Pressure induced by thermal fluctuation of an elastic filament confined within a narrow channel

Consider a flexible macro-molecule that is immersed in water at or above room temperature. As a result of thermal motion within the water, the filament is driven to undergo random fluctuations in shape. These fluctuations are a consequence of uncoordinated motion of water molecules. If the range of filament motion is restricted by nearby surfaces, the phenomenon becomes more complex. In this study, it is presumed that the filament is restricted to lie within a plane so that the motion is two dimensional. Furthermore, the range of the planar motion of the filament is confined to the region between inflexible straight boundaries lying in the plane of motion. A result of thermal fluctuation of the filament is that, when in close proximity to a boundary, a normal pressure is induced between the filament and that confining boundary. In the present development, frictional interaction of the filament with either confining boundary is presumed to be negligible. The goal is to determine the dependence of the induced pressure on the separation distance between the confining boundaries in terms of prevailing thermal conditions and physical characteristics of the system.     

Numerical simulations of a filament in a flowing soap film

Experiments concerning the properties of soap films have recently been carried out and these systems have been proposed as experimental versions of theoretical two‐dimensional liquids. A silk filament introduced into a flowing soap film, was seen to demonstrate various stable modes, and these were, namely, a mode in which the filament oscillates and one in which the filament is stationary and aligns with the flow of the liquid. The system could be forced from the oscillatory mode into the non‐ oscillatory mode by varying the length of the filament. In this article we use numerical and computational techniques in order to simulate the strongly coupled behaviour of the filament and the fluid. Preliminary results are presented for the specific case in which the filament is seen to oscillate continuously for the duration of our simulation. We also find that the filament oscillations are strongly suppressed when we reduce the effective length of the filament. We believe that these results are reminiscent of the different oscillatory and non‐oscillatory modes observed in experiment. The numerical solutions show that, in contrast to experiment, vortices are created at the leading edge of the filament and are preferentially grown in the curvature of the filament and are eventually released from the trailing edge of the filament. In a similar manner to oscillating hydrofoils, it seems that the oscillating filaments are in a minimal energy state, extracting sufficient energy from the fluid to oscillate. In comparing numerical and experimental results it is possible that the soap film does have an effect on the fluid flow especially in the boundary layer where surface tension forces are large. Copyright © 2004 John Wiley & Sons, Ltd.     

Equilibrium and stability of a z pinch in a multipole magnetic field

In the theoretical investigation of the dynamic stabilization of a current-carrying plasma filament by a high-frequency multipole magnetic field it is usually assumed that the cross section of the filament has a circular form in equilibrium [1, 2]. This considerably simplifies the calculations but it does not correspond to reality, since the surface of the plasma must be fluted in the multipole field. An attempt to estimate the influence of the deformation of the filament cross section on its stability against bending in the special case of quadrupole field was made in [3], in which the parameters were determined of the elliptical cross section corresponding to a plasma filament with current in a quadrupole field and an expression was found for the electrodynamic force acting on the filament in the case of long-wavelength kink perturbations. However, this force was calculated incorrectly in [3]. In the present paper a study is made of the equilibrium and stability of a current-carrying plasma filament against kink perturbations in the general case of a multipole stabilizing field. Under the assumption that the flute depth is small, the equilibrium form of the cross section of the current-carrying plasma filament in the multipole magnetic field is found and the components of the force exerted by the field on the perturbed filament are calculated. It is shown that the external field interacts with the current in the perturbed filament only in the case of a quadrupole field. The results are discussed in connection with the problem of multipole dynamical stabilization of a z pinch against kink perturbations.     

Dynamics of horizontal viscoelastic fluid filaments

The tension force of a thinning high-molecular polymer solution filament is measured using the filament itself as a force sensor. The axial filament stresses and the effects of fluid flow from the filament into adjacent drops are estimated. It is shown that these effects are insignificant for polymer solutions in a low-viscosity solvent (water) but substantial for solutions in a high-viscosity fluid (glycerine). A modification of the standard rheological capillary filament method is proposed. This modification makes it possible to exclude any hypotheses concerning the stress distribution pattern within the filament. Periodic transverse oscillations of the filament axis are revealed and analyzed.     

Motion of a flexible finite-length filament in a flow of a viscous fluid

The two-dimensional problem of the configuration of a flexible filament of finite length in a deformable viscous fluid is solved. The flexuural stresses in the filament and the inertial and gravitational forces are not taken into account. The equilibrium equations are obtained. The friction force that acts on the filament surface from the side of the viscous fluid is proportional to the flow rate. The specific features of the evolution of a bent filament under the conditions of pure and simple shear of a fluid are studied numerically. Analytical solutions are obtained for the evolution of a rectilinear filament; in particular, the stretching force in the filament is found. For the indicated types of flow, the stability of a rectilinear filament against small perturbations is investigated. Volzhskii Polytechnical Institute at the Volgograd State Technical University, Volgograd 404121. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 144–153, March–April, 2000.     

Numerical prediction of breakup mode of contracting gas filament in liquid

The breaking up of gas filament in liquid is important in many industrial and scientific applications. In this study, a transient axisymmetric model with the level set method is built up to examine the dynamics of a contracting gas filament, and to determine the effects of the aspect ratio, Ohnesorge (Oh) number, and viscosity ratio on its breakup mode. The filament undergoes no break, middle break, or end-pinching modes with increasing aspect ratio at either a low or a high Oh number, and one critical initial aspect ratio is observed for each Oh number. The fate of the filament is determined by the interaction of capillary waves on its surface, and can be predicted accurately by using the one-dimensional wave superposition method. The decreasing viscosity ratio of liquid over gas reduces the critical initial aspect ratio for the fate transition between the no break and breakup modes, and this effect is reduced at a low viscosity ratio. These findings may be helpful in fabricating gas bubbles and their breakup suppression.     

Measurement of relaxation times in extensional flow of weakly viscoelastic polymer solutions

The characterization of the extensional rheology of polymeric solutions is important in several applications and industrial processes. Filament stretching and capillary breakup rheometers have been developed to characterize the extensional properties of polymeric solutions, mostly for high-viscosity fluids. However, for low concentration polymer solutions, the measurements are difficult using available devices, in terms of the minimum viscosity and relaxation times that can be measured accurately. In addition, when the slow retraction method is used, solvent evaporation can affect the measurements for volatile solvents. In this work, a new setup was tested for filament breakup experiments using the slow retraction method, high-speed imaging techniques, and an immiscible oil bath to reduce solvent evaporation and facilitate particle tracking in the thinning filament. Extensional relaxation times above around 100 μs were measured with the device for dilute and semi-dilute polymer solutions. Particle tracking velocimetry was also used to measure the velocity in the filament and the corresponding elongation rate, and to compare with the values obtained from the measured exponential decay of the filament diameter.     

Experimental investigations into an isothermal spinning threadline: Extensional rheology of a Separan AP 30 solution in glycerol and water

Summary Experimental observations on a steady isothermally extending filament of a water/glycerol solution of Separan AP 30 are presented. Photographic records were analysed to give filament diameter (and hence filament speed) as a function of distance below the extrusion die (a glass capillary). Measurements of inline tension were also made. When effects of weight, surface tension and air drag were accounted for, the extensional stress at every point along the filament could be calculated. Results for stress versus extension rate are presented for various flow situations.Independent rheogoniometric measurements of simple shear viscosity, first and second normal stress differences, and of a crude relaxation time were also made at comparable rates of deformation.Comparison shows that apparent extensional viscosities are several orders of magnitude larger than corresponding simple shear viscosities. After discussion, no conclusion can be drawn about what constitutive equation is most suitable to describe the results.An analysis to predict air drag is given.With 18 figures     

A new look at extensional rheology of low-density polyethylene

The nonlinear rheology of three selected commercial low-density polyethylenes (LDPE) is measured in uniaxial extensional flow. The measurements are performed using three different devices including an extensional viscosity fixture (EVF), a homemade filament stretching rheometer (DTU-FSR) and a commercial filament stretching rheometer (VADER-1000). We show that the measurements from the EVF are limited by a maximum Hencky strain of 4, while the two filament stretching rheometers are able to probe the nonlinear behavior at larger Hencky strain values where the steady state is reached. With the capability of the filament stretching rheometers, we show that LDPEs with quite different linear viscoelastic properties can have very similar steady extensional viscosity. This points to the potential for independently controlling shear and extensional rheology in certain rate ranges.     

Transient stress and interfacial area generated during breakup of a Newtonian thread immersed in a Newtonian medium

The transient stress and the transient average orientation generated by the breakup process of a long Newtonian filament imbedded in a quiescent Newtonian viscous liquid have been calculated. Rayleigh disturbances were used to describe the relaxation of the filament and the variation of interfacial area in the absence of flow during the course of disintegration process. The effect of viscosity ratio and initial radius of the filament were discussed. It was demonstrated that the predictions of the model in terms of the time-evolution of interfacial area can be used to select the best conditions for carrying out the breaking thread experiments. The predictions of the proposed model were compared to some experimental data on polyamide/polyethylene system.     

Extensional flow of a viscoelastic filament governed by the FENE-CR model

A thin filament model is used to analyze the extensional flow of a viscoelastic thread governed by the FENE-CR model. The problem is solved numerically by finite differences using a third-order upwind scheme in space and a second order Runge-Kutta scheme in time. The behavior of the filament is controlled by the competing effects of surface tension and axial normal stresses which are characterized in terms of three-dimensional groups, the Deborah number De, the extensibility parameter L and the capillary number Ca. Surface tension has a destabilizing effect causing the filament to thin in the mid-section leading to a rupture. On the other hand normal stresses tend to stabilize the filament. If axial normal stresses are sufficiently large the filament deforms almost uniaxially due to strain hardening.     

The effect of brittle interfacial compounds on deformation and fracture of molybdenum-aluminum fiber composites

The effect of brittle intermetallic compounds at the fiber-matrix interface on the deformation characteristics of molybdenum-aluminum fiber composites was investigated. If the filament is ductile and notch-insensitive, then composite strength degradation is relatively minor and composite strength can be predicted by a modified mixture-rule which neglects the strength contribution of the brittle compound. For the case of notch-sensitive filaments, severe filament degradation occurs upon compound formation. The degradation was shown to result from cracks formed during deformation at the roots of compound nodules. The presence of 10 per cent compound by volume results in a 50 per cent decrease in tensile strength, but larger amounts of compound cause little additional strength reduction. At filament volume fractions of 25 and 34 per cent and compound volume fractions less than 10 per cent, composite fracture occurs by the statistical accumulation of fiber necks or fractures depending on the notch sensitivity of the fiber. At high fiber or compound volume fractions, composite failure occurs upon the first or the second filament fracture.     

A penny-shaped crack in a filament-reinforced matrix—II. The crack problem

Using the filament model developed in the previous paper, the elastostatic interaction problem between a penny-shaped crack and a slender inclusion or filament in an elastic matrix is formulated. For a single filament as well as multiple identical filaments located symmetrically around the crack the problem is shown to reduce to a singular integral equation. The solution of the problem is obtained for various geometries and filament to-matrix stiffness ratios, and the results relating to the angular variation of the stress intensity factor and the maximum filament stress are presented.     

Tensile behaviors of pre-twisted composite strands

In this paper, we address the tensile behaviors of pre-twisted composite strands, which consists of a pre-twisted single core filament surrounded by n-helical side filaments. Based on the extensible rod with zero bending and small twisting moduli for the core filament and inextensible rod for the side filaments, we develop the analytical method of the tensile behaviors of pre-twisted composite strands. Using a numerical continuation scheme, we elucidate the effects of microscopic factors such as initial helical angle, pre-twist of core filament, ratio of elastic modulus of core to that of side filament, and the number of the side filaments on the macroscopic tensile behavior of the strand as a whole structure. As a result, we show that the behavior is not trivial, even though the filament is a linear elastic due to the interplay of both geometrical constraint and finite deformation of the strand.     

A simple extensional viscometer

An extensional viscometer is described in which the liquid filament leaving a capillary is subjected to a stretching deformation. In order to keep the flow rate through the capillary unaltered upon inception of stretching, the pressure head at the capillary entrance has to be reduced by an amount equal to the extensional viscoelastic stress at the capillary exit. This affords a simple means of measuring small fluid forces such as those that occur in the stretching of dilute polymer solutions. Since stretch rates can be obtained from a knowledge of the mass flow rate and the filament diameter profile, extensional viscosities can be computed. The efficacy of the technique is demonstrated by obtaining the anticipated results for Newtonian liquids.     

The effect of pre-shear on the extensional rheology of wormlike micelle solutions

The effect of initial microstructural deformation, alignment, and morphology on the response of wormlike micelle solutions in transient uniaxial extensional flows is investigated using a pre-shear device attached to a filament stretching rheometer. In filament stretching experiments, increasing the strength and the duration of the pre-shear just before stretch is found to delay the onset of strain hardening. In these experiments, the wormlike micelle solution filaments fail through a rupture near the axial midplane. The value of the elastic tensile stress at rupture is found to decrease with increasing pre-shear rate and duration. The most dramatic effects are observed at shear rates for which shear banding has been independently observed. The reduction in the strain hardening suggests that pre-shear before filament stretching might break down the wormlike micelles reducing their size before stretch. Strain hardening is also observed in capillary breakup rheometry experiments; however, the pre-sheared wormlike micelle solutions strain harden faster, achieve larger steady-state extensional viscosities and an increase in the extensional relaxation time with increasing shear rate and duration. The difference between the response of the wormlike micelles in filament stretching and capillary breakup experiments demonstrates the sensitivity of these self-assembling micelle networks to pre-conditioning.     

On the motion of flexible threads in a stokes shear field

We consider the motion of a flexible threadlike filament suspended in an unbounded Stokes shear flow. As the extension of a well-established slender-body theory, we treat the case of a highly anisotropic drag relation, as reflected by a small ratio ? of transverse to longitudinal filament mobility. This provides one potentially useful model of hindered filament motion in highly concetrated filamentary suspensions. In the limit ? → 0, which corresponds to the kinematics associated with a recent model (of Doi and Edwards) for polymer chains, the filament motion is governed by a non-linear kinematic-wave equation. We show that this equation can be solved in closed form, for inextensible filaments and general time-dependent shear flows. We investigate the permanence and stability of threads having uniform axial mobility, showing that only piecewise straight threads have permanent shape. Also, we investigate the stability of fully-extended treads by means of both finite and infinitesimal stability analyses, and we provide equations for the growth rate of arbitrary intial disturbances. These are applied to simple shear and simple-extensional flows. Finally, we discuss the failure of our model for small but non-zero ?, near points of extreme curvature of “entaglement”, and the possible relevance to the rheology of fibrous composites and suspensions.     

Transient filament stretching rheometer

The filament stretching device which is used increasingly as an apparatus for measuring extensional properties of polymeric liquids is analysed. A force balance that includes the effects of inertia and surface tension is derived. The force balance may be used to correct for the effects of inertia and surface tension, provided online measurements of the filament surface shape are available. In addition, the question of initial asymmetry due to gravity is addressed.     

Numerical characterization of high-temperature filament interaction with blunt cylinder at Mach 3

Energy deposition is a robust technique for various high-speed flow control applications including drag reduction. A numerical study of energy deposition is performed for perfect gas flow approaching a blunt cylinder at Mach 3. The energy deposition is simulated by a high-temperature filament injected instantaneously in front of the cylinder. The effect of important dimensionless parameters is studied to characterize the drag modification. The results indicate a saturation effect on maximum drag reduction at higher magnitudes of energy deposition. The computations reveal that the discharge location of the filament does not significantly impact the drag. A phenomenological examination of the interaction is performed. The effectiveness and efficiency of the filament on drag reduction are investigated. A one-dimensional analytical approach is studied to describe the numerical results.     

Numerical analysis of the buckling and recuperation dynamics of flexible filament using an immersed boundary framework

The dynamics of flexible filaments in viscous shear flow is of interest to biologists and engineers in a wide variety of applications involving folding and unfolding sequence of long-chain biomolecules like DNA, non-motile sperm and microalgae. It is also helpful in understanding the deformation of natural and synthetic fibers which can be applied in areas such as biotechnology. In the present work, deformation and migration behavior of non-motile unicellular phytoplankton diatoms subjected to viscous shear flow are considered. These unicellular diatoms develop into colonies which are made up of linked chains. The complex fluid-structure interaction is solved by developing a two-dimensional numerical model with an immersed boundary framework. The simulation consists of suspending an elastic filament mimicking a diatom chain in a shear flow at low Reynolds number. The governing continuity and Navier–Stokes equations are solved on a Cartesian grid arranged in a staggered manner. A forcing term is added to the momentum equation that incorporates the presence of flexible filament in the fluid domain. The discretization of the governing equation is based on a finite volume method, and a SIMPLE algorithm is used to compute pressure and velocity. A computer code is developed to perform numerical simulations, and the model is first verified with the deformation study of a tethered flexible filament in uniform fluid flow. Next, the shape deformations for flexible filament placed freely in shear flow are compared with the studies of previous researchers. Further, the present results are validated with Jeffery's equation for particles immersed in shear flow along with classification plot for filament orbit regimes. All of these comparisons provide a reasonable validity for the developed model. The effect of bending rigidity and shear rate on the deformation and migration characteristics is ascertained with the help of parametric studies. A non-dimensional parameter called Viscous Flow Forcing value (VFF) is calculated to quantify the parametric results. An optimum Viscous Flow Forcing value is determined which indicates the transition of filaments exhibiting either a recuperative (regaining original shape past deformation) or non-recuperative (permanently deformed) behavior. The developed model is successful in capturing fluid motion, diatom buckling, shape recurrences and recuperation dynamics of diatom chains subjected to shear flow. Further, the developed computational model can successfully illustrate filament-fluid interaction for a wide variety of similar problems.