1D curve does not result in a well-posed boundary value problem for the Stokes equations in 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, allows 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.

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 , and showed that the difference between our closed-loop PDE solution and the slender body approximation is bounded by an expression proportional to | log |. 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 | log | term appearing in the closed loop setting and an endpoint term proportional to , where is now the maximum fiber radius.Contents *

We prove various estimates that relate the Ginzburg-Landau energy E ε (u) = Ω |∇u| 2 2 + (|u| 2 −1) 2 4ε 2 dx of a function u ∈ H 1 (Ω; R 2), Ω ⊂ R 2 , to the distance in the W −1,1 norm the Jacobian J(u) = det ∇u and a sum of point masses. These are interpreted as quantifying the precision with which "vortices" in a function u can be located via measure-theoretic tools such as the Jacobian; and the extent to which variations in the Ginzburg-Landau energy due to translation of vortices can be detected using the Jacobian. We give examples to show that some of our estimates are close to optimal.

We consider the problem of a one-dimensional elastic filament immersed in a two-dimensional steady Stokes fluid. Immersed boundary problems in which a thin elastic structure interacts with a surrounding fluid are prevalent in science and engineering, a class of problems for which Peskin has made pioneering contributions. Using boundary integrals, we first reduce the fluid equations to an evolution equation solely for the immersed filament configuration. We then establish local well-posedness for this equation with initial data in low-regularity Hölder spaces. This is accomplished by first extracting the principal linear evolution by a small-scale decomposition and then establishing precise smoothing estimates on the nonlinear remainder. Higher regularity of these solutions is established via commutator estimates with error terms generated by an explicit class of integral kernels. Furthermore, we show that the set of equilibria consists of uniformly parametrized circles and prove nonlinear stability of these equilibria with explicit exponential decay estimates, the optimality of which we verify numerically. Finally, we identify a quantity that respects the symmetries of the problem and controls global-in-time behavior of the system.