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Daniel Spirn - One of the best experts on this subject based on the ideXlab platform.
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Theoretical Justification and Error Analysis for Slender Body Theory with Free Ends
Archive for Rational Mechanics and Analysis, 2019Co-Authors: Yoichiro Mori, Laurel Ohm, Daniel SpirnAbstract: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 Mori et al. (Commun Pure Appl Math, 2018. arXiv:1807.00178 ), we developed a PDE framework for analyzing the error introduced by the Slender Body approximation for closed-loop fibers with constant radius $$\varepsilon $$ ε , and showed that the difference between our closed-loop PDE solution and the Slender Body approximation is bounded by an expression proportional to $$\varepsilon |\log \varepsilon |$$ ε | log ε | . 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 $$\varepsilon |\log \varepsilon |$$ ε | log ε | term appearing in the closed loop setting and an endpoint term proportional to $$\varepsilon $$ ε , where $$\varepsilon $$ ε is now the maximum fiber radius.
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Theoretical justification and error analysis for Slender Body Theory with free ends
Archive for Rational Mechanics and Analysis, 2019Co-Authors: Yoichiro Mori, Laurel Ohm, Daniel SpirnAbstract: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 $\epsilon$, and showed that the difference between our closed-loop PDE solution and the Slender Body approximation is bounded by an expression proportional to $\epsilon|\log\epsilon|$. 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 $\epsilon|\log\epsilon|$ term appearing in the closed loop setting and an endpoint term proportional to $\epsilon$, where $\epsilon$ is now the maximum fiber radius.
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theoretical justification and error analysis for Slender Body Theory
arXiv: Analysis of PDEs, 2018Co-Authors: Yoichiro Mori, Daniel SpirnAbstract: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.
Daniel R. Givan - One of the best experts on this subject based on the ideXlab platform.
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A Wave Resistance Formulation for Slender Bodies With Fine Bow Shapes
29th International Conference on Ocean Offshore and Arctic Engineering: Volume 3, 2010Co-Authors: Brandon M. Taravella, William S. Vorus, Daniel R. GivanAbstract:Ogilvie (1972) investigated a Green’s function method for calculating the wave profile of Slender ships with fine bows. He recognized that near a Slender ship’s bow, rates of change of flow variables axially should be greater than those typically assumed in Slender Body Theory. Ogilvie’s (1972) result is still a Slender Body Theory in that the rates of change in the near field are different transversely (a half-order) than axially; however, the difference in order of magnitude between them is less than in the usual Slender Body Theory. Typical of Slender Body Theory, this formulation results in a downstream stepping solution (along the ship’s length) in which downstream effects are not reflected upstream. Ogilvie (1972), however, only developed a solution for wedge-shaped bodies. In this paper, a general solution for arbitrary Slender ships has been developed based on Ogilvie’s (1972) formulation. The results for wave resistance have been calculated and are compared to previously published model test data and calculated results. The free surface profile has also been calculated.Copyright © 2010 by ASME
Brandon M. Taravella - One of the best experts on this subject based on the ideXlab platform.
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A Wave Resistance Formulation for Slender Bodies at Moderate to High Speeds
Journal of Ship Research, 2012Co-Authors: Brandon M. Taravella, William S. VorusAbstract:T. Francis Ogilvie (1972) developed a Green's function method for calculating the wave profile of Slender ships with fine bows. He recognized that near a Slender ship's bow, rates of change of flow variables axially should be greater than those typically assumed in Slender Body Theory. Ogilvie's result is still a Slender Body Theory in that the rates of change in the near field are different transversely (a half-order different) than axially; however, the difference in order of magnitude between them is less than in the usual Slender Body Theory. Typical of Slender Body Theory, this formulation results in a downstream stepping solution (along the ship's length) in which downstream effects are not reflected upstream. Ogilvie, however, developed a solution only for wedge-shaped bodies. Taravella, Vorus, and Givan (2010) developed a general solution to Ogilvie's formulation for arbitrary Slender ships. In this article, the general solution has been expanded for use on moderate to high-speed ships. The wake trench has been accounted for. The results for wave resistance have been calculated and are compared with previously published model test data.
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A Wave Resistance Formulation for Slender Bodies With Fine Bow Shapes
29th International Conference on Ocean Offshore and Arctic Engineering: Volume 3, 2010Co-Authors: Brandon M. Taravella, William S. Vorus, Daniel R. GivanAbstract:Ogilvie (1972) investigated a Green’s function method for calculating the wave profile of Slender ships with fine bows. He recognized that near a Slender ship’s bow, rates of change of flow variables axially should be greater than those typically assumed in Slender Body Theory. Ogilvie’s (1972) result is still a Slender Body Theory in that the rates of change in the near field are different transversely (a half-order) than axially; however, the difference in order of magnitude between them is less than in the usual Slender Body Theory. Typical of Slender Body Theory, this formulation results in a downstream stepping solution (along the ship’s length) in which downstream effects are not reflected upstream. Ogilvie (1972), however, only developed a solution for wedge-shaped bodies. In this paper, a general solution for arbitrary Slender ships has been developed based on Ogilvie’s (1972) formulation. The results for wave resistance have been calculated and are compared to previously published model test data and calculated results. The free surface profile has also been calculated.Copyright © 2010 by ASME
Donald L Koch - One of the best experts on this subject based on the ideXlab platform.
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Slender Body Theory for particles with non-circular cross-sections
Journal of Fluid Mechanics, 2019Co-Authors: Neeraj Sinai Borker, Donald L KochAbstract:This paper presents a Theory to obtain the force per unit length acting on a Slender filament with a non-circular cross-section moving in a fluid at low Reynolds number. Using a regular perturbation of the inner solution, we show that the force per unit length has $O(1/ln(2A))$ + $O(\alpha/ln^2(2A))$ contributions driven by the relative motion of the particle and the local fluid velocity and an $O(\alpha/(ln(2A)A))$ contribution driven by the gradient in the imposed fluid velocity. Here, the aspect ratio ($A=l/a_0$) is defined as the ratio of the size of the particle ($l$) and the cross-sectional dimension ($a_0$); and $\alpha$ is the amplitude of the non-circular perturbation. Using thought experiments, we show that two-lobed and three-lobed cross-sections affect the response to relative motion and velocity gradients, respectively. A two-dimensional Stokes flow calculation is used to extend the perturbation analysis to cross-sections that deviate significantly from a circle (i.e., $\alpha \sim O(1)$). We demonstrate the ability of our method to accurately compute the resistance to translation and rotation of a Slender triaxial ellipsoid. Furthermore, we illustrate novel dynamics of straight rods in a simple shear flow that translate and rotate quasi-periodically if they have two-lobed cross-section; and rotate chaotically and translate diffusively if they have a combination of two- and three-lobed cross-sections. Finally, we show the remarkable ability of our Theory to accurately predict the motion of rings, retaining great accuracy for moderate aspect ratios ($\sim 10$) and cross-sections that deviate significantly from a circle, thereby making our Theory a computationally inexpensive alternative to other Stokes flow solvers.
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Slender Body Theory for transient heat conduction theoretical basis numerical implementation and case studies
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2015Co-Authors: Koenraad F Beckers, Donald L Koch, Jefferson W TesterAbstract:Slender-Body Theory (SBT) for transient heat transfer from bodies whose lengths are much larger than their radius into a conductive medium is derived. SBT uses matched asymptotic expansions of inner and outer solutions. An analytical inner solution for heat transfer from a circular cross section is matched to an outer solution obtained using Green’s functions. An efficient numerical implementation is obtained based on a judicious choice of the discrete elements used to represent the Body and implementation of the fast multipole method (FMM). The SBT requires a one-dimensional spatial discretization only along the axis of the Body in contrast to the three-dimensional discretization for finite-element models. Two case studies, heat transfer from two parallel cylinders and heat transfer from a slinky-coil heat exchanger, are used to show the speed and accuracy of the SBT model and its ability to model interacting Slender bodies of finite length and bodies with centreline curvature and internal advective heat flow.
Yoichiro Mori - One of the best experts on this subject based on the ideXlab platform.
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Accuracy of Slender Body Theory in approximating force exerted by thin fiber on viscous fluid
arXiv: Analysis of PDEs, 2020Co-Authors: Yoichiro Mori, Laurel OhmAbstract: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 $\delta$-regularization of Tornberg and Shelley (2004). For a straight, periodic fiber with constant radius $\epsilon>0$, 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 $\sim1/\epsilon$. For both the truncated and $\delta$-regularized approximations, we obtain similar convergence results to the PDE solution as $\epsilon\to0$: a fiber velocity with $H^1$ regularity gives $O(\epsilon)$ convergence, while a fiber velocity with at least $H^2$ regularity yields $O(\epsilon^2)$ convergence. Moreover, we determine the dependence of the $\delta$-regularized error estimate on the regularization parameter $\delta$.
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Theoretical Justification and Error Analysis for Slender Body Theory with Free Ends
Archive for Rational Mechanics and Analysis, 2019Co-Authors: Yoichiro Mori, Laurel Ohm, Daniel SpirnAbstract: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 Mori et al. (Commun Pure Appl Math, 2018. arXiv:1807.00178 ), we developed a PDE framework for analyzing the error introduced by the Slender Body approximation for closed-loop fibers with constant radius $$\varepsilon $$ ε , and showed that the difference between our closed-loop PDE solution and the Slender Body approximation is bounded by an expression proportional to $$\varepsilon |\log \varepsilon |$$ ε | log ε | . 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 $$\varepsilon |\log \varepsilon |$$ ε | log ε | term appearing in the closed loop setting and an endpoint term proportional to $$\varepsilon $$ ε , where $$\varepsilon $$ ε is now the maximum fiber radius.
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Theoretical justification and error analysis for Slender Body Theory with free ends
Archive for Rational Mechanics and Analysis, 2019Co-Authors: Yoichiro Mori, Laurel Ohm, Daniel SpirnAbstract: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 $\epsilon$, and showed that the difference between our closed-loop PDE solution and the Slender Body approximation is bounded by an expression proportional to $\epsilon|\log\epsilon|$. 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 $\epsilon|\log\epsilon|$ term appearing in the closed loop setting and an endpoint term proportional to $\epsilon$, where $\epsilon$ is now the maximum fiber radius.
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theoretical justification and error analysis for Slender Body Theory
arXiv: Analysis of PDEs, 2018Co-Authors: Yoichiro Mori, Daniel SpirnAbstract: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.