Average Lift Research Articles

CFD Analysis of Takeoff from a Water Surface for an Insect-Scale Aerial/Aquatic Robot

To develop an insect-scale aerial/aquatic robot, we analyzed takeoff mechanisms to counteract surface tension, such as paddling, slapping, and clap-and-fling. Because a diving beetle, Eretes griseus, takes off directly from the water surface, a flapping-wing robot is promising as an alternative to a drone with multiple rotary wings. In this study, we first investigated diving beetle flight with a three-dimensional high-speed camera system and analyzed the motion characteristics. Subsequently, we developed a computational fluid dynamics method that tracked the water surface using a volume of fluid method, reproduced the motion with a multibody model, treated the deformation of the elastic membrane wing with the phase delay of the joint angle functions, and simulated takeoff, that is, the transition from water to air, and hovering near the water surface. The simulation result showed that during the transition, the slapping motion exerted the maximum and average lift per unit of body weight of 18 and 9.2, respectively, while those of paddling produced 0.46 and 0.23, respectively. The water surface effect improved the lift by 25% at the normalized height of less than 0.44 and disappeared at a height greater than 0.7. During hovering, while the clap-and-fling motion improved lift by 2.6% and the water surface effect was 9.8%, the synergy effect was 22%. In addition, the former enhanced it significantly after the fling, while the latter was remarkable during the wing acceleration phase. In contrast to ground effects, flapping reduced the water level and caused the ripples, dynamically changing the water surface effect.

Aerodynamic performance characteristics of low Re airfoils: A parametric and multi criteria decision study

This research aims at investigating the aerodynamic performance of 51 low Re airfoils and applying a Multi Criteria Decision Making (MCDM) approach for selecting airfoils that align with the specific requirements of small wind turbines (SWTs). The study was carried out by varying the Re from 5 × 104 to 5 × 105; angle of attack (AOA) from −10° to 20°, thickness to chord ratio (0.1–0.9), camber ratio (0.1–0.9), and camber position (10 %–80 %) to evaluate the impact of these parameters on the airfoils’ performance. Furthermore, MCDM approach was applied to identify suitable airfoils optimized for their average lift to drag ratio, stall angle, operating range of AOA, strength, manufacturability, and material cost. The findings showed that of the teste parameters, the AOA and airfoil profile have the most pronounced effects on the airfoil performance. Among the tested airfoils, the E63 provided superior performance at 105 Re with LDR of 76.32, a 37.7 % improvement over that of NACA 4412 which is commonly used for wind turbine blade design. Furthermore, the MCDM results revealed that SG6043, FX63-137, E216 and E62 airfoils were the top-ranked airfoils suitable for SWTs application. These findings can help to produce optimized airfoil designs for increased performance and efficiency of SWTs.

"Effects of the Rotation Phase on the Average Lift of an Insect-Like Flapping Wing"

"Effects of the Rotation Phase on the Average Lift of an Insect-Like Flapping Wing"

Improving the energy harvesting performance of clamped flexible plates using vortex shedding behind bluff bodies: A computational study

This study assesses flow-induced vibrations of clamped flexible plates with the objective of improving their energy harvesting performance. Toward this, a rectangular bluff body is placed between the two clamped flexible plates to harness the vortices generated behind the bluff body. The strain energy of the plate is used as a measure of energy harvesting performance. Computational studies are performed for different parameters of interest, such as dimensions and material properties of the plate and dimensions and locations of the bluff body. The effects of these stated parameters on flow-induced vibration response and vortical structures are investigated, and the optimal values for the location and geometry of the bluff body, as well as the aspect ratio and Young's modulus of the plate for energy harvesting performance, are determined. The results show that vortex shedding from the bluff body strongly influences the dynamic behavior and energy output of the flexible structures inside the wake region of the bluff body at different locations. Additionally, the aspect ratio and its effect on vorticity and energy harvesting are discussed in detail, along with the average displacement and average lift force experienced by the plates. The outcomes of this work offer significant insights into optimizing the design of clamped flexible plates for optimal energy generation by cleverly exploiting the vortex shedding behind fixed bluff bodies.

Design and performance analysis of different cambered wings for flapping-wing aerial vehicles based on wind tunnel test

PurposeBionic flapping-wing aerial vehicles (FWAVs) mimic natural flyers to generate the lift and thrust, such as birds, bats and insects. As an important component of the FWAVs, the flapping wings are crucial for the flight performance. The aim of this paper is to study the effects of different wings on aerodynamic performance.Design/methodology/approachInspired by the wings structure of birds, the authors design four cambered wings to analyze the effect of airfoils on the FWAVs aerodynamic performance. The authors design the motor-driven mechanism of flapping wings, and realize the control of flapping frequency. Combined with the wind tunnel equipment, the authors build the FWAVs force test platform to test the static and dynamic aerodynamic performance of different flapping wings under the state variables of flapping frequency, wind speed and inclined angle.FindingsThe results show that the aerodynamic performance of flapping wing with a camber of 20 mm is the best. Compared with flat wing, the average lift can be improved by 59.5%.Originality/valueDifferent from the traditional flat wing design of FWAVs, different cambered flapping wings are given in this paper. The influence of airfoils on aerodynamic performance of FWAVs is analyzed and the optimal flapping wing is obtained.

Numerical simulations on cables attached with different shapes of lights and its mitigations by using perforated shrouds

To reduce the cable vibration caused by rectangular lights, the light shape is modified to be ellipsoidal and circular sections, together with the use of a perforated shroud. Numerical simulations on six types of cable-light models were conducted using computational fluid dynamics. The mean drag coefficient CD, mean and mean lift coefficient CL, mean of the ellipsoidal lights, circle lights, and perforated shroud cable are reduced compared to the rectangular light. CD, mean can be reduced by a maximum of 50%; CL, mean can be reduced by a maximum of 43%. The maximum fluctuating drag and lift coefficient are also significantly reduced. This indicates that the perforated shrouds might be useful to mitigate the vortex-induced vibration. The ellipsoidal and circular lights are significantly better than rectangular lights in terms of galloping critical wind speed. Furthermore, the perforated shroud is very beneficial for suppressing galloping vibration. The variation of Strouhal number St with the wind attack angle for the shroud rectangular light cable model is relatively smooth, all around 0.1. The wake vortices of rectangular light, ellipsoidal light, and circular light are shed alternately in a typical 2S pattern. However, the alternately shedding vortex of the shroud cable was interrupted by the flow of air through the holes. When the wind attack angle exceeds the most dangerous wind attack angle, the incoming stream cannot be reattached after separation, and the force on the cable is greatly reduced, resulting in a sudden drop in the average lift coefficient. Wind pressure in the gap between the shroud and the cable is negative. The fluctuating wind pressure coefficients on the shrouded cables are significantly lower than those of rectangular lights because of the interference effect of the outer perforated shrouds. In total, the perforated shroud is the best to reduce the galloping caused by the rectangular light.

Aerodynamic Analysis of Hovering Flapping Wing Using Multi-Plane Method and Quasi-Steady Blade Element Theory

In the design of flapping-wing micro-size air vehicles capable of hovering, wings serve as the primary source of hovering power, making the analysis of aerodynamics and aerodynamic efficiency crucial. Traditional quasi-steady models treat the wings as single rigid plane, neglecting the deformable characteristics of flexible wings. This paper proposes a multi-plane method that, in conjunction with various design parameters of flexible wings in a two-dimensional plane, analyzes their deformation characteristics under the assumption of multiple planes in three-dimensional space, and describes the deformation of wings during flapping. By combining the quasi-steady aerodynamic model, aerodynamic analysis of the deformed wings can be conducted. The relationship between the slack angle, wing flapping position, and wing deformation are analyzed, along with their effects on aerodynamics and aerodynamic efficiency. Experiments validate the deformation patterns of wings during flapping and compare the simulated aerodynamic forces with measured ones. The results indicate that wing deformation can be accurately described by adjusting the parameters in the multi-plane method and that the aerodynamic analysis using this method closely approximates the average lift results. Additionally, the multi-plane method establishes a connection between wing morphology and aerodynamic forces and efficiency, providing valuable insights for aerodynamic analysis.

Flow past a random array of statistically homogeneously distributed stationary Platonic polyhedrons: Data analysis, Probability maps and Deep Learning models

We perform particle-resolved direct numerical simulations (PR-DNS) of the flow past a random array of stationary Platonic polyhedrons seeded in a tri-periodic box at Reynolds number Re=1, 10, and 100 and solid volume fraction 0.05, 0.1 and 0.2. Using Platonic polyhedrons enables us to examine the effects of particle angularity on the hydrodynamic force and torque exerted on the stationary particles in a controlled manner. We report the average drag, lift, and torque coefficients as well as their respective distribution. We explore the relationship between the particle angularity, particle angular position, flow disturbances created by the neighboring particles, and the hydrodynamic force and torque exerted on each individual particle. We attempt to extend our probability map based model (MPP), our Physics Informed neural network (PINN) model previously developed for spheres to Platonic polyhedrons, and propose an additional NN architecture. We show that ignoring the angular position and actual shape of the reference and neighboring particles constitutes a reasonable first order approximation that significantly simplifies the construction of deterministic models of hydrodynamic forces and torques at low Re=1 to moderate Re=10. At high Re=100, both the MPP model and the PINN model without any additional knowledge of the particle shape and angular position fail to properly predict the force and torque fluctuations. We then design a Convolutional Neural Network (CNN) architecture that is able to maintain a satisfactory performance at Re=100 but requires additional input data in the form of a coarse-grained velocity field around each particle. Overall, predicting the torque fluctuation remains a significant challenge.

Fluid dynamics of a flapping wing interacting with the boundary layer at a flat wall

In this paper, we consider the fluid dynamics of a flapping wing interacting with a boundary layer developed at a no-slip flat wall. Direct numerical simulations are carried out via implementing the non-iterative immersed boundary-lattice Boltzmann method, over a Reynolds number range of 10≤Re≤1000, for a fixed Strouhal number of St = 0.3 and for a given symmetric plunging and pitching flapping motion. The interactions between the wing and the boundary layer are modulated by varying the mean distance of the wing to the wall H0. The results indicate that the presence of the boundary layer at the wall amplifies the fluctuations in both lift and drag due to the boundary layer separation, in contrast to the pure ground effect. This separation also leads to the decrease in both average lift and average drag over one flapping cycle when H0 is low. When it comes to the flow patterns in the wake, it generally gets more complex for a low H0 and/or a high Re. Secondary vortices can be observed for Re≥500 in the present configuration, which either evolve by themselves or interact with the vortices in the wake while being convected downstream and dissipated via viscosity. In the end, a dynamic mode decomposition analysis is performed to explore further the flow structures in the wake. One observes the sheltering effect of the boundary layer that the vortices in the wake are prevented from penetrating the boundary layer, while this effect will not hold if the vortex intensity is sufficiently high, such as the low order mode of the case for Re≥1000 in this study.

Aerodynamic and Flow Characteristics of a Rough Airfoil: A Numerical Study

A computational study explores the aerodynamic and flow properties of rough-surfaced NACA-0018 airfoils using turbulent K-omega SST methodology and the control volume method. Ansys Fluent 2022 R2 was used to do the simulations, and Reynolds numbers between 5000 and 10000 were used. The analysis of how the rough surface affects the performance of the airfoil was the main goal of the study. The skin friction coefficient, drag force, lift force, pressure contours, and stream functions were among the many variables assessed. To have a thorough understanding of the flow behavior, these parameters were studied at various Reynolds numbers. The findings revealed a clear pattern: along the wall of the airfoil, the average drag force, lift force, and skin friction coefficient all increased linearly as the Reynolds number rose from 5000 to 10000. This discovery sheds important light on how the rough surface affects the overall aerodynamic performance of the airfoil. This study advances knowledge of the aerodynamic behavior of rough airfoils by illuminating the flow characteristics and their relationship to surface roughness. Significant results impact aeronautics, wind turbine design, and other airfoil applications.

Wing design optimization and stall analysis with Co-flow Jet active control

Coupled with Co-flow Jet (CFJ) technology, the Non-dominated Sorting Genetic Algorithm II was utilized for the multi-objective combination optimization of an optimized Co-flow Jet wing, based on National Advisory Committee for Aeronautics (NACA) 6421. A high-precision numerical simulation using the delayed detached eddy simulation model was performed on the optimized wing to investigate the three-dimensional flow separation characteristics after static stall. The stall improvement was investigated by adjusting the momentum coefficient of the injection. The results show that the optimized wing exhibits significant improvements in aerodynamic performance and corrected aerodynamic efficiency. At an angle of attack of 10°, the average lift increased by 16.25% and the drag decreased by 27.23% compared to the CFJ6421 wing, while effectively addressing the problem of low modified aerodynamic efficiency of the CFJ wing at lower angles of attack. By utilizing higher momentum and improving the boundary layer control capability, flow separation is effectively suppressed, thus achieving the goal of stall recovery of the CFJ wing.

Cause-Related Marketing as Sales Promotion

This study presents the first field investigation of the sales impact of cause-related marketing promotions (CMPs) in retail settings. Whereas prior work primarily studies CMPs in simplified experimental settings, actual fast-moving consumer goods markets are considerably more complex; ergo, consumers are unlikely to consider and evaluate all brands and CMPs in detail. In this analysis based on 63 CMPs across 20 categories, the authors therefore investigate the short-term sales impact of CMPs as a function of the brand and category context in which they are executed. On average, CMPs run 11 weeks and donate 3.2% of product price, resulting in an average sales lift of 4.9% per week. The findings suggest that a necessary precondition for CMP success is that consumers notice it at the point of sale and hence have considered the CMP brand for reasons other than the CMP itself. Accordingly, the sales impact of CMP can more than double when the category assortment is smaller, the category has less price dispersion, the brand is a category leader, or the brand is priced below the category average. Brands operating in less favorable market conditions can still achieve above-average CMP impact by combining CMPs with price promotions to ensure consumer consideration.

Remodeling of facial soft tissue induced by simultaneous application of HIFES and synchronized radiofrequency provides nonsurgical lift of facial soft tissues.

The application of radiofrequency (RF) and HIFES on the body provides improvement in skeletal muscle tissue, reduction in fatty tissue, reorganization of connective tissue, and skin texture improvement. However, overall facial appearance relies on both skin and underlying structures, specifically muscles and connective tissue which have to be treated as one unit to achieve proper care while preserving fatty layers that define youth facial appearance. The aim of this study is to find whether the effect of novel RF + HIFES is safe and can induce the lifting of soft tissue and overall improvement in facial appearance. In this study, 21 subjects were enrolled. The therapy was administered in four 20-min treatments on the forehead and cheeks. Photographs were evaluated by a Global Aesthetic Improvement Score (GAIS) and linear measurements of facial tissue lifting at 1- and 3-month follow-ups. Volumetric changes in the cheek area were investigated as well. The patients' satisfaction, safety, and comfort were documented throughout the study. The data indicated improvement in overall facial appearance, and 23% of average lifting was found in brows (p = 3.14 × 10-12 ) and cheeks (p = 6.00 × 10-15 ). The assessment of digital photographs showed an improvement in 100% of patients at 3-month follow-up. The treatments were safe, accompanied by high therapy comfort and subject satisfaction of 98%. The treatment by simultaneous RF and HIFES technology produces significant changes to the overall facial appearance, characterized by the lifting of facial tissues.

Direct drive or slider-crank? Comparing motor-actuated flapping-wing micro aerial vehicles

For flapping-wing micro aerial vehicles, the common approach to converting the rotational motion of a DC motor to the reciprocal flapping motion is using a slider-crank mechanism. However, frictional losses in sliders and rotational joints can hinder the performance of such a system. An alternative is a direct drive system where the wings are directly connected to a DC motor that has been driven by an AC signal. These two approaches are compared in this paper, to evaluate their performances and assess which one provides a better solution for flapping-wing micro drones. The electromechanical model of the two systems is used in this paper to compare their performances. System parameters for both types of drones were derived through a multi-variable optimisation process using the same DC motor. The comparisons are made in terms of input power requirement, aerodynamic power, system efficiency, and lift. The direct drive model can generate about 16% higher average lift at 5 V with 50% lower input electrical power. It has 29% larger aerodynamic power and the system efficiency is 16.0% higher than that of the slider-crank model.

Impact of large-scale free-stream turbulence on a pitching airfoil

This paper investigates the impact of large-scale turbulence on the aerodynamic characteristics of a pitching wind turbine blade at Reynolds number 135 000, whose cross-section is a NACA0012 airfoil with constant chord length. Large-eddy simulations at reduced frequencies, $k_{red} = 0.05$ and 0.1, were validated against reference data from the literature. An efficient method capable of generating synthetic large-scale turbulence at the inlet was applied by using two streamwise integral length scales $L_{x} = 1c$ and 1.5 $c$ , which represent energetic turbulence eddies at the height where the wind turbine operates. For $k_{red}= 0.1$ , the change in the maximum lift coefficient at the dynamic stall angle near the maximum angle of attack is on average 20 % lower and during the downstroke it is on average 22 % lower, compared with the smooth inflow. A higher reduced frequency ( $k_{red} = 0.2$ ) apparently does not further change the lift, drag and moment coefficients, and the inflow turbulence disordered leading-edge vortices. The turbulent shear stress and the phase-averaged dispersive shear stress in the wake are of the same magnitude, but with negative and positive signs, respectively, suggesting that the large-scale phase-averaged fluctuations transfer momentum in the opposite direction compared with the turbulent fluctuations, reducing the drag on the suction-side flow, and subsequently increasing the averaged lift coefficient. This demonstrates the critical importance of the large-scale non-turbulent unsteadiness in the wake of a pitching wing.

A novel dragonfly dual-wing hovering flight model

During the hovering flight of dragonflies, the coupling interaction between the forewings and hindwings leads to a reduction in the lift of each wing. Numerous scholars have reached a unanimous conclusion that under the coupling effect, the lift of the hindwings is significantly decreased. Meanwhile, the coupling of the forewings and hindwings enhances the controllability of dragonfly flight. In this article, a novel hovering flight model termed the partial advanced dual-wing model (PADM) is proposed. This model is capable of increasing the lift of both the forewings and hindwings. The maximum average lift of the forewings is increased by 18.09%, and the maximum average lift of the hindwings is increased by 41.58%. In addition to the shared advantage of enhanced positive pressure on the rear half of the wing surface due to the advanced rotation, the superior performance of the hindwings compared to the forewings is attributed to the hindwings cutting off the trailing-edge vortex ring formed by the coupling of the fore and hind wings during the downstroke phase. The vertical force and energy consumption exhibit a linear relationship with the partially advanced time, independent of the coupled aerodynamic effects. The PADM model not only sustains the weight of the dragonfly but also plays a controlling role in transitioning from a hovering flight model to a vertical leap flight model. Furthermore, it enables dragonflies and micro air vehicles to maintain hovering flight while carrying additional loads.

Development and Evaluation of Refreshable Braille Display and Active Touch-Reading System for Digital Reading of the Visually Impaired.

The traditional way of reading through Braille books is constraining the reading experience of blind or visually impaired (BVI) in the digital age. In order to improve the reading convenience of BVI, this paper proposes a low-cost and refreshable Braille display device, and solves the problems of high energy consumption and low latching force existing in existing devices. Further, the Braille display device was combined with the 3D Systems Touch device to develop an active Braille touch-reading system for digital reading of BVI with the help of the CHAI3D virtual environment. Firstly, according to the actual needs of BVI to touch and read the Braille dots, this paper utilizes the beam structure to provide a full latching function for the raised Braille dot without energy consumption. Through theoretical derivation and finite element analysis, the performance of the Braille dot actuator is optimized to provide sufficient feedback force and latching force for finger's touch-reading. Then, this paper designs a virtual Braille interactive environment based on the CHAI3D, and combines the sense of touch with audio to effectively improve the recognition accuracy and reading efficiency of BVI for Braille through the multi-modal presentation of Braille information. The performance test results of the device show that the average lifting force of the Braille dot actuator is 101.67 mN, the latching force is over 5 N, and the average refresh frequency is 17.1 Hz, which meets the touch-reading needs of BVI. User experiments show that the average accuracy rate of BVI subjects in identifying digitized Braille is 95.5%, and subjects have a high subjective evaluation of the system.

Leading-Edge Vortex Controller (LEVCON) Influence on the Aerodynamic Characteristics of a Modern Fighter Jet

The purpose of this paper is to assess the influence of a novel type of vortex creation device called the leading-edge vortex controller (LEVCON) on the aerodynamic characteristics of a fighter jet. LEVCON has become a trending term in modern military aircraft in recent years and is a continuation of an existing and widely used aerodynamic solution called the leading-edge root extension (LERX). LEVCON is designed to operate on the same principles as LERX, but its aim is to generate lift-augmenting vortices, i.e., vortex lift, at higher angles of attack than LERX. To demonstrate the methodology, a custom delta wing fighter aircraft is introduced, and details about its aerodynamic configuration are provided. The LEVCON geometry is designed and then incorporated into an existing three-dimensional (3D) model of the aircraft in question. The research is conducted using OpenFOAM 8, a high-fidelity computational fluid dynamics (CFD) open-source software. The computational cases are designed to simulate the aircraft’s flight at stall velocities within a high range of angles of attack. The results are assessed and discussed in terms of aerodynamic characteristics. A conclusion is drawn from the analysis regarding the perceived improvements in fighter jet aerodynamics. The analysis reveals that both lift and critical angle of attack can be manipulated positively. With the addition of LEVCON, the average lift gain in the high angle of attack (α) range is between 8.5% and 10%, while the peak gain reaches 19.4%. The critical angle of attack has also increased by 2°, and a flatter stall characteristic has been achieved.

Wind tunnel experimental study of aerodynamic characteristics of co-flow jet wing with built-in air source

To further investigate the aerodynamic characteristics and control mechanisms of co-flow jet wings, this study presents the design of a co-flow jet wing (CFJ-CLARKY18) with an embedded micro centrifugal compressor. The CFJ-CLARKY18 wing, based on the CLARKY18 airfoil, incorporates a miniature centrifugal compressor positioned horizontally within the wing's cavity, with its rotor axis perpendicular to the wing chord line, thus alleviating geometric constraints and the compressor's exhaust swirl. The integration of the co-flow jet device with the wing is achieved through a split-flow channel design to ensure uniform co-flow jet injection, followed by verification of jet uniformity and calibration of six jet momentum coefficients. Subsequently, static and dynamic control experiments of the co-flow jet wing are conducted in an open-return wind tunnel, utilizing force measurement systems and high-speed particle image velocimetry tools to investigate its aerodynamic characteristics and control mechanisms. Static stall tests indicate that, compared to the original wing, the CFJ-CLARKY18 wing exhibits a significant increase in lift and a noticeable delay in stall angle. At low angles of attack, an increase in co-flow jet momentum leads to a reduction in drag coefficient, with the possibility of achieving negative drag (thrust) even at high co-flow jet momentum coefficients. The results of dynamic stall tests with force measurements indicate that, while keeping the reduced frequency constant, an increase in co-flow jet momentum coefficient results in higher maximum lift and average lift coefficients, accompanied by a significant reduction in average drag coefficient and the hysteresis loop area of the lift coefficient. Despite the increased reduced frequency exacerbating the wing's stall effect, the stall effect weakens significantly once the co-flow jet momentum coefficient is increased.

Inertial migration of a neutrally buoyant spheroid in plane Poiseuille flow

We study the cross-stream inertial migration of a torque-free neutrally buoyant spheroid, of an arbitrary aspect ratio $\kappa$ , in wall-bounded plane Poiseuille flow for small particle Reynolds numbers ( $Re_p\ll 1$ ) and confinement ratios ( $\lambda \ll 1$ ), with the channel Reynolds number, $Re_c = Re_p/\lambda ^2$ , assumed to be arbitrary; here $\lambda =L/H$ , where $L$ is the semi-major axis of the spheroid and $H$ denotes the separation between the channel walls. In the Stokes limit ( $Re_p =0)$ , and for $\lambda \ll 1$ , a spheroid rotates along any of an infinite number of Jeffery orbits parameterized by an orbit constant $C$ , while translating with a time-dependent speed along a given ambient streamline. Weak inertial effects stabilize either the spinning ( $C=0$ ) or tumbling orbit ( $C=\infty$ ), or both, depending on $\kappa$ . The asymptotic separation of the Jeffery rotation and orbital drift time scales, from that associated with cross-stream migration, implies that migration occurs due to a Jeffery-averaged lift velocity. Although the magnitude of this averaged lift velocity depends on $\kappa$ and $C$ , the shape of the lift profiles are identical to those for a sphere, regardless of $Re_c$ . In particular, the equilibrium positions for a spheroid remain identical to the classical Segre–Silberberg ones for a sphere, starting off at a distance of about $0.6(H/2)$ from the channel centreline for small $Re_c$ , and migrating wallward with increasing $Re_c$ . For spheroids with $\kappa \sim O(1)$ , the Jeffery-averaged analysis is valid for $Re_p\ll 1$ ; for extreme aspect ratio spheroids, the regime of validity becomes more restrictive being given by $Re_p \kappa /\ln \kappa \ll 1$ and $Re_p/\kappa \ll 1$ for $\kappa \rightarrow \infty$ (slender fibres) and $\kappa \rightarrow 0$ (flat disks), respectively.