Kinematic irreversibility in surfactant-laden interfaces

The surface shear viscosity of an insoluble surfactant monolayer often depends strongly on its surface pressure. Here, we show that a particle moving within a bounded monolayer breaks the kinematic reversibility of low-Reynolds-number flows. The Lorentz reciprocal theorem allows such irreversibilities to be computed without solving the full nonlinear equations, giving the leading-order contribution of surface-pressure-dependent surface viscosity. In particular, we show that a disk translating or rotating near an interfacial boundary experiences a force in the direction perpendicular to that boundary. In unbounded monolayers, coupled modes of motion can also lead to non-intuitive trajectories, which we illustrate using an interfacial analog of the Magnus effect. This perturbative approach can be extended to more complex geometries, and to 2D suspensions more generally

Absolute Entropy and Energy of Carbon Dioxide Using the Two-Phase Thermodynamic Model

The two-phase thermodynamic (2PT) model is used to determine the absolute entropy and energy of carbon dioxide over a wide range of conditions from molecular dynamics trajectories. The 2PT method determines the thermodynamic properties by applying the proper statistical mechanical partition function to the normal modes of a fluid. The vibrational density of state (DoS), obtained from the Fourier transform of the velocity autocorrelation function, converges quickly, allowing the free energy, entropy, and other thermodynamic properties to be determined from short 20-ps MD trajectories. The anharmonic effects in the vibrations are accounted for by the broadening of the normal modes into bands from sampling the velocities over the trajectory. The low frequency diffusive modes, which lead to finite DoS at zero frequency, are accounted for by considering the DoS as a superposition of gas-phase and solid-phase components (two phases). The analytical decomposition of the DoS allows for an evaluation of properties contributed by different types of molecular motions. We show that this 2PT analysis leads to accurate predictions of entropy and energy of CO_2 over a wide range of conditions (from the triple point to the critical point of both the vapor and the liquid phases along the saturation line). This allows the equation of state of CO_2 to be determined, which is limited only by the accuracy of the force field. We also validated that the 2PT entropy agrees with that determined from thermodynamic integration, but 2PT requires only a fraction of the time. A complication for CO_2 is that its equilibrium configuration is linear, which would have only two rotational modes, but during the dynamics it is never exactly linear, so that there is a third mode from rotational about the axis. In this work, we show how to treat such linear molecules in the 2PT framework

Hydromechanics of low-Reynolds-number flow. Part 5. Motion of a slender torus

In order to elucidate the general Stokes flow characteristics present for slender bodies of finite centre-line curvature the singularity method for Stokes flow has been employed to construct solutions to the flow past a slender torus. The symmetry of the geometry and absence of ends has made a highly accurate analysis possible. The no-slip boundary condition on the body surface is satisfied up to an error term of O(E^2 ln E), where E is the slenderness parameter (ratio of cross-sectional radius to centre-line radius). This degree of accuracy makes it possible to determine the force per unit length experienced by the torus up to a term of O(E^2). A comparison is made between the force coefficients of the slender torus to those of a straight slender body to illustrate the large differences that may occur as a result of the finite centre-line curvature

The Effects of Wing Rotation on Unsteady Aerodynamic Performance at Low Reynolds Numbers

The downstroke-to-upstroke transition of many insects is characterized by rapid wing rotation. The aerodynamic consequences of these rapid changes in angle of attack have been investigated using a mechanical model dynamically scaled to the Reynolds number appropriate for the flight of small insects such as Drosophila. Several kinematic parameters of the wing flip were examined, including the speed and axis of rotation, as well as the duration and angle of attack during the wing stroke preceding rotation. Alteration of these kinematic parameters altered force generation during the subsequent stroke in a variety of ways. 1. When the rotational axis was close to the trailing edge, the model wing could capture vorticity generated during rotation and greatly increase aerodynamic performance. This vortex capture was most clearly manifested by the generation of lift at an angle of attack of 0°;. Lift at a 0°; angle of attack was also generated following rotation about the leading edge, but only if the downstroke angle was large enough to generate a von Karman street. The lift may be due to an alteration in the effective angle of attack caused by the inter-vortex stream in the downstroke wake. 2. The maximum lift attained (over all angles of attack) was substantially elevated if the wing translated backwards through a wake generated by the previous stroke. Transient lift coefficient values of nearly 4 were obtained when the wing translated back through a von Karman street generated at a 76.5°; angle of attack. This effect might also be explained by the influence of the inter-vortex stream, which contributes a small component to fluid velocity in the direction of translation. 3. The growth of lift with angle of attack was significantly elevated following a 7.5 chord stroke with a 76.5°; angle of attack, although it was relatively constant under all other kinematic conditions. 4. The results also indicate the discrepancies between transient and time-averaged measures of performance that arise when unsteady mechanisms are responsible for force generation. Although the influence of wing rotation was strong during the first few chords of translation, averaging the performance over as little as 6.5 chords of motion greatly attenuated the effects of rotation. 5. Together, these modeling results suggest that the unsteady mechanisms generated by simple wing flips could provide an important source for the production of aerodynamic forces in insect flight. Furthermore, the extreme sensitivity to small variations in almost all kinematic parameters could provide a foundation for understanding the aerodynamic mechanisms underlying active flight control

Variational formulation of ideal fluid flows according to gauge principle

On the basis of the gauge principle of field theory, a new variational formulation is presented for flows of an ideal fluid. The fluid is defined thermodynamically by mass density and entropy density, and its flow fields are characterized by symmetries of translation and rotation. The rotational transformations are regarded as gauge transformations as well as the translational ones. In addition to the Lagrangians representing the translation symmetry, a structure of rotation symmetry is equipped with a Lagrangian $\Lambda_A$ including the vorticity and a vector potential bilinearly. Euler's equation of motion is derived from variations according to the action principle. In addition, the equations of continuity and entropy are derived from the variations. Equations of conserved currents are deduced as the Noether theorem in the space of Lagrangian coordinate \ba. Without $\Lambda_A$, the action principle results in the Clebsch solution with vanishing helicity. The Lagrangian $\Lambda_A$ yields non-vanishing vorticity and provides a source term of non-vanishing helicity. The vorticity equation is derived as an equation of the gauge field, and the $\Lambda_A$ characterizes topology of the field. The present formulation is comprehensive and provides a consistent basis for a unique transformation between the Lagrangian \ba space and the Eulerian \bx space. In contrast, with translation symmetry alone, there is an arbitrariness in the ransformation between these spaces.Comment: 34 pages, Fluid Dynamics Research (2008), accepted on 1st Dec. 200

Particle motion between parallel walls: Hydrodynamics and simulation

The low-Reynolds-number motion of a single spherical particle between parallel walls is determined from the exact reflection of the velocity field generated by multipoles of the force density on the particle’s surface. A grand mobility tensor is constructed and couples these force multipoles to moments of the velocity field in the fluid surrounding the particle. Every element of the grand mobility tensor is a finite, ordered sum of inverse powers of the distance between the walls. These new expressions are used in a set of Stokesian dynamics simulations to calculate the translational and rotational velocities of a particle settling between parallel walls and the Brownian drift force on a particle diffusing between the walls. The Einstein correction to the Newtonian viscosity of a dilute suspension that accounts for the change in stress distribution due to the presence of the channel walls is determined. It is proposed how the method and results can be extended to computations involving many particles and periodic simulations of suspensions in confined geometries

Exact solutions for hydrodynamic interactions of two squirming spheres

We provide exact solutions of the Stokes equations for a squirming sphere close to a no-slip surface, both planar and spherical, and for the interactions between two squirmers, in three dimensions. These allow the hydrodynamic interactions of swimming microscopic organisms with confining boundaries, or each other, to be determined for arbitrary separation and, in particular, in the close proximity regime where approximate methods based on point singularity descriptions cease to be valid. We give a detailed description of the circular motion of an arbitrary squirmer moving parallel to a no-slip spherical boundary or flat free surface at close separation, finding that the circling generically has opposite sense at free surfaces and at solid boundaries. While the asymptotic interaction is symmetric under head-tail reversal of the swimmer, in the near field microscopic structure can result in significant asymmetry. We also find the translational velocity towards the surface for a simple model with only the lowest two squirming modes. By comparing these to asymptotic approximations of the interaction we find that the transition from near- to far-field behaviour occurs at a separation of about two swimmer diameters. These solutions are for the rotational velocity about the wall normal, or common diameter of two spheres, and the translational speed along that same direction, and are obtained using the Lorentz reciprocal theorem for Stokes flows in conjunction with known solutions for the conjugate Stokes drag problems, the derivations of which are demonstrated here for completeness. The analogous motions in the perpendicular directions, i.e. parallel to the wall, currently cannot be calculated exactly since the relevant Stokes drag solutions needed for the reciprocal theorem are not available.Comment: 27 pages, 7 figure

Similarity transformations for the two-dimensional, unsteady, stream-function equation

The methods described by Bluman & Cole (1974) are used to derive the infinitesimals of the general invariance group of the unsteady, two-dimensional, stream-function equation for the case where the kinematic viscosity v is equal to a constant and the case where v = 0. The infinitesimals in each case involve ten independent parameters, seven of which appear explicitly and three of which are contained implicitly in three arbitrary functions of time. The various finite groups and similarity transformations which may be derived from the infinitesimals are discussed through examples. Two of the arbitrary functions of time are non-trivial and represent invariance of the stream-function equation under a transformation to a co-ordinate system which moves in a non-uniform irrotational fashion. A general similarity form is derived for which the equations dx/dt = u(x, y, t) and dy/dt = v(x, y, t) for the particle paths may be reduced to an autonomous system. This form is general enough to suggest the hypothesis that, under certain restrictions, the entrainment processes of unsteady flows dominated by two-dimensional large-scale motions may be displayed diagrammatically on a phase-plane plot of particle trajectories