Linear Momentum Equations Research Articles

A theoretical model for the noninvasive assessment of the transmitral pressure-flow relation

The purpose of this paper is to formulate from the equations of fluid mechanics an equation which describes the transmitral pressure-flow relationship. According to the linear momentum equation applied to the atrioventricular coupling, the left-atrium-left-ventricle pressure difference ( P a−P v) can be written as P a−P v=A∂v∂t+ Bv 2 + Cv, where v is the transmitral blood velocity and A, B, and C are variables related to the geometry of the atrium, ventricle and mitral orifice, respectively. Based on this theory, P a−P v is calculated noninvasively in a patient with a nonobstructive mitral valve. Mitral flow and cardiac dimensions recorded by Doppler echocardiography are digitized and analyzed. Calculation shows that P a−P v reaches its peak value at the time of flow peak acceleration and has already considerably decreased at the time of peak velocity. The time course of calculated P a−P v is in close agreement with the published experimental catherization data. Numerical computation of early diastolic left atrium and left ventricle pressure curves based on the experimental data of others for the time constant of left ventricular relaxation, left atrial and ventricular chambers stiffness constants, combined with sine-waveform-simulated mitral flow, verifies the time course and the magnitude of P a−P v as predicted from flow equations. This paper provides a theoretical method for the noninvasive assessment of the transmitral pressure-flow relationship using ultrasound technique and might help to achieve a better understanding of the diastolic function as assessed by Doppler echocardiography.

A gauge field theory of dislocations and disclinations with irreversible processes

We, basing on a variation principle, proposed a gauge field theory of dislocations and disclinations with irreversible processes. The linear momentum equation, the dislocation and disclination balance equation, the energy balance equation, and the entropy balance equation have been derived. It shows that the internal production of entropy is due to three parts i.e. heat conduction, plastic deformation and the motions of dislocations and disclinations, and viscous dissipation in general situation.

Sound wave propagation through emulsions, colloids, and suspensions using a generalized Fick’s law

A multiphase, continuum theory, that uses volume-averaged balance equations for mass, linear momentum, and energy, is developed to predict the speed and attenuation of small-amplitude sound waves propagating through a suspension of rigid spherical particles. Both a total and a diffusive momentum equation are used rather than the linear momentum equations for each individual phase. Furthermore, this approach admits constitutive equations that satisfy the second law of thermodynamics and enables phoretic effects, such as thermophoresis and barophoresis, to be expressed explicitly as flux terms for the production of linear momentum in Fick’s law. Including the phoretic terms in the viscothermal acoustics problem results in a bicubic algebraic equation for the complex propagation coefficient whose roots can be determined numerically. A low-frequency expansion provides the interpretation that attenuation is due to viscous, thermal conduction, phoretic and Dufour effects. Furthermore, it has been found that by using only the definition of the diffusion coefficient in the constitutive equation for the linear momentum supply, and the hydrodynamic formula for the drag on a particle given by Stokes, one obtains the Einstein formula for the diffusion coefficient.

The dynamics of avalanches of granular materials from initiation to runout. Part I: Analysis

This paper describes a model to predict the flow of an initially stationary mass of cohesion-less granular material down rough curved beds. This work is of interest in connection with the motion of rock and ice avalanches and dense flow snow avalanches. The constitutive behaviour of the material making up the pile is assumed to be described by a Mohr-Coulomb criterion while the bed boundary condition is treated by a similar Coulomb-type basal friction law assumption. By depth averaging the incompressible conservation of mass and linear momentum equations that are written in terms of a curvilinear coordinate system aligned with the curved bed, we obtain evolution equations for the depthh and the depth averaged velocityū. Three characteristic length scales are defined for use in the non-dimensionalization and scaling of the governing equations. These are a characteristic avalanche lengthL, a characteristic heightH, and a characteristic bed radius of curvatureR. Three independent parameters emerge in the non-dimensionalized equations of motion. One, which is the aspect ratio e-H/L, is taken to be small. By choosing different orderings for the other two, the tangent of the bed friction angle δ and the characteristic non-dimensional curvature λ=L/R, we can obtain different sets of equations of motion which appropriately display the desired importance of bed friction and bed curvature effects. The equations, correct to order e for moderate curvature, are discretized in the form of a Lagrangian-type finite difference representation which proved to be successful in the earlier studies of Savage and Hutter [24] for granular flow down rough plane surfaces. Laboratory experiments were performed with plastic particles flowing down a chute having a bed made up of a straight, inclined portion, a curved part and a horizontal portion. Numerical solutions are presented for conditions corresponding to the laboratory experiments. It is found that the predicted temporal-evolutions of the rear and front of the pile of granular material as well as the shape of the pile agree quite well with the laboratory experiments.

Pressure and flow in fluid films constrained between translating and vibrating flexible surfaces

In this paper, the dynamic pressure and flow developed in a two-dimensional, viscous fluid film constrained between flexible surfaces are analyzed. The problem formulation assumes that the response of the flexible surface is governed by linear equations of motion, and the fluid motion is governed by linearized momentum equations including the unsteady inertia. Three states of the model are developed to describe the coupled fluid-structural response problem. The fluid dynamic pressure is derived in the frequency domain as a function of the fluid impedances and the surface transverse vibrations. The perturbed, coupled problem is described by an integral equation (in state vector form) that governs the coupled responses of the flexible surfaces. The integral equation is solved by a discretization method. The analysis is applied to a rigid slider bearing with a flexible, translating plate surface under the excitation of a harmonic point load. The accuracy of the discretization method is evaluated, and numerical results for the dynamic pressure and the plate response are presented.

Radial inflow within two flat disks

Introduction T HE characteristics of flows confined in narrow gaps formed by two flat circular plates are of interest in industry. Disk-type devices have been widely used in many practical applications.' Two basic nonswirling types of flow can be produced inside the gap: one is developed by supplying the fluid through a centrally located inlet port, and the other is realized by admitting the flow radially through the circumference. For the former case, the streamlines diverge and the gradual decrease of the local Reynolds number based on the gap height gives rise to the interesting phenomenon of relaminarization. In the latter case, the streamlines converge, resulting in a drastic pressure drop similar to that produced in a converging nozzle. The present work focuses on the second type of flow. Previous theoretical studies have concentrated on solutions to the integral equations of motion.~ Lee and Lin attempted a differential solution of the linearized momentum equation to obtain the radial pressure distribution. Their method of treating the equation led them to an expression for pressure that was amenable only to a numerical treatment. Nonetheless, the results obtained have shown a good agreement with the experiment of Ref. 6. In a recent paper, the author, convinced also that only a numerical result for the linearized momentum equation was possible, simplified further the inertial terms. With the additional simplification, he was able to obtain a closed-form solution for the pressure which agreed closely with the numerical results of Ref. 7. In the present Note, closed-form solutions of the linearized momentum equations of Lee and Lin are presented. The inviscid and creeping-flow solutions are shown to be the asymptotes of the general problem. The results have been found to confirm the observation. Analysis and Discussion Consider the nonswirling, steady laminar flow between the disks shown in Fig. 1. Under the assumption that the flow is purely radial, the equations of motion, in nondimensional form, are Continuity:

Sound wave propagation through multiphase materials

Theories for infinitesimal, planar sound wave propagation in a dilute suspension of rigid particles in a viscous fluid has been investigated by generally three approaches: (1) wave scattering, (2) hydrodynamic, and (3) ad hoc approaches specific to particular systems. Here, a hydrodynamic development that uses spatially averaged continuum balance and constitutive equations for multiphase materials is presented. Two alternative approaches derive equivalent results (i.e., a bicubic polynomial equation) for the complex propagation constant χ = − (a + ik), where a is the spatial attenuation coefficient and k is the wavenumber. One approach uses linear momentum equations for each individual phase, while the other uses an overall linear momentum equation and a relative momentum equation, obtained by taking the sum and difference of the individual momentum equations for the continuous and particulate phases. The advantage of this latter aproach is that it expresses the results in the form of a generalized Fick's law, and expresses the diffusion of particles, as well as the supply terms such as barophoresis, thermophoresis, and pycnophoresis, explicitly. Furthermore, an Einstein relation can be obtained by simply defining a diffusion coefficient, and interpreting this coefficient in the linear momentum supply term that is proportional to the relative velocity as a Stokes viscous drag force.

Ferrohydrodynamic torque-driven flows

Ferrofluid motion driven by a travelling wave magnetic field can be in the same or opposite direction to the direction of wave propagation. A net time average body force on a ferrofluid is produced when there is a time phase lag between the ferrofluid magnetization M and the driving magnetic field H . With M and H not collinear, there is also a dipole torque exerted on each magnetic domain which is balanced by the fluid viscous drag. Conservation of linear and angular momentum equations are solved in the steady state for a ferrofluid layer stressed by a travelling wave magnetic field. This configuration can be easily set up as an experiment, allows closed form solutions for the time average flow and spin fields, and exemplifies the essential physics of ferrohydrodynamic interactions.

A constitutive theory for snow as a continuous multiphase mixture

Snow on the ground is viewed in this formulation, as a saturated two-phase granular material comprised of small grains of ice with interstitial pores filled by a single vapor. The snow is considered as a continuous mixture in which the ice and vapor constituents are themselves treated as individual but interacting continua. Mathematical modeling of the Snow is accomplished using a relatively recent continuum theory for mixtures where the individual constituents are physically separate. This approach considers the volume fraction occupied by each constituent as an additional kinematic variable. Therefore, in addition to the balance equations for mass, linear momentum, angular momentum and energy, usually applied in continuum mechanics, an equation which accounts for changes in the volume fraction, called the balance of equilibrated force, is included. Balance equations for each constituent as well as for the mixture are considered. The immiscible nature of the constituents allows constitutive equations to be developed which depend only on those variables which pertain to that constituent. Exchange between the ice and vapor is aecounted for by interaction terms which enter the theory through the balance equations for the constituents. Forms for these interaction terms are used which guarantee that the entropy inequality is not violated. A one-dimensional analysis of an isothermal homogeneous snow cover suddenly subjected to a colder surface temperature reveals a thermodynamically active zone associated with a large temperature gradient, initially located near the top surface, but which moves downward with time in a wavelike fashion decreasing in intensity. Slight differences in constituent temperatures are calculated during the more active transient phase in conjunction with a decrease in snow density.

A three‐dimensional coupled ice‐ocean model of coastal circulation

A coupled ice‐ocean model is employed to examine roles of ocean circulation on sea ice formation and distribution along the east coasts of North America and Greenland. The model has a flat bottom and a rectangular domain with a coastal boundary to the west, an artificial solid boundary to the north, and open eastern and southern boundaries. The model is driven by idealized alongshore wind stress and atmospheric cooling; the wind stress decays northward and southward with a maximum amplitude at the middle. The open water heat flux is locally specified with both seaward and southward decay, and the heat flux through ice is taken to be inversely proportional to the ice thickness. Ice internal stresses are simplified under the assumption that the major strain is cross‐shore shear deformation. The ocean model has three levels: the surface mixed layer and two levels in the lower ocean. Linear momentum equations are used, and a geostrophic cross‐shore momentum balance is assumed. Vertical viscosity is represented only by the surface and bottom Ekman flow. Equations for temperature and salinity retain advection and diffusion terms in both vertical and horizontal directions. The model reveals the following important roles of the ocean: northerly wind induces shoreward Ekman flow and southward barotropic flow, which develops only to the southern (downstream in the sense of long coastal‐trapped wave propagation) side of wind forcing region and merges westward owing to the β‐effect. Coastal downwelling produces additional baroclinic alongshore flow, intensifying (weakening) the southward flow in the upper (lower) ocean. Cyclonic circulation is generated by differences in Ekman flow between the ice‐covered onshore area and open‐water offshore area. Ice, which occupies the coastal area, is narrowed dynamically by the shoreward Ekman flow and thermodynamically by the warm offshore water in the Ekman flow. Ice is also advected southward by the alongshore current, associated with the coastal flow and the cyclonic circulation, as well as direct wind stress. When a lower ocean is warmer than the freezing point, convective overturning carries heat upward and reduces ice formation. Ice grows faster over downwelled and southward advected cold anomalies in the lower ocean.

Surface Rheology II. The Curved Fluid Surface

The surface rheological theory presented in the first part of this series for the planar fluid surface is extended to curved fluid surfaces with curvature approaching that which is observed in microemulsion systems. Both the surface excess linear momentum equation and the moment of linear momentum equation are developed. The balances are for the first time generalized to include large surface curvature stresses, introducing new first and second moments of surface excess shear and dilatational viscosity. The classical “zeroth order” surface excess shear and dilatational viscosities are shown to be curvature dependent. The curvature dependency of each zeroth moment surface viscosity coefficient is shown completely defined by the respective first and second moment viscosity coefficients.

Yangtze Brackish water plume—circulation and diffusion

A system of 3-D linearized momentum equations, the continuity equation and a simplified equation of state for sea water are analytically solved. The solution comprises three parts. The first part is a wind-driven current, the second a jet-like current caused by Yangtze River outflow and the last a density flow. The computed current velocity is obtained by solving an advection-diffusion equation of salinity by a finite-difference method. The results show that the Yangtze brackish water plume spreads as a large low salinity tongue. At the surface, the turning of the tongue axis is mainly controlled by wind stress, Yangtze run-off and bottom topography. Near the bottom, the axis changes little all the year around. At about 10–15 m depth, there is a sharp halocline in summer, but at deeper layers the vertical distribution is almost always uniform. The computed salinity distribution agrees fairly well with the observed one.

Sound wave propagation in viscoelastic fluids with simultaneous chemical reactions

The viscothermal problem of infinitesimal, planar acoustic wave propagation in nondiffusive, multicomponent materials that are viscoelastic and reacting is studied. The attenuation and dispersion of the sound wave are determined by solving the linearized (first-order) equations of mass, linear momentum, energy, and chemical kinetics. The analysis assumes the simple fluid theory for the stress tensor that incorporates the memory of the material. General results are obtained in the form of a biquadratic characteristic equation (the modified Kirchhoff–Langevin equation) for the complex propagation constant χ=−(α+iω/c), where α is the attenuation coefficient, c is the phase speed of the progressive wave, and ω is the angular frequency. Approximations needed to interpret results in terms of a linear superposition of chemical reaction and viscoelastic relaxations are examined. Calculations of the absorption and dispersion in cyclohexanol were made and compared to experimental measurements reported by Rasulmukhamedova et al. [Sov. Phys. Acoust. 22, 422 (1976)] over an extended range of frequencies.

A complete stress-function representation of stresses for a continuum of second grade

At present, Cauchy's equations of linear momentum and of moment of momentum seem to demand a modification, advancing forward Cauchy's classical considerations in the sense of incorporating the concept of couple-stresses in those equations. This concept is well-known in the theory of Cosserat-continua, less so in the theory of point-continua. However, the theory of point-continua is widely known and has been applied with great success e.g. to such materials as steel, water or air in engineering mechanics. Incorporating the concept of couple-stresses in the theory of point-continua, however, involves many serious problems. One such problem is the type of structure of the couple-stress tensor. An-other problem is the question of the completeness of a stress-function representation of stresses. In the following paper the author presents a complete stress-function representation of stresses for his «refined theory» of the point-continuum. The main features of incorporation of the concept of couple-stresses in that theory—e.g. the reason why the couple-stress tensor must be of skew-symmetric structure—have already been presented in [1].

The Reverse Spaghetti Problem: Drooping Motion of an Elastica Issuing from a Horizontal Guide

The nonlinear equations of motion of an elastica that moves out of a horizontal guide at a constant velocity are expressed in terms a dimensionless weight-to-stiffness ratio and a dimensionless velocity. The equations are written in horizontal-vertical directions rather than tangential-normal directions to minimize algebraic complexities. The introduction of deformation potentials allows each of the linear momentum equations to be integrated once. This simplifies the remaining equations. A series solution of the equations, useful for small motions—and perhaps useful for design—is given. To facilitate numerical solution, the triangular space-time domain of the problem is transformed into a square domain in pseudo space-time. Finally, some solutions based on the finite element method are presented for typical values of the dimensionless weight-to-stiffness and velocity parameters.

The pressure correction method

Abstract The pressure correction technique for solving the Navier-Stokes and Euler equations is studied. A general pressure correction analysis is presented using the full form of the linearized momentum equations. It is shown how the SIMPLE procedure results from adopting the most approximate form of the momentum equations. Two extended pressure correction techniques are derived using higher-order approximations and shown to be up to 50% faster than SIMPLE on a test example. Finally, a number of alternative approaches are considered and their performance evaluated against SIMPLE.

Macroscopic Balance Equations in Soils and Aquifers: The Case of Space‐ and Time‐Dependent Instrumental Response

Macroscopic balance equations for mass and linear momentum in soils and aquifers are derived in the relativist concept under the condition that all physical variables are associated with a single instrumental weighting function that depends on both position and time. The resulting balance equations differ in concept from expressions obtained recently in the representative elementary volume (REV) approach by exhibiting additional terms that depend on the time and space derivatives of the instrumental weighting function. These terms indicate that mass and linear momentum balance at the macroscopic scale are not preserved unless the weighting function has space‐time invariance properties over the period and spatial domain of measurement.

Sound wave propagation in a viscoelastic fluid with coupled chemical reactions

The classical viscothermal problem of forced, infinitesimal, planar acoustic wave propagation in a single‐component Newtonian fluid is extended to a viscoelastic fluid with coupled chemical reactions. The response of the fluid is represented by Noll's simple‐fluid theory which incorporates the memory of the material. In addition, the treatment accounts for multiple‐step independent chemical reactions. The attenuation and dispersion of the sound wave are determined by solving the linearized (first‐order) equations of mass, linear momentum, energy, and chemical kinetics. General results for the complex propagation coefficient, χ = −(α + iω/c), were obtained in the form of a biquadratic characteristic equation. α is the attenuation coefficient, c is the phase speed of the progressive wave, and ω is the angular frequency. The theory has been applied to interpret measurements reported on cyclohexanol [D. A. Rasulmukhamedova, P. K. Khabibullaev, and M. G. Khaliulin, Sov. Phys. Acoust. 22, 422–426 (1976)] over a...

The Operational Significance of the Continuum Hypothesis in the Theory of Water Movement Through Soils and Aquifers

The operational meaning of the representative elementary volume (REV) concept, on which current foundational theories of water movement through porous media are based, is analyzed critically. It is concluded that the REV concept as applied to real porous media is both unnecessarily restrictive and experimentally unverifiable. In its place a relativist concept is proposed in which macroscopic physical variables are defined as convolution products of microscopic properties of a porous medium with weighting functions that represent the appropriate measuring instruments. The effects of resolution and precision differences among experimental measuring devices, now recognized as critical in the assessment of spatial variability in the properties of soils and aquifers, can be incorporated naturally within the relativist concept but are excluded a priori in the REV concept. The relativist point of view, unlike the REV concept, has a clear operational meaning and, in principle, extends theoretical validity to all local measurements made on porous media. It is sufficient mathematically to derive conventional macroscopic balance equations for mass and linear momentum as well as a differential equation for the isothermal transport of water through an unsaturated, anisotropic, deformable soil or aquifer.

The turn function and vorticity method for numerical fluid dynamics

A numerical method is presented that solves in a consistent fashion, conservation equations for both vorticity and linear momentum in multidimensional fluid-dynamics calculations. The equations are given in both two- and three-dimensional Cartesian geometry, and it is shown how the method can be easily implemented in a two-dimensional Eulerian fluid-dynamics code. The results of example calculations, which were performed with and without the new method, show the large errors that can arise when the vorticity equation is not solved in compressible flow calculations.