Small-amplitude Deformations Research Articles

A new solution for the deformations of an initially elliptical elastic-walled tube

SummaryWe investigate the small-amplitude deformations of a long thin-walled elastic tube having an initially axially uniform elliptical cross-section. The tube is deformed by a (possibly non-uniform) transmural pressure. At leading order, its deformations are shown to be governed by a single partial differential equation (PDE) for the azimuthal displacement as a function of the axial and azimuthal co-ordinates and time. Previous authors have obtained solutions to this PDE by making ad hoc approximations based on truncating an approximate Fourier representation. In this article, we instead write the azimuthal displacement as a sum over the azimuthal eigenfunctions of a generalised eigenvalue problem and show that we are able to derive an uncoupled system of linear PDEs with constant coefficients for the amplitude of the azimuthal modes as a function of the axial co-ordinate and time. This results in a formal solution of the whole system being found as a sum over the azimuthal modes. We show that the $n$th mode’s contribution to the tube’s relative area change is governed by a simplified second-order PDE and examine the case in which the tube’s deformations are driven by a uniform transmural pressure. The relative errors induced by truncating the series solution after the first and second terms are then evaluated as a function of both the ellipticity and pre-stress of the tube. After comparing our results with Whittaker et al. (A rational derivation of a tube law from shell theory, Q. J. Mech. Appl. Math. 63 (2010) 465–496), we find that this new method leads to a significant simplification when calculating contributions from the higher-order azimuthal modes, which in turn makes a more accurate solution easier to obtain.

Fluctuating nonlinear spring theory: Strength, deformability, and toughness of biological nanoparticles from theoretical reconstruction of force-deformation spectra

We developed the Fluctuating Nonlinear Spring (FNS) model to describe the dynamics of mechanical deformation of biological particles, such as virus capsids. The theory interprets the force-deformation spectra in terms of the “Hertzian stiffness” (non-linear regime of a particle's small-amplitude deformations), elastic constant (large-amplitude elastic deformations), and force range in which the particle's fracture occurs. The FNS theory enables one to quantify the particles’ elasticity (Young's moduli for Hertzian and bending deformations), and the limits of their strength (critical forces, fracture toughness) and deformability (critical deformations) as well as the probability distributions of these properties, and to calculate the free energy changes for the particle's Hertzian, elastic, and plastic deformations, and eventual fracture. We applied the FNS theory to describe the protein capsids of bacteriophage P22, Human Adenovirus, and Herpes Simplex virus characterized by deformations before fracture that did not exceed 10–19% of their size. These nanoshells are soft (~1–10-GPa elastic modulus), with low ~50–480-kPa toughness – a regime of material behavior that is not well understood, and with the strength increasing while toughness decreases with their size. The particles’ fracture is stochastic, with the average values of critical forces, critical deformations, and fracture toughness comparable with their standard deviations. The FNS theory predicts 0.7-MJ/mol free energy for P22 capsid maturation, and it could be extended to describe uniaxial deformation of cylindrical microtubules and ellipsoidal cellular organelles.

Моделирование аккумуляции кинетической энергии внутренних волн в областях с большим отношением горизонтального и вертикального масштабов

Tidal forcing excites internal waves in the bulk of the ocean. Deep ocean is an example of a system with continuous stratification subject to large-scale periodic forcing. Owing to specific dispersion relation of internal waves, the domains bounded by sloping boundaries may support wave patterns with wave rays converging to closed trajectories (geometric attractors) as result of iterative focusing reflections. Previously the behavior of kinetic energy in wave attractors has been investigated in two-dimensional domain with comparable depth and length. As the geometric aspect ratio of the domain increases, the dynamic pattern of energy focusing may significantly evolve both in laminar and turbulent regimes. The present paper shows that the energy density in domains with large aspect ratio can significantly increase. In numerical simulations the input forcing has been introduced at global scale by prescribing small-amplitude deformations of the upper bound of the liquid domain. The evolution of internal wave motion in such system has been computed numerically for different values of the forcing amplitude. The behavior of the large-aspect-ratio system has been compared to the well-studied case of the system with depth-to-length ratio of order unity. A number of most typical situations has been analysed in terms of behavior of integral mechanical quantities such as total dissipation, mean kinetic energy and energy fluctuations in laminar and turbulent cases.

Boundary element methods for particles and microswimmers in a linear viscoelastic fluid

The consideration of viscoelasticity within fluid dynamical boundary element methods has traditionally required meshing over the whole flow domain. In turn, a major advantage of the boundary element method is lost, namely the need to consider only surface boundary integrals. Here, using a generalised reciprocal relation and viscoelastic force singularities, a boundary element method is developed for linear viscoelastic flows. We proceed to explore finite-deformation microswimming in a linear Maxwell fluid. We firstly deduce a finite-amplitude generalisation of a previously reported result that the flow field is unchanged between a Newtonian and linear Maxwell fluid for prescribed small-amplitude deformations. Hence Purcell’s theorem holds for a linear Maxwell fluid. We proceed to consider deformation swimming in a linear Maxwell fluid given an external forcing. Boundary scattering trajectories for an exemplar squirmer approaching a surface are observed to exhibit a weak dependence on the Deborah number, while the trajectories of a sperm and monotrichous bacterium near a surface are predicted to be essentially unaffected at moderate Deborah number. In turn, the latter supports the common simplification of using Newtonian Stokes flows for studying flagellate swimming in linear Maxwell media. In addition, the motion of a magnetic helix under the influence of an external magnetic field is considered, and highlights that linear viscoelasticity can significantly impact the propagation of the helix, in turn demonstrating that even linear rheology is important to consider for forced swimmers. Finally, the presented framework requires minimalistic adjustments to Newtonian boundary element codes, enabling rapid implementation, and is more generally applicable, for instance to studies of particle interactions in active linear rheology on the microscale.

The effect of gait on swimming in viscoelastic fluids

In this paper, we give formulas for the swimming of simplified two-dimensional bodies in complex fluids using the reciprocal theorem. By way of these formulas we calculate the swimming velocity due to small-amplitude deformations on the simplest of these bodies, a two-dimensional sheet, to explore general conditions on the swimming gait under which the sheet may move faster, or slower, in a viscoelastic fluid compared to a Newtonian fluid. We show that in general, for small amplitude deformations, a speed increase can only be realized by multiple deformation modes in contrast to slip flows. Additionally, we show that a change in swimming speed is directly due to a change in thrust generated by the swimmer.

A shear-relaxation boundary layer near the pinned ends of a buckled elastic-walled tube

We consider small-amplitude deformations of a thin-walled elastic tube, which initially has a uniform elliptical cross-section and is subject to a large axial pre-stress. We derive a boundary-layer model for the deformations near an end of such a tube that is pinned to a rigid elliptical support. The model is appropriate in the limit in which F ≡ d2F/[24πaK(1−ν2)]≪ 1, where d is the wall thickness, F is the axial tension that gives rise to the pre-stress, 2πa is the tube circumference, K is the bending stiffness of the tube wall, and ν is its Poisson ratio. In particular, the model takes into account in-plane shear forces arising because of geometrical constraints. These forces are asymptotically small outside the boundary layer, and so were not present in the previous tube-law model of Whittaker et al. (2010; Q. J. Mech. Appl. Math. 64(4), pp465–492). Deformation profiles from the boundary-layer model are matched to solutions for the interior arising from the tube-law model of Whittaker et al. (2010). The net effect is to modify the previous tube-end boundary condition on the interior solution, from zero normal displacement to a Robin type condition. The predictions from the matched models compare favourably with full numerical simulations of the tube wall deformations. While the additional shear forces are only important in the boundary layer near the end, they can have a significant effect on the global solution when F ≪ 1.

Nonlinear Behavior of Nematic Platelet Dispersions in Shear Flow

Dispersions of platelets in the nematic phase are submitted to large amplitude oscillatory shear flow and probed by high temporal resolution small angle x-ray scattering. The response displays rich dynamic and structural behavior. Under small amplitude deformations we observe an elastic response, while structurally symmetry is broken: a preferential direction of deformation is selected which induces off-plane orientation of the platelets. We associate the elastic responses with the tilting director of the platelets towards the flow direction at all strain amplitudes. At large strain amplitudes there is a yielding transition between elastic and plastic deformation, accompanied by a flipping of the director. At intermediate strain amplitudes the director has a rich dynamic behavior, illustrating the complex motion of platelets in shear flow. These observations are confirmed by steady-shear flow reversal experiments, which underline the unique character of sheared nematic platelet dispersions.

Hydrodynamic interactions of spherical particles in a fluid confined by a rough no-slip wall

In this paper we develop a theoretical framework to study the hydrodynamic interactions in the presence of a nonflat and no-slip boundary. We calculate the influence of a small amplitude and sinusoidal deformations of a boundary wall in the self-mobility and the two-body hydrodynamic interactions for spherical particles. We show that the surface roughness enhances the self-mobility of a sphere in a way that, for motion in front of a local hump of the surface, the mobility strength decreases while it increases for the motion above a local deep of the rough surface. The influence of the surface roughness in the two-body hydrodynamic interactions is also analyzed numerically.

A Rational Derivation of a Tube Law from Shell Theory

We consider small-amplitude deformations of a long thin-walled elastic tube having an initially axially uniform elliptical cross section. The tube is subject to an axial pre-stress, and the deformations result from an applied transmural pressure. An approximate tube law (linking the transmural pressure with the cross-sectional area and its axial derivatives) is derived from shell theory in the distinguished asymptotic limit in which the tube's behaviour is dominated by the restoring forces from the axial pre-stress and azimuthal bending. This is possible because the deformations of the tube induced by both the transmural pressure and the axial forces can be described, to very good approximation, by a single azimuthal mode of deformation of axially varying amplitude. The resulting tube law is compared with numerical solutions of the full shell equations and good agreement is found (provided the tube is sufficiently long and the wall not too thin so that in-plane shearing is negligible). We discuss the applications of our results to the modelling of flow in collapsible tubes.

Modulation equations for strongly nonlinear oscillations of an incompressible viscous drop

Large-amplitude oscillations of incompressible viscous drops are studied at small capillary number. On the long viscous time scale, a formal perturbation scheme is developed to determine original modulation equations. These two ordinary differential equations comprise the averaged condition for conservation of energy and the averaged projection of the Navier–Stokes equations onto the vorticity vector. The modulation equations are applied to the free decay of axisymmetric oblate–prolate spheroid oscillations. On the long time scale, only the modulation equation for energy is required. In this example, the results compare well with linear viscous theory, weakly nonlinear inviscid theory and experimental observations. The new results show that previous experimental observations and numerical simulations are all manifestations of a single-valued relationship between dimensionless decay rate and amplitude. Moreover, if the amplitude of the oscillations does not exceed 30% of the drop radius, this decay rate may be approximated by a quadratic. The new results also show that, when the amplitude of the oscillations exceeds 20% of the drop radius, fluid in the inviscid bulk of the drop is undergoing abrupt changes in its acceleration in comparison to the acceleration during small-amplitude deformations.

Crawling beneath the free surface: Water snail locomotion

Land snails move via adhesive locomotion. Through muscular contraction and expansion of their foot, they transmit waves of shear stress through a thin layer of mucus onto a solid substrate. Since a free surface cannot support shear stress, adhesive locomotion is not a viable propulsion mechanism for water snails that travel inverted beneath the free surface. Nevertheless, the motion of the freshwater snail, Sorbeoconcha physidae, is reminiscent of that of its terrestrial counterparts, being generated by the undulation of the snail foot that is separated from the free surface by a thin layer of mucus. Here, a lubrication model is used to describe the mucus flow in the limit of small-amplitude interfacial deformations. By assuming the shape of the snail foot to be a traveling sine wave and the mucus to be Newtonian, an evolution equation for the interface shape is obtained and the resulting propulsive force on the snail is calculated. This propulsive force is found to be nonzero for moderate values of the capillary number but vanishes in the limits of high and low capillary number. Physically, this force arises because the snail’s foot deforms the free surface, thereby generating curvature pressures and lubrication flows inside the mucus layer that couple to the topography of the foot.

Liquid jet response to internal modulated ultrasonic radiation pressure and stimulated drop production

Experimental evidence shows that a liquid jet in air is an acoustic waveguide having a cutoff frequency inversely proportional to the jet diameter. Ultrasound applied to the jet supply liquid can propagate within the jet when the acoustic frequency is near to or above the cutoff frequency. Modulated radiation pressure is used to stimulate large amplitude deformations and the breakup of the jet into drops. The jet response to the modulated internal ultrasonic radiation pressure was monitored along the jet using (a) an optical extinction method and (b) images captured by a video camera. The jet profile oscillates at the frequency of the radiation pressure modulation and where the response is small, the amplitude was found to increase in proportion to the square of the acoustic pressure amplitude as previously demonstrated for oscillating drops [P.L. Marston and R.E. Apfel, J. Acoust. Soc. Am. 67, 27-37 (1980)]. Small amplitude deformations initially grow approximately exponentially with axial distance along the jet. Though aspects of the perturbation growth can be approximated from Rayleigh's analysis of the capillary instability, some detailed features of the observed jet response to modulated ultrasound are unexplained neglecting the effects of gravity.

Light-induced deformation and instability of a liquid interface. II. Dynamics

We study the dynamics of the deformation of a soft liquid-liquid interface by the optical radiation pressure of a focused cw Gaussian laser beam. We measured the temporal evolution of both the hump height and the hump curvature by direct observation and by detecting the focusing effect of the hump acting as a lens. Extending the results of Yoshitake [J. Appl. Phys. 97, 024901 (2005)] to the case of liquid-liquid interfaces and to the Bo approximately =1 regime [Bo=(omega0/lc)2, , where omega0 is the beam waist and lc the capillary length], we show that, in the Bo<<1 and Bo approximately =1 ranges, the small-amplitude deformations are correctly described by a linear hydrodynamic theory predicting an overdamped dynamics. We also study the dynamics of the large-amplitude interface deformations at the onset of optohydrodynamic instability [Phys. Rev. Lett. 90, 144503 (2003)]. Using a simple, phenomenological model for the nonlinear evolution of the hump height, we interpret the observed interface dynamics at the instability onset as the signature of an imperfect subcritical instability.

Surface partition of large clusters

The surface partition of large clusters is derived analytically for a simple statistical model by using the Laplace-Fourier transformation method. In the limit of small amplitude deformations, a suggested "hills and dales model" reproduces the leading term of the Fisher result for the surface entropy to within a few percent. The model also gives the degeneracy prefactor of large clusters. The surface partition of finite clusters is also discussed.

Optical detection of the response of liquid jets to internal modulated ultrasonic radiation pressure

A liquid jet in air is an acoustic waveguide having a cutoff frequency inversely proportional to the jet diameter. Ultrasound applied to the jet supply liquid can propagate down the jet when the carrier frequency is near-to or above the cutoff frequency. The jet response to the internal acoustic radiation pressure of amplitude modulated ultrasound was monitored along the jet using an optical pseudo-extinction method. The jet profile oscillates at the frequency of the radiation pressure modulation and, where the response is small, the amplitude was found to increase in proportion to the square of the acoustic pressure amplitude as previously demonstrated for oscillating drops [P. L. Marston and R. E. Apfel, J. Acoust. Soc. Am. 67, 27–37 (1980)]. Small amplitude deformations initially grow approximately exponentially with axial distance along the jet. Modulated radiation pressure can be used to stimulate large amplitude deformations and the breakup of the jet into drops. [Work supported by NASA.]

A Surface Forces Study of DNA Hybridization

We report the first use of a surface forces apparatus to study the hybridization of DNA. We study the regime of very high ionic strength (≈1 M) at which commercial DNA chip operations are performed. Using a thiol end-attached single-stranded oligonucleotide, we find that exposure to the complementary strand resulted in larger thickness. In addition, the resistance to small-amplitude shear deformations when opposed to a nonadsorbing surface (mica) increased significantly after hybridization. The thickness at onset of significant shear resistance was, for single-stranded DNA, considerably less than that of hybridized DNA. This could provide a new method to detect DNA hybridization efficiency.

The effect of fluid pressure on wave speeds in a cracked solid

In a porous or cracked elastic solid, the effective stress (defined in terms of the loads applied to the solid part of the outer boundary) and effective strain (defined in terms of the displacements at the solid part of the outer boundary) occurring in small-amplitude deformations are connected by a linear relation along with the pressure within the fluid occupying the pores and cracks. We derive here a formula of this kind for a static system in which enough time is allowed for pressure to be equalized throughout the fluid (on the assumption that all pockets of fluid are connected in some way). The formula depends on the overall stiffnesses relating stress to strain for the same material with the fluid removed (dry or empty cracks and pores). For undrained conditions where no fluid is allowed to enter or leave the body, the pressure is directly related to the effective stress and strain, and the Gassmann relations are obtained relating the stiffnesses for an isotropic material in dry and undrained conditions. For an anisotropic material, the Brown–Korringa relations are recovered. Externally imposed stresses and fluid pressure distort the material structure and influence the wave speeds of elastic waves. The main way in which this occurs is in changing the aspect ratios of flat cracks, the most compliant part of the microstructural geometry. This effect on the wave speeds is studied here both in terms of crack closure, with corresponding changes in crack number density, and in variations in crack aspect ratios. The principal way in which the latter influences the wave speeds is through the fluid incompressibility factor in the formula for the properties of materials with connected cracks. An increase in aspect ratio of the cracks is equivalent to a reduction in the bulk modulus of the fluid. This effect is apparent in the limits of both high frequencies, when the material behaves as if the cracks were isolated, and low frequencies, when undrained conditions apply.

The elasticity of a contact line

We calculate the work required to deform a three-phase contact line between two fluid phases and one solid phase using an interface displacement model. This includes the effects of both surface and line tensions. The leading-order dependence on the wavenumber q of the distortion has been described by Joanny and de Gennes (J. Chem. Phys. 81 (1984) 552 ). We examine the next-to-leading term, which contrary to expectation is not the line tension but a quantity of similar magnitude. Cases of divergent line tension are also examined, which demonstrate a regularization exhibiting scaling on a length scale set by q. Negative line tension does not lead to instability of the contact line to small amplitude deformations at any wavenumber.

Quantitative Prediction of Impact Forces In Elastomers

We measure the impact forces and deflections resulting from drop tests of a mass with a flat impact surface onto flat pads of various elastomeric materials, and show that the forces can be predicted quantitatively with no adjustable parameters by using a theory whose only inputs are the linear viscoelastic characteristics of the materials, measured in small-amplitude oscillatory deformations. The theory, which models the elastomer as a nonlinear neo-Hookean material, is accurate for several elastomeric solids including polyurethanes, polynorbomene, and poly-vinyl-chlorides (PVCs), over a wide range of impact velocities, masses, temperatures and pad thicknesses. Some steps are taken to extend the model to surfaces which are not flat. The application in mind is the rational design of elastomeric components in impact-tolerant portable electronic equipment.

Theory of Surface Deformations of Polymer Brushes in Solution

The excess free energy of small-amplitude deformations of a wet “parabolic” brush consisting of polymer chains attached to a solid plane by one end is calculated. The brush is swollen by a good (marginal) solvent. A generic deformation involving both compression normal to the grafting surface and lateral shear is considered. It is shown that the free energy attains a minimum as a function of the wave vector of the deformation, q. A shallow maximum of the scattering function S(q) is thus predicted. The theoretical scattering functions calculated for an arbitrary orientation of the scattering vector are compared with evanescent-wave light scattering data.