Volume 26, Issue 2, February 2014
Index of content:

From flight data obtained on a fruit bat, Cynopterus brachyotis, a kinematic model for straightline flapping motion is extracted and analyzed in a computational fluid dynamics (CFD) framework to gain insight into the complexity of bat flight. The intricate functional mechanics and architecture of the bat wings set it apart from other vertebrate flight. The extracted kinematic model is simulated for a range of Reynolds numbers, to observe the effect these phenomena have on the unsteady transient mechanisms of the flow produced by the flapping wings. The Strouhal number calculated from the data is high indicating that the oscillatory motion dominates the flow physics. From the obtained data, the bat exhibits fine control of its mechanics by actively varying wing camber, wing area, torsional rotation of the wing, forward and backward translational sweep of the wing, and wing conformation to dictate the fluid dynamics. As is common in flapping flight, the primary force generation is through the attached unsteady vortices on the wing surface. The bat through varying the wing camber and the wing area modulates this force output. The power requirement for the kinematics is analyzed and correlated with the aerodynamic performance.
 LETTERS


Critical point anomalies include expansion shock waves
View Description Hide DescriptionFrom firstprinciple fluid dynamics, complemented by a rigorous state equation accounting for critical anomalies, we discovered that expansion shock waves may occur in the vicinity of the liquidvapor critical point in the twophase region. Due to universality of nearcritical thermodynamics, the result is valid for any common pure fluid in which molecular interactions are only shortrange, namely, for socalled 3dimensional Isinglike systems, and under the assumption of thermodynamic equilibrium. In addition to rarefaction shock waves, diverse nonclassical effects are admissible, including composite compressive shockfanshock waves, due to the change of sign of the fundamental derivative of gasdynamics.

Characteristics of the viscous superlayer in shear free turbulence and in planar turbulent jets
View Description Hide DescriptionDirect numerical simulations of a planar jet and of shear free turbulence at Re λ = 115–140 using very fine resolutions allow the first direct identification and characterisation of the viscous superlayer (VSL) that exists at the edges of mixing layers, wakes, jets, and boundary layers, adjacent to the turbulent/nonturbulent interface. For both flows the VSL is continuous with higher local thicknesses forming near the larger intense vorticity structures. The mean thickness of the VSL is of the order of the Kolmogorov microscale and agrees well with an estimate based on the Burgers vortex model.

 ARTICLES


Biofluid Mechanics

Straightline climbing flight aerodynamics of a fruit bat
View Description Hide DescriptionFrom flight data obtained on a fruit bat, Cynopterus brachyotis, a kinematic model for straightline flapping motion is extracted and analyzed in a computational fluid dynamics (CFD) framework to gain insight into the complexity of bat flight. The intricate functional mechanics and architecture of the bat wings set it apart from other vertebrate flight. The extracted kinematic model is simulated for a range of Reynolds numbers, to observe the effect these phenomena have on the unsteady transient mechanisms of the flow produced by the flapping wings. The Strouhal number calculated from the data is high indicating that the oscillatory motion dominates the flow physics. From the obtained data, the bat exhibits fine control of its mechanics by actively varying wing camber, wing area, torsional rotation of the wing, forward and backward translational sweep of the wing, and wing conformation to dictate the fluid dynamics. As is common in flapping flight, the primary force generation is through the attached unsteady vortices on the wing surface. The bat through varying the wing camber and the wing area modulates this force output. The power requirement for the kinematics is analyzed and correlated with the aerodynamic performance.

Lattice Boltzmann simulations of a pitchup and pitchdown maneuver of a chordwise flexible wing in a free stream flow
View Description Hide DescriptionA rapid pitchup and pitchdown maneuver of a chordwise flexible wing in a steady free stream is studied by using a lattice Boltzmann flexible particle method in a threedimensional space at a chord based Reynolds number of 100. The pitching rates, flexibility, and wing density are systematically varied, and their effects on aerodynamic forces are investigated. It is demonstrated that the flexibility can be utilized to significantly improve lift forces. The flexible wing has a larger angular momentum due to elasticity and inertia and generates a larger leading edge vortex as compared with a rigid wing. Such lift enhancement occurs mainly during the pitchdown motion while a large stall angle is produced during the pitchup motion. At a low pitch rate, the flexibility cannot improve lift.

Micro and Nanofluid Mechanics

Thermodiffusion in concentrated ferrofluids: Experimental and numerical results on magnetic thermodiffusion
View Description Hide DescriptionFerrofluids consist of magnetic nanoparticles dispersed in a carrier liquid. Their strong thermodiffusive behaviour, characterised by the Soret coefficient, coupled with the dependency of the fluid's parameters on magnetic fields is dealt with in this work. It is known from former experimental investigations on the one hand that the Soret coefficient itself is magnetic field dependent and on the other hand that the accuracy of the coefficient's experimental determination highly depends on the volume concentration of the fluid. The thermally driven separation of particles and carrier liquid is carried out with a concentrated ferrofluid (φ = 0.087) in a horizontal thermodiffusion cell and is compared to equally detected former measurement data. The temperature gradient (1 K/mm) is applied perpendicular to the separation layer. The magnetic field is either applied parallel or perpendicular to the temperature difference. For three different magnetic field strengths (40 kA/m, 100 kA/m, 320 kA/m) the diffusive separation is detected. It reveals a sign change of the Soret coefficient with rising field strength for both field directions which stands for a change in the direction of motion of the particles. This behaviour contradicts former experimental results with a dilute magnetic fluid, in which a change in the coefficient's sign could only be detected for the parallel setup. An anisotropic behaviour in the current data is measured referring to the intensity of the separation being more intense in the perpendicular position of the magnetic field: S T‖ = −0.152 K^{−1} and S T⊥ = −0.257 K^{−1} at H = 320 kA/m. The ferrofluiddynamicstheory (FFDtheory) describes the thermodiffusive processes thermodynamically and a numerical simulation of the fluid's separation depending on the two transport parameters ξ‖ and ξ⊥ used within the FFDtheory can be implemented. In the case of a parallel aligned magnetic field, the parameter can be determined to ξ‖ = {2.8; 9.1; 11.2} × 10^{−11} · D ‖ kg/(A^{2}m) for the different field strengths and in dependence on the magnetic diffusion coefficient D ‖. An adequate fit in the perpendicular case is not possible, by ξ⊥ = 1 × 10^{−17} kg/(Am^{2}) a rather good agreement between numerical and experimental data can be found for a field strength of 40 kA/m, a change in the coefficient's sign in the perpendicular setup is not numerically determinable via this theory. The FFDtheory is only partly applicable to calculate the concentration profile in concentrated magnetic fluids established due to a temperature gradient and magnetic field applied.

Mobilization of a trapped nonwetting fluid from a threedimensional porous medium
View Description Hide DescriptionWe use confocal microscopy to directly visualize the formation and complex morphologies of trapped nonwetting fluid ganglia within a model 3D porous medium. The wetting fluid continues to flow around the ganglia after they form; this flow is characterized by a capillary number, Ca. We find that the ganglia configurations do not vary for small Ca; by contrast, as Ca is increased above a threshold value, the largest ganglia start to become mobilized and are ultimately removed from the medium. By combining our 3D visualization with measurements of the bulk transport, we show that this behavior can be quantitatively understood by balancing the viscous forces exerted on the ganglia with the porescale capillary forces that keep them trapped within the medium. Our work thus helps elucidate the fluid dynamics underlying the mobilization of a trapped nonwetting fluid from a 3D porous medium.

Interfacial Flows

On the stabilizing effect of a liquid film on a cylindrical core by oscillatory motions
View Description Hide DescriptionLiquid films on cylindrical bodies like wires or fibres disintegrate if their length exceeds a critical size (PlateauRayleigh instability). Stabilization can be achieved by an axial oscillation of the solid core provided that a suitable combination of forcing amplitude and frequency is given. To investigate the stabilizing effect, direct numerical simulations (DNS) of the axisymmetric problem are conducted with a height function based solver. It is found that the mechanism of film stabilization is caused by the interaction between an inertia dominated region (high film thickness) and a viscosity dominated region (low film thickness). Replenishing of the thin film region is thereby supported while depleting is suppressed, finally leading to a stable film flow on an oscillating cylinder. To the end, a systematic variation of the main system parameters, e.g., the Weber number, the ratio between the radius of the inner core and the average film coating thickness, and the oscillation frequency is presented and the influence of the parameters discussed.

The interaction between viscous fingering and wrinkling in elasticwalled HeleShaw cells
View Description Hide DescriptionThe development of viscous fingers in circular HeleShaw cells is a classical and widely studied fluid mechanical problem. The introduction of wall elasticity (via the replacement of one of the bounding plates by an elastic membrane) can weaken or even suppress the fingering instability, but it also makes the system susceptible to additional solidmechanical instabilities. We show that in elasticwalled HeleShaw cells that are bounded by sufficiently thin elastic sheets the (fluidbased) viscous fingering instability can arise concurrently with a (solidbased) wrinkling instability. We study the interaction between these distinct instabilities, using a theoretical model that couples the depthaveraged lubrication equations for the fluid flow to the Föpplvon Kármán equations, which describe the deformation of the thin elastic sheet. We employ a linear stability analysis to determine the growth rate of nonaxisymmetric perturbations to the axisymmetrically expanding bubble, and perform direct numerical simulations to study the nonlinear interactions between the instabilities. We show that the system's behaviour may be characterised by a nondimensional parameter that indicates the strength of the fluidstructure interaction. For small [large] values of this parameter, the system's behaviour is dominated by viscous fingering [wrinkling], with strong interactions between the two instabilities arising in an intermediate regime.

Spectral response of a droplet in pulsating external flow field
View Description Hide DescriptionA droplet introduced in an external convective flow field exhibits significant multimodal shape oscillations depending upon the intensity of the aerodynamic forcing. In this paper, a theoretical model describing the temporal evolution of normal modes of the droplet shape is developed. The fluid is assumed to be weakly viscous and Newtonian. The convective flow velocity, which is assumed to be incompressible and inviscid, is incorporated in the model through the normal stress condition at the droplet surface and the equation of motion governing the dynamics of each mode is derived. The coupling between the external flow and the droplet is approximated to be a oneway process, i.e., the external flow perturbations effect the droplet shape oscillations and the droplet oscillation itself does not influence the external flow characteristics. The shape oscillations of the droplet with different fluid properties under different unsteady flow fields were simulated. For a pulsatile external flow, the frequency spectra of the normal modes of the droplet revealed a dominant response at the resonant frequency, in addition to the driving frequency and the corresponding harmonics. At driving frequencies sufficiently different from the resonant frequency of the prolateoblate oscillation mode of the droplet, the oscillations are stable. But at resonance the oscillation amplitude grows in time leading to breakup depending upon the fluid viscosity. A line vortex advecting past the droplet, simulated as an isotropic jump in the far field velocity, leads to the resonant excitation of the droplet shape modes if and only if the time taken by the vortex to cross the droplet is less than the resonant period of the P 2 mode of the droplet. A train of two vortices interacting with the droplet is also analysed. It shows clearly that the time instant of introduction of the second vortex with respect to the droplet shape oscillation cycle is crucial in determining the amplitude of oscillation.

Taylor dispersion in heterogeneous porous media: Extended method of moments, theory, and modelling with tworelaxationtimes lattice Boltzmann scheme
View Description Hide DescriptionThis article describes a generalization of the method of moments, called extended method of moments (EMM), for dispersion in periodic structures composed of impermeable or permeable porous inclusions. Prescribing precomputed steady state velocity field in a single periodic cell, the EMM sequentially solves specific linear stationary advectiondiffusion equations and restores anyorder moments of the resident time distribution or the averaged concentration distribution. Like the pioneering Brenner's method, the EMM recovers mean seepage velocity and Taylor dispersion coefficient as the first two terms of the perturbative expansion. We consider two types of dispersion: spatial dispersion, i.e., spread of initially narrow pulse of concentration, and temporal dispersion, where different portions of the solute have different residence times inside the system. While the first (mean velocity) and the second (Taylor dispersion coefficient) moments coincide for both problems, the higher moments are different. Our perturbative approach allows to link them through simple analytical expressions. Although the relative importance of the higher moments decays downstream, they manifest the nonGaussian behaviour of the breakthrough curves, especially if the solute can diffuse into less porous phase. The EMM quantifies two principal effects of bimodality, as the appearance of sharp peaks and elongated tails of the distributions. In addition, the moments can be used for the numerical reconstruction of the corresponding distribution, avoiding timeconsuming computations of solute transition through heterogeneous media. As illustration, solutions for Taylor dispersion, skewness, and kurtosis in Poiseuille flow and open/impermeable stratified systems, both in rectangular and cylindrical channels, powerlaw duct flows, shallow channels, and Darcy flow in parallel porous layers are obtained in closed analytical form for the entire range of Péclet numbers. The highorder moments and reconstructed profiles are compared to their predictions from the advectiondiffusion equation for averaged concentration, based on the same averaged seepage velocity and Taylor dispersion coefficient. In parallel, we construct LatticeBoltzmann equation (LBE) tworelaxationtimes scheme to simulate transport of a passive scalar directly in heterogeneous media specified by discontinuous porosity distribution. We focus our numerical analysis and assessment on (i) truncation corrections, because of their impact on the moments, (ii) stability, since we show that stable Darcy velocity amplitude reduces with the porosity, and (iii) interface accuracy which is found to play the crucial role. The task is twofold: the LBE supports the EMM predictions, while the EMM provides nontrivial benchmarks for the numerical schemes.

On the control and suppression of the RayleighTaylor instability using electric fields
View Description Hide DescriptionIt is shown theoretically that an electric field can be used to control and suppress the classical RayleighTaylor instability found in stratified flows when a heavy fluid lies above lighter fluid. Dielectric fluids of arbitrary viscosities and densities are considered and a theory is presented to show that a horizontal electric field (acting in the plane of the undisturbed liquidliquid surface), causes growth rates and critical stability wavenumbers to be reduced thus shifting the instability to longer wavelengths. This facilitates complete stabilization in a given finite domain above a critical value of the electric field strength. Direct numerical simulations based on the NavierStokes equations coupled to the electrostatic fields are carried out and the linear theory is used to critically evaluate the codes before computing into the fully nonlinear stage. Excellent agreement is found between theory and simulations, both in unstable cases that compare growth rates and in stable cases that compare frequencies of oscillation and damping rates. Computations in the fully nonlinear regime supporting finger formation and rollup show that a weak electric field slows down finger growth and that there exists a critical value of the field strength, for a given system, above which complete stabilization can take place. The effectiveness of the stabilization is lost if the initial amplitude is large enough or if the field is switched on too late. We also present a numerical experiment that utilizes a simple onoff protocol for the electric field to produce sustained time periodic interfacial oscillations. It is suggested that such phenomena can be useful in inducing mixing. A physical centimetersized model consisting of stratified water and olive oil layers is shown to be within the realm of the stabilization mechanism for field strengths that are approximately 2 × 10^{4} V/m.

Viscous and NonNewtonian Flows

The aqueous viscous drag of a contracting open surface
View Description Hide DescriptionA problem for fluid flow around an axisymmetric spherical surface with a hole is presented to characterize pore dynamics in liposomes. A rotational stream function for the contraction of a punctured plane region is obtained and is used in the perturbation expansion for a stream function in the case of a spherical surface with a hole of small radius compared to the spherical radius. The Rayleigh dissipation function is calculated and used to infer the aqueous friction induced by the contraction of the hole. The theoretical aqueous friction coefficient is compared with one derived from experimental data, and they are in agreement.

Numerical simulation of the settling behaviour of particles in thixotropic fluids
View Description Hide DescriptionA numerical study on the settling behaviour of particles in shear‑thinning thixotropic fluids has been conducted. The numerical scheme was based on the volume of fluid model, with the solid particle being likened to a fluid with very high viscosity. The validity of this model was confirmed through comparisons of the flow field surrounding a sphere settling in a Newtonian fluid with the analytical results of Stokes. The rheology model for the fluid was time‑dependent, utilising a scalar parameter that represents the integrity of a “structural network,” which determines its shear thinning and thixotropic characteristics. The results of this study show that the flow field surrounding the settling sphere is highly localised, with distinct regions of disturbed/undisturbed fluids. The extension of these regions depends on the relaxation time of the fluid, as well as its shear thinning characteristics, and reflects the drag force experienced by the sphere. As the sphere settles, a region of sheared fluid that has significantly lower values of viscosity is formed above the sphere. This region slowly recovers in structure in time. As a result, a sphere that falls in a partially recovered domain (e.g., due to the shearing motion of an earlier sphere) tends to attain a greater velocity than the terminal velocity value. This was found to be true even in cases where the “resting time” of the fluid was nearly twice the relaxation time of the fluid. The results of this study could provide a framework for future analysis on the time‑dependent settling behaviour of particles in thixotropic shear‑thinning fluids.

Particulate, Multiphase, and Granular Flows

Multiphase effects on spherical RayleighTaylor interfacial instability
View Description Hide DescriptionA spherical shocktube model is implemented to focus the attention on the flow instability produced by the release of the driver mixture of gasparticles into the cold driven pure gas. Four discontinuous spherical surfaces are produced which are in order from outward to inward the Primary Shock, gas Contact Interface, Particle Interface, and Secondary Shock. An appropriate methodology is developed to capture the base flows and the physics of RayleighTaylorbased instabilities. The interaction forces between the two phases and the heat transfer are modeled for both the base and the perturbation flows. The parametric space is explored by varying the particle characteristics in order to reveal the mechanisms involved. The results indicate that the gasgas contact interface remains unstable for the multiphase cases; however, the growth rate of the instability is dampened due to the inclusion of the particles. Results are compared with theoretical models to explain the mechanisms involved.

Erosion threshold of a liquid immersed granular bed by an impinging plane liquid jet
View Description Hide DescriptionErosion threshold of a model granular bed by a jet in a quasi bidimensional configuration has been studied experimentally in both laminar and turbulent regimes. The jet is a liquid sheet which impinges normally a packing of immersed beads monodisperse in size and density. The erosion threshold has been characterized at different impact distances of the jet on the sediment and for different grain size and fluid viscosity. In the explored range of parameters, we show that the erosion threshold is well described by a critical inertial Shields number based on the local flow velocity at the impinging point. This has been done by a careful analysis of the different jet flow regimes taking into account the position of the virtual origin of the jet.

Binary droplet collision simulations by a multiphase cascaded lattice Boltzmann method
View Description Hide DescriptionThreedimensional binary droplet collisions are studied using a multiphase cascaded lattice Boltzmann method (LBM). With this model it is possible to simulate collisions with a Weber number of up to 100 and a Reynolds number of up to 1000, at a liquid to gas density ratio of over 100. This is made possible by improvements to the collision operator of the LBM. The cascaded LBM in three dimensions is introduced, in which additional relaxation rates for higher order moments, defined in a comoving reference frame, are incorporated into the collision operator. It is shown that these relaxation rates can be tuned to reduce spurious velocities around curved phase boundaries, without compromising the accuracy of the simulation results. The range of attainable Reynolds numbers is therefore increased. Different outcomes from both headon and offcentre collisions are simulated, for both equal and unequal size droplets, including coalescence, headon separation, and offcentre separation. For headon collisions the critical Weber number between coalescence and separation is shown to decrease with decreasing ambient gas pressure. The variation of critical Weber number with droplet size ratio is also studied. Comparisons are made with the theoretical predictions of Tang et al. [“Bouncing, coalescence, and separation in headon collision of unequalsize droplets,” Phys. Fluids24, 022101 (2012)], and the effect of ambient gas pressure is again considered. For offcentre collisions, boundaries between different collision outcomes are accurately defined and quantitative comparisons are made with the theoretical predictions of Rabe et al. [“Experimental investigation of water droplet binary collisions and description of outcomes with a symmetric Weber number,” Phys. Fluids22, 047101 (2010)]. While general agreement between the simulated and theoretical boundaries is presented, deviations due to varying liquid viscosity are observed. Finally, the prediction of the independence of regime boundaries with varying droplet size ratio, when using the symmetric Weber number as defined by Rabe et al., is discussed. Simulation results showing qualitative agreement are presented, although some discrepancies are reported.

Attached cavitation at a small diameter ultrasonic horn tip
View Description Hide DescriptionUltrasonic horn transducers are frequently used in applications of acoustic cavitation in liquids, for instance, for cell disruption or sonochemical reactions. They are operated typically in the frequency range up to about 50 kHz and have tip diameters from some mm to several cm. It has been observed that if the horn tip is sufficiently small and driven at high amplitude, cavitation is very strong, and the tip can be covered entirely by the gas/vapor phase for longer time intervals. A peculiar dynamics of the attached cavity can emerge with expansion and collapse at a selfgenerated frequency in the subharmonic range, i.e., below the acoustic driving frequency. Here, we present a systematic study of the cavitation dynamics in water at a 20 kHz horn tip of 3 mm diameter. The system was investigated by highspeed imaging with simultaneous recording of the acoustic emissions. Measurements were performed under variation of acoustic power, air saturation, viscosity, surface tension, and temperature of the liquid. Our findings show that the liquid properties play no significant role in the dynamics of the attached cavitation at the small ultrasonic horn. Also the variation of the experimental geometry, within a certain range, did not change the dynamics. We believe that the main two reasons for the peculiar dynamics of cavitation on a small ultrasonic horn are the higher energy density on a small tip and the inability of the big tip to “wash” away the gaseous bubbles. Calculation of the somewhat adapted Strouhal number revealed that, similar to the hydrodynamic cavitation, values which are relatively low characterize slow cavitation structure dynamics. In cases where the cavitation follows the driving frequency this value lies much higher – probably at Str > 20. In the spirit to distinguish the observed phenomenon with other cavitation dynamics at ultrasonic transducer surfaces, we suggest to term the observed phenomenon of attached cavities partly covering the full horn tip as “acoustic supercavitation.” This reflects the conjecture that not the sound field in terms of acoustic (negative) pressure in the liquid is responsible for nucleation, but the motion of the transducer surface.

Laminar Flows

Onset of laminar separation and vortex shedding in flow past unconfined elliptic cylinders
View Description Hide DescriptionThis article presents the numerical studies on predicting onset of flow separation and vortex shedding in flow past unconfined twodimensional elliptical cylinders for various Axis Ratios (AR) and a wide range of Angles of Attack (AOA). An efficient Cartesian grid technique based immersed boundary method is used for numerical simulations. The laminar separation Reynolds number (Re s ) that marks separation of flow from surface and the critical Reynolds number (Re cr ) which represents transition from steady to unsteady flow are determined using diverse methods. A stability analysis which uses StuartLandau equation is also performed for calculating Re cr . The shedding frequency (St cr ) that corresponds to Re cr is calculated using Landau constants. The simulated results for circular cylinder are found to be in good agreement with the literature. The effects of AR and AOA on Re s , Re cr , and St cr are studied. It is observed that the Re s , Re cr , and St cr exhibit a direct/inverse relationship with AR depending upon the given AOA. Correlations of Re s , Re cr , and St cr with respect to AR and AOA are proposed with good accuracy.

Axisymmetric flow within a torsionally oscillating sphere
View Description Hide DescriptionThe flow of an incompressible Newtonian fluid inside a torsionally oscillating spherical cavity is considered. The threedimensional NavierStokes and continuity equations are solved by means of a Galerkin projection spectral method, based on a secondorder incremental fractionalstep approach. Legendre and Jacobi polynomial expansions are used in the zenithal and radial directions, respectively. Axisymmetric solutions are sought for a relatively wide set of the parameters controlling the flow, namely, the Rossby and the Womersley numbers. In particular, the behaviour of the flow for relatively large amplitudes of oscillation is studied, with emphasis on the generation of centrifugal instabilities. Numerical results are compared with experimental observations and semianalytical solutions in the smallamplitude regime, showing good agreement.

Instability and Transition

Liquid supercoiling
View Description Hide DescriptionSupercoiling is the largescale secondary coiling or buckling of a structure that is already coiled at a smaller scale. Here, we show experimentally that a fluidmechanical analog of supercoiling can occur when a thin “rope” of viscous fluid falls vertically from a great height onto a surface. For appropriate values of the viscosity ν, the flow rate Q, and the fall height H, a primary coiling instability of the rope forms a hollow coiled cylinder that then experiences a secondary buckling instability in the form of periodic folding accompanied by slow rotation of the folding plane. To delineate the conditions under which this supercoiling state appears, we carry out systematic laboratory experiments over wide ranges of Q and H using several fluids with different viscosities. We find that five different states of the rope are possible: supercoiling (SC), periodic collapse of the fluid cylinder formed by a primary coiling instability (PC), periodic folding (F), and steady coiling (C) of the rope itself, and axisymmetric stagnation flow (S). Up to three of these states can be realized for a given set of experimental conditions, and we determine detailed state diagrams showing which combinations are observed as a function of ν, Q, and H. The selection of the states is controlled by the dimensionless parameter gHQ ^{2}/ν^{4} (g is the gravitational acceleration), which is directly related to the ratio of the rope radius a to the coil radius R in steady primary coiling with the parameters ν, Q, and H.
