Hydrodynamic Memory Effects Research Articles

A Maneuvering Model for an Underwater Vehicle Near a Free Surface—Part I: Motion Without Memory Effects

In this paper, we propose a Lagrangian mechanics framework for modeling the motion of a submerged vehicle operating near a free surface in calm water. By augmenting the system Lagrangian used to derive Kirchhoff's equations for a rigid body moving in an unbounded fluid, we directly incorporate the free surface into the derivation of the equations of motion. This is done using a free surface Lagrangian, which accounts for the energy stored within the free surface due to the motion of a submerged vehicle. The system Lagrangian then enables us to derive the six-degrees-of-freedom nonlinear equations of motion using the Euler–Lagrange equations. The model structure is similar to standard maneuvering models for surface ships, though additional complexities are present since the hydrodynamic parameters are shown to depend on the vessel position and orientation relative to the free surface. We also examine a fundamental tradeoff between model simplicity and model fidelity: we choose to model only the instantaneous fluid response and neglect the hydrodynamic memory effects. Without the memory effects, the proposed model does not incorporate wave-resistance forces. However, the simplified representation of the hydrodynamics does enable a control-oriented representation of the nonlinear radiation forces and moments commonly associated with added mass and free surface suction effects.

Microreversibility and driven Brownian motion with hydrodynamic long-time correlations

A nonequilibrium fluctuation theorem is established for a colloidal particle driven by an external force within the hydrodynamic theory of Brownian motion, describing hydrodynamic memory effects such as the t−3∕2 power-law decay of the velocity autocorrelation function. The generalized Langevin equation is obtained for the general case of slip boundary conditions between the particle and the fluid. The Gaussian probability distributions for the particle to evolve in position–velocity space are deduced. It is proved that the joint probability distributions of forward and time-reversed paths have a ratio depending only on the work performed by the external force and the fluid temperature, in spite of the non-Markovian character of the generalized Langevin process.

Fluid-induced resonances in vibrational and Brownian dynamics of a shear oscillator

We generally describe vibrational and Brownian dynamics of a shear oscillator interacting with a confined Newtonian fluid. We show that the shear oscillator exhibits three characteristic dynamics in viscous, weak inertial, and strong inertial regimes, and the dynamics are controlled by two system parameters, effective confining height and oscillator-fluid coupling strength. While resonances of oscillators are usually deteriorated in fluids, we interestingly find the resonances arisen in originally overdamped oscillators, originated from hydrodynamic memory effects in strong inertial regime. The present theory could be exploited for improving designs and performances of shear oscillators in fluids, and reciprocally for investigating fluid properties by using shear oscillatory probes such as atomic force microscope in shear-mode.

Thermal Noise and Hydrodynamic Memory Effects on Force Measurements at Microscales

The paper is devoted to the problem of the determination of regular forces acting on microscopic and smaller objects in fluids when the presence of thermal noise affects the results of measurements. One of the methods on how these forces are determined is the measurement of the drift velocity of Brownian particles. Traditionally such experiments are interpreted on the basis of the overdamped Langevin equation. An exact expression has been obtained for this velocity within the linear hydrodynamic theory of the Brownian motion in incompressible fluids, when the external force is constant and when the particle is in a harmonic potential well. It is shown that for sufficiently short experimental times the influence of the hydrodynamic memory effects in the drift velocity determination is significant. The obtained solutions contain algebraic long-time tails due to which the results expected from the standard theory are approached very slowly with the increase of time.

High-resolution detection of Brownian motion for quantitative optical tweezers experiments

We have developed an in situ method to calibrate optical tweezers experiments and simultaneously measure the size of the trapped particle or the viscosity of the surrounding fluid. The positional fluctuations of the trapped particle are recorded with a high-bandwidth photodetector. We compute the mean-square displacement, as well as the velocity autocorrelation function of the sphere, and compare it to the theory of Brownian motion including hydrodynamic memory effects. A careful measurement and analysis of the time scales characterizing the dynamics of the harmonically bound sphere fluctuating in a viscous medium directly yields all relevant parameters. Finally, we test the method for different optical trap strengths, with different bead sizes and in different fluids, and we find excellent agreement with the values provided by the manufacturers. The proposed approach overcomes the most commonly encountered limitations in precision when analyzing the power spectrum of position fluctuations in the region around the corner frequency. These low frequencies are usually prone to errors due to drift, limitations in the detection, and trap linearity as well as short acquisition times resulting in poor statistics. Furthermore, the strategy can be generalized to Brownian motion in more complex environments, provided the adequate theories are available.

Motion of a colloidal particle in an optical trap

Thermal position fluctuations of a colloidal particle in an optical trap are measured with microsecond resolution using back-focal-plane interferometry. The mean-square displacement <Delta(x)2(t)> and power spectral density are in excellent agreement with the theory for a Brownian particle in a harmonic potential that accounts for hydrodynamic memory effects. The motion of a particle is dominated at short times by memory effects and at longer times by the potential. We identify the time below which the particle's motion is not influenced by the potential, and find it to be approximately tau(k)/20 , where tau(k) is the relaxation time of the restoring force of the potential. This allows us to exclude the existence of free diffusive motion, <Delta(x)2(t)> proportional to t, even for a sphere with a radius as small as 0.27 microm in a potential as weak as 1.5 microN/m. As the physics of Brownian motion can be used to calibrate an optical trap, we show that neglecting memory effects leads to an underestimation of more than 10% in the detector sensitivity and the trap stiffness for an experiment with a micrometer-sized particle and a sampling frequency above 200kHz . Furthermore, these calibration errors increase in a nontrivial fashion with particle size, trap stiffness, and sampling frequency. Finally, we present a method to evaluate calibration errors caused by memory effects for typical optical trapping experiments.

Direct Observation of Nondiffusive Motion of a Brownian Particle

The thermal position fluctuations of a single micron-sized sphere immersed in a fluid were recorded by optical trapping interferometry with nanometer spatial and microsecond temporal resolution. We find, in accord with the theory of Brownian motion including hydrodynamic memory effects, that the transition from ballistic to diffusive motion is delayed to significantly longer times than predicted by the standard Langevin equation. This delay is a consequence of the inertia of the fluid. On the shortest time scales investigated, the sphere's inertia has a small, but measurable, effect.

Dynamics of ships running aground

A comprehensive dynamic model is presented for analysis of the transient loads and responses of the hull girder of ships running aground on relatively plane sand, gravel, or rock sea bottoms. Depending on the seabed soil characteristics and the geometry of the ship bow, the bow will plow into the seabed to some extent. The soil forces are determined by a mathematical model based on a theory for frictional soils in rupture and dynamic equilibrium of the fluid phase in the saturated soil. The hydrodynamic pressure forces acting on the decelerated ship hull are determined by taking into account the effect of shallow water. Hydrodynamic memory effects on the transient hull motions are modeled by application of an impulse response technique. The ship hull is modeled as an elastic beam to determine the structural response in the form of flexural and longitudinal stress waves caused by the transient ground reaction and hydrodynamic forces. A number of numerical analysis results are presented for a VLCC running aground. The results include bow trajectory in the seabed, time variation of the grounding force, and the maximum values of the sectional shear forces and bending moments in the hull girder.