A two-dimensional cochlear fluid model based on conformal mapping

Using conformal mapping, fluid motion inside the cochlear duct is derived from fluid motion in an infinite half plane. The cochlear duct is represented by a two-dimensional half-open box. Motion of the cochlear fluid creates a force acting on the cochlear partition, modeled by damped oscillators. The resulting equation is one-dimensional, more realistic, and can be handled more easily than existing ones derived by the method of images, making it useful for fast computations of physically plausible cochlear responses. Solving the equation of motion numerically, its ability to reproduce the essential features of cochlear partition motion is demonstrated. Because fluid coupling can be changed independently of any other physical parameter in this model, it allows the significance of hydrodynamic coupling of the cochlear partition to itself to be quantitatively studied. For the model parameters chosen, as hydrodynamic coupling is increased, the simple resonant frequency response becomes increasingly asymmetric. The stronger the hydrodynamic coupling is, the slower the velocity of the resulting traveling wave at the low frequency side is. The model's simplicity and straightforward mathematics make it useful for evaluating more complicated models and for education in hydrodynamics and biophysics of hearing.     

Auditory localization of ground-borne vibrations in snakes

Interaural time differences allow many animals to perform azimuthal sound localization. Snakes lack a tympanic membrane, external ear openings, and any other superficial indication of an auditory mechanism. They do, however, possess an inner ear with functional cochlea. The oval window is connected through a loss-free osseous lever system to the two, de facto independent, sides of the lower jaw, which typically rest on the substrate. The footfall of prey generates small-amplitude, low propagation velocity, Rayleigh waves in the soil. This type of wave can be described as fluid motion. Accordingly we apply naval-engineering techniques to show that lower-jaw motion gives rise to a neuronal representation of the auditory world with realistic sensitivity and stereo precision.     

Flows of a generalized second grade fluid in a cylinder due to a velocity shock

Unsteady axial flows of second grade fluids with generalized fractional constitutive equation in a circular cylinder are studied. Flows are generated by a time-dependent pressure gradient in the axial direction, an external magnetic field perpendicular on the flow direction and by the cylinder motion. Two different problems are analyzed; one in which the cylinder velocity supports a shock at the instant t = 0 and another in which the cylinder motion is a translation with time-dependent velocity along the axis of cylinder. The generalized fractional constitutive equation of second grade fluid is described by the Caputo time-fractional derivative. Analytical solutions for the velocity field are obtained by using the Laplace transform with respect to time variable and the finite Hankel transform of order zero with respect to the radial coordinate. The influence of the fractional parameter of Caputo derivative on the fluid velocity has been studied by numerical simulations and graphical illustrations. It is found that the fractional fluid flows are faster than the ordinary second grade fluid.     

Three-dimensional numerical modeling for global cochlear dynamics

A hybrid analytical-numerical model using Galerkin approximation to variational equations has been developed for predicting global cochlear responses. The formulation provides a flexible framework capable of incorporating morphologically based mechanical models of the cochlear partition and realistic geometry. The framework is applied for a simplified model with an emphasis on application of hybrid methods for three-dimensional modeling. The resulting formulation is modular, where matrices representing fluid and cochlear partition are constructed independently. Computational cost is reduced using two methods, a modal-finite-element method and a boundary element-finite-element method. The first uses a cross-mode expansion of fluid pressure (2.5D model) and the second uses a waveguide Green's-function-based boundary element method (BEM). A novel wave number approach to the boundary element formulation for interior problem results in efficient computation of the finite-element matrix. For the two methods a convergence study is undertaken using a simplified passive structural model of cochlear partition. It is shown that basilar membrane velocity close to best place is influenced by fluid and structural discretization. Cochlear duct pressure fields are also shown demonstrating the 3D nature of pressure near best place.     

Force multipole moments for a spherically symmetric particle in solution

We present a general theorem for the force multipole moments of arbitrary order induced in a spherically symmetric particle immersed in a fluid whose motion satisfies the linear Navier-Stokes equation for steady incompressible viscous flow. The multipole moments are expressed in terms of the unperturbed fluid velocity field. It is shown that for a particle with a finite extension only a few terms give rise to fluid perturbations which are not confined to the interior of the particle. We give explicit results for a polymer satisfying the Debye-Bueche-Brinkman equations and for a hard sphere with mixed slip-stick boundary conditions.     

Basilar membrane mechanics at the base of the chinchilla cochlea. II. Responses to low-frequency tones and relationship to microphonics and spike initiation in the VIII nerve

Low-frequency stimuli (40- to 1000-Hz tones) have been used to correlate the motion of the 8-to 9-kHz place of the chinchilla basilar membrane with the cochlear microphonics recorded at the round window and with the responses of auditory nerve fibers with appropriate characteristic frequency. At the lowest stimulus frequencies, maximum displacement of the basilar membrane toward scala tympani occurs in near synchrony with maximum rarefaction at the eardrum and maximum negativity at the round window; at higher frequencies, the mechanical and microphonic response phases progressively lag rarefaction, reaching - 240 deg at 1000 Hz. At most frequencies (40-1000 Hz) near-threshold neural responses, once corrected for neural travel-time and synaptic delays, somewhat lead (by some 40 deg) maximal scala tympani displacement and maximal negativity of the round window microphonics. The variation of sensitivity with frequency is similar for basilar membrane displacement and microphonic responses: Under open-bulla conditions, sensitivity is constant for frequencies between 100 and 1000 Hz; below 100 Hz, sensitivity decreases at rates close to 12 dB/oct toward lower frequencies. Neural response sensitivity matches BM displacement more closely than BM velocity.     

Fluid volume displacement at the oval and round windows with air and bone conduction stimulation

The fluids in the cochlea are normally considered incompressible, and the fluid volume displacement of the oval window (OW) and the round window (RW) should be equal and of opposite phase. However, other channels, such as the cochlear and vestibular aqueducts, may affect the fluid flow. To test if the OW and RW fluid flows are equal and of opposite phase, the volume displacement was assessed by multiple point measurement at the windows with a laser Doppler vibrometer. This was done during air conduction (AC) stimulation in seven fresh human temporal bones, and with bone conduction (BC) stimulation in eight temporal bones and one human cadaver head. With AC stimulation, the average volume displacement of the two windows is within 3 dB, and the phase difference is close to 180 degrees for the frequency range 0.1 to 10 kHz. With BC stimulation, the average volume displacement difference between the two windows is greater: below 2 kHz, the volume displacement at the RW is 5 to 15 dB greater than at the OW and above 2 kHz more fluid is displaced at the OW. With BC stimulation, lesions at the OW caused only minor changes of the fluid flow at the RW.     

Nonlinear and active two-dimensional cochlear models: time-domain solution

A numerical solution method for two-dimensional (2-D) cochlear models in the time domain is presented. The method has particularly been designed for models with a cochlear partition having nonlinear and active mechanical properties. The 2-D cochlear model equations are reformulated as an integral equation for the acceleration of the basilar membrane (BM). This integral equation is discretized with respect to the spatial variable to yield a system of ordinary differential equations in the time variable. To solve this system, the variable step-size, fourth-order Runge-Kutta method described in Diependaal et al. [J. Acoust. Soc. Am. 82, 1655-1666 (1987)] is used. This method is robust and computationally efficient. The incorporation of a simple middle-ear model can be handled by this method. The method can also be extended to models in which the cochlear partition at each point along its length is represented by more than one degree of freedom.     

The anisorrhopic guiding-center fluid

A study is made of so-called “finite-orbit effects” in a two-dimensional guiding-center plasma. The macroscopic mass motion of the plasma is represented on the basis of a simple incompressible one-fluid model (so-called “representative fluid”), and the guiding-center motions of single particles are then referred to a Lagrangian coordinate network comoving with the representative fluid. The fluid motion defines the network motion. It turns out, however, to have no effect on the guiding-center motion relative to the network (autonomy theorem). It is found, in other words, that the relative trajectories of guiding centers are determinable in advance independently of the network motion (or the fluid motion), and this provides the necessary information to determine all the state parameters of the representative fluid (density of mass, density of gyrational angular momentum, etc.) as functions of the time, t, at any given point of the network. Once this information is available, the fluid motion is then completely determined by the remaining hydrodynamic equations (equation of motion, equation of incompressibility). The so-called “finite-orbit effects” take the form of gyroscopic-quasielastic forces in the equation of motion. No special isorrhopy condition is assumed. (This refers to a special initial condition assumed in an earlier work, for the sake of analytical simplicity. Here, the special initial condition is dropped.) Much attention is devoted to problems of wave propagation and stability. There are two independent sets of wave modes (if a nonvanishing anisorrhopy is allowed): so-called fluid modes, and so-called drift modes, respectively defined as first-order perturbations in the network motion (or the fluid motion) relative to the fixed coordinate frame, and in the guiding-center motion relative to the network. The stability conditions against both sets of modes are found to be quite stringent, much more so than in the earlier isorrhopic case. Nonetheless, a reasonably extensive class of stable solutions is shown to exist.     

Fractal behavior in quantum statistical physics

The properties of an ideal gas of spinless particles are investigated by using the path integral formalism. It is shown that the quantum paths exhibit a fractal character which remains unchanged in the relativistic domain provided the creation of new particles is avoided, and the Brownian motion remains the stochastic process associated with the quantum paths. These results are obtained by using a special representation of the Klein-Gordon wave equation. On the quantum paths the relation between velocity and momentum is not the usual one. The mean square value of the velocity depends on the time needed to define the velocity and its value shows the interplay between pure quantum effects and thermodynamics. The fractal character is also investigated starting from wave equations by analyzing the evolution of a Gaussian wave packet via the Hausdorff dimension. Both approaches give the same fractal character in the same limit. It is shown that the time that appears in the path integral behaves like an ordinary time, and the key quantity is the time interval needed for the thermostat to give to the particles a thermal action equal to the quantum of action. Thus, the partition function calculated via the path integral formalism also describes the dynamics of the system for short time intervals. For low temperatures, it is shown that a time-energy uncertainty relation is verified at the end of the calculations. The energy involved in this relation has not a thermodynamic meaning but results from the fact that the particles do not follow the equations of motion along the paths. The results suggest that the density matrix obtained by quantification of the classical canonical distribution function via the path integral formalism should not be totally identical to that obtained via the usual route.     

Analysis of peristaltic flow of Jeffrey six constant nano fluid in a vertical non-uniform tube

Peristaltic flow of non-Newtonian nano fluid through a non-uniform surface has been investigated in this paper. The fluid motion along the wall of the surface is caused by the sinusoidal wave traveling with constant speed. The governing equations are converted into cylindrical coordinate system and assuming low Reynolds number and long wave length partial differential equations are simplified. Analytically solutions of the problem are obtained by utilizing the homotopy perturbation method (HPM). In order to insight the impact of embedded parameters on temperature, concentration and velocity some graphs are plotted for different peristaltic waves. At the end, some observations were made from the graphical presentation that velocity, pressure rise and nano particle concentration are increasing function of thermophoresis parameter N_t while temperature and frictional forces show opposite trend.     

Efficient numerical methods for multiple surfactant-coated bubbles in a two-dimensional stokes flow

We present efficient and highly accurate numerical methods to compute the deformation of surfactant-coated, two-dimensional bubbles in a slow viscous flow. Surfactant acts to locally alter the surface tension and thereby change the nature of the interface motion. In this paper, we restrict our attention to the case of a dilute insoluble surfactant. The convection–diffusion equation for the surfactant concentration on the interface is coupled with the Stokes equations in the fluid domain through a boundary condition based on the Laplace-Young condition. The Stokes equations are first recast as an integral equation and then solved using a fast-multipole accelerated iterative procedure. The computational cost per time-step is only O(N log N) operations, with N being the number of discretization points on the interface. The bubble interfaces are described by a spectral mesh and is advected according to the fluid velocity in such a manner so as to preserve equal arc length spacing of marker points. This equal arc length framework has the dual advantage of dynamically maintaining the spatial mesh and allowing efficient, implicit treatment of the stiffest terms in the dynamics. Several phenomenologically different examples are presented.     

On dynamics of velocity vector potential in incompressible fluids

An elegant quaternionic formulation is given for the Lagrangian advection equation for velocity vector potential in fluid dynamics. At first we study the topological significance of a restricted conserved quantity viz., stream-helicity and later more realistic configuration of open streamlines is figured out. Also, using Clebsch parameterisation of the velocity vector potential yet another physical significance for the stream-helicity is provided. Finally we give a Nambu-Poisson formalism of the Lagrangian advection equation for velocity vector potential.     

Two-dimensional Euler flows in slowly deforming domains

We consider the evolution of an incompressible two-dimensional perfect fluid as the boundary of its domain is deformed in a prescribed fashion. The flow is taken to be initially steady, and the boundary deformation is assumed to be slow compared to the fluid motion. The Eulerian flow is found to remain approximately steady throughout the evolution. At leading order, the velocity field depends instantaneously on the shape of the domain boundary, and it is determined by the steadiness and vorticity-preservation conditions. This is made explicit by reformulating the problem in terms of an area-preserving diffeomorphism g_Λ which transports the vorticity. The first-order correction to the velocity field is linear in the boundary velocity, and we show how it can be computed from the time derivative of g_Λ.The evolution of the Lagrangian position of fluid particles is also examined. Thanks to vorticity conservation, this position can be specified by an angle-like coordinate along vorticity contours. An evolution equation for this angle is derived, and the net change in angle resulting from a cyclic deformation of the domain boundary is calculated. This includes a geometric contribution which can be expressed as the integral of a certain curvature over the interior of the circuit that is traced by the parameters defining the deforming boundary.A perturbation approach using Lie series is developed for the computation of both the Eulerian flow and geometric angle for small deformations of the boundary. Explicit results are presented for the evolution of nearly axisymmetric flows in slightly deformed discs.     

FREE VIBRATION OF A PARTIALLY LIQUID-FILLED AND SUBMERGED, HORIZONTAL CYLINDRICAL SHELL

The dynamic characteristics (i.e., natural frequencies and mode shapes) of a partially filled and/or submerged, horizontal cylindrical shell are examined. In this investigation, it is assumed that the fluid is ideal, and fluid forces are associated with inertial effects only: namely, the fluid pressure on the wetted surface of the structure is in phase with the structural acceleration. The in vacuo dynamic characteristics of the cylindrical shell are obtained using standard finite element software. In the “wet” part of the analysis, it is assumed that the shell structure preserves its in vacuo mode shapes when in contact with the contained and/or surrounding fluid and that each mode shape gives rise to a corresponding surface pressure distribution of the shell. The fluid-structure interaction effects are calculated in terms of generalized added masses, using a boundary integral equation method together with the method of images in order to impose an appropriate boundary condition on the free surface. To assess the influence of the contained and/or surrounding fluid on the dynamic behaviour of the shell structure, the wet natural frequencies and associated mode shapes were calculated and compared with available experimental measurements.     

Equations of hydrodynamics in general relativity in the slow motion approximation

By means of a formal solution to the Einstein gravitational field equations a slow motion expansion in inverse powers of the speed of light is developed for the metric tensor. The formal solution, which satisfies the deDonder coordinate conditions and the Trautman outgoing radiation condition, is in the form of an integral equation which is solved iteratively. A stress-energy tensor appropriate to a perfect fluid is assumed and all orders of the metric needed to obtain the equations of motion and conserved quantities to the 21/2post-Newtonian approximation are found. The results are compared to those obtained in another gauge by S. Chandrasekhar. In addition, the relation of the fast motion approximation to the slow motion approximation is examined.     

Can the photon velocity be derived from the Klein-Gordon equation?

In their most recent article, Grado-Caffaro et al. have addressed the question of the ‘photon velocity’. They have expressed the photon velocity in terms of the wavefunctions of the Klein-Gordon equation (Grado-Caffaro and Grado-Caffaro [4]). In this note, we closely follow their work and explicitly obtain the photon velocity using the free solutions of the Klein-Gordon equation. It is shown that the plane wave solutions give rise to six possible values of the photon velocity. Two of these solutions are the most expected (v=±c). The remaining four solutions, the real pair ±0.786c and the imaginary pair ±1.272ic are difficult to comprehend.     

Synchrotron motion with radiation reaction

It is shown that the longitudinal velocity of a charged particle moving in a uniform magnetic field, and obeying Dirac-Lorentz relativistic equation of motion with radiation reaction is constant. Suitable approximate methods, which give fairly accurate results, are used to obtain the expression for velocity and displacement along the transverse section. They describe the motion completely up to a correcting factor $$1 + 0\left\{ {\left( {\frac{{e^3 B}}{{m^3 c^4 }}} \right)^2 } \right\}; \frac{{e^3 B}}{{m^3 c^4 }} \simeq 10^{ - 16} B$$ for electrons,B inG.     

Intracochlear pressure and derived quantities from a three-dimensional model

Intracochlear pressure is calculated from a physiologically based, three-dimensional gerbil cochlea model. Olson [J. Acoust. Soc. Am. 103, 3445-3463 (1998); 110, 349-367 (2001)] measured gerbil intracochlear pressure and provided approximations for the following derived quantities: (1) basilar membrane velocity, (2) pressure across the organ of Corti, and (3) partition impedance. The objective of this work is to compare the calculations and measurements for the pressure at points and the derived quantities. The model includes the three-dimensional viscous fluid and the pectinate zone of the elastic orthotropic basilar membrane with dimensional and material property variation along its length. The arrangement of outer hair cell forces within the organ of Corti cytoarchitecture is incorporated by adding the feed-forward approximation to the passive model as done previously. The intracochlear pressure consists of both the compressive fast wave and the slow traveling wave. A Wentzel-Kramers-Brillowin asymptotic and numerical method combined with Fourier series expansions is used to provide an efficient procedure that requires about 1 s to compute the response for a given frequency. Results show reasonably good agreement for the direct pressure and the derived quantities. This confirms the importance of the three-dimensional motion of the fluid for an accurate cochlear model.