Propulsion of microorganisms by a helical flagellum

We study the propulsion matrix of bacterial flagella numerically using slender body theory and the regularized Stokeslet method in a biologically relevant parameter regime. All three independent elements of the matrix are measured by computing propulsive force and torque generated by a rotating flagellum, and the drag force on a translating flagellum. Numerical results are compared with the predictions of resistive force theory, which is often used to interpret micro-organism propulsion. Neglecting hydrodynamic interactions between different parts of a flagellum in resistive force theory leads to both qualitative and quantitative discrepancies between the theoretical prediction of resistive force theory and the numerical results. We improve the original theory by empirically incorporating the effects of hydrodynamic interactions and propose new expressions for propulsive matrix elements that are accurate over the parameter regime explored.

Flagellar hydrodynamics. A comparison between resistive-force theory and slender-body theory

*# A method for analyzing cambered bodies has been incorporated into Missile Datcom. Existing methods based on mean line theory, slender body theory, viscous crossflow theory were assessed. They were compared with wind tunnel data from subsonic to supersonic speeds for cambered bodies with axi-symmetric cross-sections. The comparison cases consisted of three configurations with discrete camber changes (deflectable nose missile) and twelve configurations with continuous camber changes. A combination of slender body and viscous crossflow theory was found to provide the best agreement with experimental data. Nomenclature

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 ℝ3. 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. © 2019 Wiley Periodicals, Inc.

Study on the hydrodynamics of the helix with non-uniform helical radius in viscoelastic fluid at low Reynolds number

본 논문에서는 박테리아 편모를 모사한 스프링 모델을 이용하여 박테리아의 편모의 추진적 거동에 관한 연구를 수행하였다. 본 해석에서는 상용프로그램을 사용하였으며, 별도의 회전영역 설정에 따른 수치기법의 타당성 확인과 더불어 파라미터 연구를 수행하였다. 수치해석 결과는 전반적으로 Resistive force theory와는 잘 일치하지 않았지만, Slender body theory와는 잘 일치하였다. 그리고 스프링의 회전속도, 피치, 나선반경 및 유체의 점성의 영향을 확인하였다. 또한 벽과의 거리에 따른 효과도 분석하였다.

Micro-swimmers with hydrodynamic interactions

Compressible flow past thin wings and slender bodies are studied. First, subsonic flow past thin wings are analyzed by means of potential flow theory. The kernel function method is introduced for arbitrary planforms undergoing simple harmonic oscillations. The spanwise polynomial approximation and the chordwise trigonometric function approximations which automatically satisfy the Kutta condition and inherit the leading edge singularity are used. The Doublet-Lattice method as a more general numerical approach is given for the analysis of the flow past additional surfaces like tail or store surfaces, which are not necessarily in plane with the wing surface or surfaces having spanwise deflections. Airfoil response to the arbitrary unsteady motion is also given for subsonic flows. A brief review of shock waves and Mach waves in a supersonic flow is given. Afterwards, unsteady supersonic potential flow is studied for a simple harmonically oscillating point source. First, unsteady flow past an airfoil is considered. Then, supersonic flow about thin wings are analyzed using the Mach box technique. Introduction to supersonic kernel function method is briefly provided. Arbitrary unsteady motion of an airfoil in a supersonic flow is presented. Slender body theory is introduced to analyze the cross flow past bodies of considerable fineness where the cross flow is shown to be approximately incompressible under certain conditions. In connection with the slender body approximation, the Munk’s airship theory is utilized to predict the stability derivatives of missile like bodies.

Geometry of locomotion of the generalized Purcell’s swimmer: Modelling, controllability and motion primitives

In this paper, we analyze the inverse dynamics and control of a bacteria-inspired uniflagellar robot in a fluid medium at low Reynolds number. Inspired by the mechanism behind the locomotion of flagellated bacteria, we consider a robot comprising a flagellum—a flexible helical filament—connected to a spherical head. The flagellum rotates about the head at a controlled angular velocity and generates a propulsive force that moves the robot forward. When the angular velocity exceeds a threshold value, the hydrodynamic force exerted by the fluid can cause the soft flagellum to buckle, characterized by a dramatic change in its shape. In this computational study, a fluid–structure interaction model that combines Discrete Elastic Rods algorithm with Lighthill's Slender Body Theory is employed to simulate the locomotion and deformation of the robot. We demonstrate that the robot can follow a prescribed path in three-dimensional space by exploiting buckling of the flagellum. The control scheme involves only a single (binary) scalar input—the angular velocity of the flagellum. By triggering the buckling instability at the right moment, the robot can follow the path in three-dimensional space. We also show that the complexity of the dynamics of the helical filament can be captured using a deep neural network, from which we identify the input–output functional relationship between the control input and the trajectory of the robot. Furthermore, our study underscores the potential role of buckling in the locomotion of natural bacteria.

We study the effects of hydrodynamic interactions between a wall and the Purcell three-link swimmer in the two-dimensional case. After deriving the equations of motion in a low Reynolds number regime using Resistive Force Theory with suitably modified drag coefficients, we show, by means of criteria from Geometric Control Theory, that the system is controllable at configurations that are nearly parallel to the wall. Furthermore, we study configurations that are tilted, and we show net displacement with respect to the initial orientation. Some numerical experiments illustrate the analytical results.

A model of feeding currents in encrusting bryozoans shows interference between zooids within a colony

Turbulent heat transfer in channel flows is an important area of research due to its simple geometry and diverse industrial applications. Reynolds-Averaged Navier–Stokes (RANS) models are the most-affordable simulation methodology and are often the only viable choice for investigating industrial flows. However, accurate modelling of wall-bounded flows is challenging in RANS, and the assessment of the performance of RANS models for heated turbulent channel flow has not been sufficiently investigated for a wide range of Reynolds and Prandtl numbers. In this study, five RANS models are assessed for their ability to predict heat transfer in channel flows across a wide range of Reynolds and Prandtl numbers (Pr) by comparing the RANS results with respect to the corresponding Direct Numerical Simulation data. The models include three Eddy Viscosity Models (EVMs): standard k−ϵ, low Reynolds number k−ϵLS, and k−ωSST, as well as two Reynolds Stress Models (RSMs): Launder–Reece–Rodi and Speziale–Sarkar–Gatski models. The study analyses the Reynolds number effects on turbulent heat transfer in a channel flow at a Pr of 0.71 for friction Reynolds number values of 180,395,640, and 1020. The results show that all models accurately predict velocity across all Reynolds numbers, but the accuracy of mean temperature prediction drops with increasing Reynolds number for all models, except for the k−ωSST model. The study also analyses the Pr effects on turbulent heat transfer in a channel flow with Pr values between 0.025 and 10.0. An error analysis is performed on the results obtained from different turbulence models, and it is shown that the k−ωSST model has the smallest error for the predictions of the mean temperature and Nusselt number for high-Prandtl-number flows, while the low Reynolds number k−ϵLS model shows the smallest errors for low-Prandtl-number flows at different Reynolds numbers. An analytical solution is utilised to identify Pr effects on forced convection in a channel flow into three different regimes: analytical region, transitional region, and turbulent diffusion-dominated region. These regimes are helpful to discuss the validity of the models in relation to the Pr. The findings of this paper provide insights into the performance of different RANS models for heat transfer predictions in a channel flow.

Hydrodynamic interactions play a key role in the swimming behavior and power consumption of bio‐inspired and biomimetic micro‐swimmers, cybernetic or artificial alike. Bio‐inspired robotic micro‐swimmers require fast and reliable numerical models for robust control in order to carry out demanding therapeutic tasks as envisaged for more than 60 years. The fastest known numerical model, the resistive force theory (RFT), incorporates local viscous force coefficients with the local velocity of slender bodies in order to find the resisting hydrodynamic forces, however, omitting the induced far‐field. As a result, the power requirement cannot be predicted accurately. The question of predicting and supplying the necessary power is one of the obstacles impeding the micro‐robotic efforts. In this study, a novel strategy is proposed to improve the RFT‐based analysis, particularly for spermatozoa and spermatozoa‐inspired micro‐swimmers with elastic slender tails, in order to present a practical solution to the problem. The postulated analytical improvement and the associated correction coefficients are based on hydrodynamic impedance analysis of the time‐dependent solution of three‐dimensional (3D) Navier–Stokes equations incorporated with deforming mesh and subject to conservation of mass.

SummaryWard’s slender-body theory is extended to derive first approximations to the external forces on slender bodies of general cross section with discontinuous profile slope. Two classes of body are considered: bodies whose profile (typified by the local radius) is continuous between the nose and base, and certain bodies whose profile is discontinuous, such as bodies with annular or side air intakes and wing-bodies on which the wing has an unswept leading edge. (Where air intakes are concerned, it is assumed that they are sharp-edged and that there is no “ spillage ” of the internal flow).The following conclusions apply to the former class of bodies. The variation of drag with Mach number is found to depend only on the discontinuities in the longitudinal rate of change of the cross-sectional area, and is thus independent of cross-sectional shape. The drag itself is unchanged if the direction of the flow is reversed. The expressions for lift and moment assume the same forms as for smooth pointed bodies, the lift depending only on conditions at the base of the body.The general theory is applied to winged bodies of revolution with an unswept wing leading edge: the results bear a marked resemblance to those obtained by Ward. The results for wings alone are seen to be applicable, with one modification, to subsonic as well as to supersonic speeds.