Hydrodynamic Equations for Coastal Boulder Movement: Reflections on a Recent Review

Numerical modeling of salinity changes in marine environment of Persian Gulf is investigated in this paper. Computer simulation of the problem is performed by the solution of a convection-diffusion equation for salinity concentration coupled with the hydrodynamics equations. The hydrodynamic equations consist of shallow water equations of continuity and motion in horizontal plane. The effects of rain and evaporations are considered in the continuity equation and the effects of bed slope and friction, as well as the Coriolis effects are considered in two equations of motion. The cell vertex finite volume method is applied for solving the governing equations on triangular unstructured meshes. Using unstructured meshes provides great flexibility for modeling the flow problems in arbitrary and complex geometries, such as Persian Gulf domain. The results of evaporation and Coriolis effects, as well as imposing river and tidal boundary conditions to the hydrodynamic model of Persian Gulf (considering variable topology rough bed) are compared with predictions of Admiralty Tide Table, which are obtained from the harmonic analysis. The performance of the developed computer model is demonstrated by simulation of salinity changes due to inflow effects and diffusion effects as well as computed currents.

A derivation of the hydrodynamic equations from the equations of motion of collective coordinates

The anisorrhopic guiding-center fluid

Several experiments have been reported that indicate a significant difference between the hydrodynamic properties of suspensions and conventional viscous fluids. According to measurements made with different types of viscometers, the viscosity of the suspension in many cases exhibits anomalous behavior that is not compatible with descriptions of the suspension as a conventional Newtonian or non-Newtonian fluid [1–3]. Stratification into solid and liquid phases also plays an important role in the flow of suspensions. In particular, in the significant practical case of Poiseuille flow of an equidense suspension in a circular pipe, wall and core effects were observed. The wall effect, i. e., the migration of suspended particles toward the pipe centerline and the corresponding reduction of the solid phase concentration near the walls, was observed in suspensions of different nature (particles of different nature and form in a viscous liquid [4] and in blood, i.e., suspensions of blood corpuscles in blood plasma [3, 5]) over a wide range of values of average concentration. In contrast with the wall effect, the core effect, which involves an increase of the solid phase concentration in an annular region at a distance of 0.5–0.7 R from the pipe centerline (R is the pipe radius), was observed only for small values of the average concentration [6, 7]. An experiment [8] showed that the presence of the core effect is associated with rotation of the solid particles. When the center of gravity of each of the particles was shifted, their rotation was hindered. In the absence of rotation the particles always migrated toward the tube centerline. Many studies have been devoted to explanation of the wall effect. The magnitude of the transverse force acting on a sphere rotating in a translational viscous liquid flow was calculated in [9] (the limits of applicability of the resulting formula do not permit examination of those values of the parameters for which the core effect is observed in practice). If we substitute the expression for force into the particle equation of motion and integrate, we find that the particle trajectory approaches the tube axis asymptotically [10], regardless of the initial position of the particle. Another approach is the averaged description of the suspension behavior. The Navier-Stokes equation, with a coefficient of viscosity that depends in known fashion on the local solid phase concentration, was written as the equation of motion for the steady flow of a suspension in a circular tube. The variational principle (the principle of minimum energy dissipation) was formulated to find the concentration distribution. The wall effect was also obtained using this approach [11, 12]. An analogous study using the Kessonmodel yielded practically no results [13]. The complete system of equations of two-fluid hydrodynamics was recently constructed in which the fluid and the dispersed phase are considered as two interpenetrating, interacting continua, with rotation and deformation of the dispersed phase particles being taken into consideration (Nguen Van D'ep, Candidate's dissertation: Some questions on the Hydrodynamics of Structured Fluids [in Russian], Voronezh State University, Voronezh, 1968). However, the practical use of this system for solving, for example, the Poiseuille problem appears to be difficult, since it is necessary to know in detail the interaction forces between the phases. In the present study we use the single-fluid approximation, i.e., the quantities introduced (velocity, density, and so on) characterize the motion of the suspensions as a whole, and not any single phase. Only two quantities characterize the solid phase proper: the bulk concentration and the local angular velocity of the solid particles. This approach does not require knowledge of the interaction forces between the phases, and the final equations are considerably simpler than those for the two-fluid description. The thermodynamics of irreversible processes is used to construct a closed system of equations that includes the diffusion equations and the generalized moment of momentum equation, which make it possible to find the three-dimensional distribution of the concentration and the angular velocity of the solid particles. The generalized Fick law, which contains three additional diffusion coefficients, is obtained. In contrast with classical models, the viscous stress tensor is asymmetric, and its antisymmetric part is proportional to the difference in the angular rates of rotation of the solid particles and the suspension as a whole. From these equations follow under certain particular assumptions the equations of the theory of a fluid with internal rotation [14]. As an example of the application of the theory, the Poiseuille problem of flow in a flat channel is solved. The concentration distributions obtained agree well with the experimental data described above.

This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to ‘ordinary’ partial differential equations, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, u=u(x,t), these equations contain the same function at a past time, w=u(x,t−τ), where τ>0 is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments, w=u(px,qt), where p and q are scaling parameters. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”.

Singularities motion equations in 2-dimensional ideal hydrodynamics of incompressible fluid

† This article applies methods of statistical thermodynamics to analysis of fluid transport in minichannels and microchannels. The hydrodynamic equations are linearized and the governing equation of gas motion is reduced to the Stokes equation. The reduced mass flow through the microchannel is expressed in terms of the molecular mass, Boltzmann constant and rarefaction parameter. A key variable in estimating pressure differences within the microchannel is the apparent viscosity and thermal conductivity. In this regard, kinetic theory is applied with a Lenard–Jones potential to derive formulations of the microfluidic viscosity and thermal conductivity. The ChungLee-Starling model is applied to calculate viscosity and thermal conductivity. Numerical results are presented for property variations in fully developed flow through a long rectangular microchannel.

Vortex motion is a complex problem due to the interplay between the short-range physics at the vortex core level and the long-range hydrodynamical effects. Here we show that the hydrodynamic equations of vortex motion in a compressible superfluid can be solved asymptotically in a model ‘slab’ geometry. Starting from an exact solution for an incompressible fluid, the hydrodynamic equations are solved with a series expansion in a small tunable parameter provided by the ratio of the healing length, characterising the vortex cores, to the slab width. The key dynamical properties of the vortex, the inertial and physical masses, are well defined and renormalizable. They are calculated at leading order beyond the logarithmic accuracy that has limited previous approaches. Subtracting the asymptotic solutions of the universal hydrodynamic problem from experimental observations of vortex motion exposes the physics of the vortex core and provides a window into interesting many-body phenomena that are currently poorly understood including the role of quantum pressure. Our results provide a solid framework for further detailed study of the vortex mass and vortex forces in strongly correlated and exotic superfluids.

The main stages of the early evolution of theoretical fluid dynamics are presented and the relevant achievements in the field are described. The year of birth of the new fluid dynamics is considered 1751, when Leonhard Euler presented a broad theory of friction in fluids to the Academy of Sciences in Berlin. It is shown that the failure of the fluid friction theory, as well as the permanent search to reconcile the theory with the experiment, motivated Euler to introduce the concepts of „fluid particle" and „perfect fluid" and to formulate, in 1755, the equations of motion of ideal fluids, devoid of viscosity and adhesion. Considering the effects of viscosity allowed the formulation of the equations of motion for incompressible viscous fluids first, and then the equations of motion for compressible viscous fluids. The difficulties encountered in formulating the fundamental equations of hydrodynamics and the role of prominent researchers are shown.

The article presents the results of the analysis of the properties of disordered structures in hydrodynamic acoustics, associated with the process of detecting physical phenomena and marine objects based on the results of their mechanical impact on the marine environment, in which acoustic vibrations propagate. If vortices, attractors, fractals arise as a result of complex interactions of forces of nature (upwellings, seiches, Coriolis forces, currents, convection flows, rotation of the Earth) and are essentially mechanical effects on the environment of formation and propagation of an acoustic field, then mechanical sources of sound introduced into the hydrosphere (water) should repeat fractal iterations on a smaller scale at the sound field level. Recognizing the equations of hydrodynamics (the equation of motion, the equation of continuity, and the equation of state) as the fundamental equations of hydroacoustics, the nonlinearity of these equations is proposed to be considered the theory of the hydroacoustic field as nonlinear, and the linearity of the processes in this study is considered a special case. The principle of superposition also becomes a special case, and the Fourier transform, remaining necessary, loses its sufficiency. Fractal analysis in combination with wavelet analysis should be involved to help him

Aims. We describe various expansion schemes that can be used to study gravitational clustering. Obtained from the equations of motion or their path-integral formulation, they provide several perturbative expansions that are organized in a different fashion or involve different partial resummations. We focus on the two-point and three-point correlation functions, but these methods also apply to all higher-order correlation and response functions.Methods. We present the general formalism, which holds for the gravitational dynamics and for any similar models, such as the Zeldovich dynamics, that obey similar hydrodynamical equations of motion with a quadratic nonlinearity. We give our explicit analytical results up to one-loop order for the simpler Zeldovich dynamics. For the gravitational dynamics, we compare our one-loop numerical results with numerical simulations.Results. We check that the standard perturbation theory is recovered from the path integral by expanding over Feynman's diagrams. However, the latter expansion is organized in a different fashion, and it contains some UV divergences that cancel out as we sum all diagrams of a given order. Resummation schemes modify the scaling of tree and one-loop diagrams, which exhibit the same scaling over the linear power spectrum (contrary to the standard expansion). However, they do not significantly improve over standard perturbation theory for the bispectrum, unless one uses accurate two-point functions (e.g. a fit to the nonlinear power spectrum from simulations). Extending the range of validity to smaller scales, to reach the range described by phenomenological models, seems to require at least two-loop diagrams.

The quantum hydrodynamic equations associated with the de Broglie–Bohm formulation of quantum mechanics are solved using a meshless method based on a moving least squares approach. An arbitrary Lagrangian–Eulerian frame of reference is used which significantly improves the accuracy and stability of the method when compared to an approach based on a purely Lagrangian frame of reference. A regridding algorithm is implemented which adds and deletes points when necessary in order to maintain accurate and stable calculations. It is shown that unitarity in the time evolution of the quantum wave packet is significantly improved by propagating using averaged fields. As nodes in the reflected wave packet start to form, the quantum potential and force become very large and numerical instabilities occur. By introducing artificial viscosity into the equations of motion, these instabilities can be avoided and the stable propagation of the wave packet for very long times becomes possible. Results are presented for the scattering of a wave packet from a repulsive Eckart barrier.

An outstanding difficulty in the theory of fluid motions is the fact that the differential equations of motion are not linear, Their Solutions are difficult and sometimes impossible. Workers therefore are frequently forced to use approximate methods of solution, or to make assumptions which are not always sound. For example, in a nümber of cases it is assumed that two distinct Solutions of the hydrodynamical equations of motion are lin e a rly s u p e r p o s a b l e , i.e. their sum is also a solution. The same assumption is used again and again, for instance, in Lamb’s hydrodynamics.

lt is shown how two fluid dynamical equations for superfluid He' are derived from quantum hydrodynamical Hamiltonian, represented in terms of the quantized canonical con­ jugate variables, mass density p(x) and velocity field v(x). An essential assumption is that expectation value of v(x) should be identified with the superfluid velocity. The random phase approximation is applied to higher order dynamical correlation function of p and v appearing in the equations of motion. This approximation implies to decouple the correlation into the hydrodynamical degrees of freedom and the incoherent one of thermal phonon. It is found that a set of equations thus obtained is consistent with the two fluid dynamical one, having been widely used in the phenomenological theories. § I. Introduction Two fluid theory for superfluid helium has considerably succeeded in describing the thermodynamical and hydrodynamical phenomena at finite temperature. Nor­ mal component of fluid at low temperatures can well be described in terms of phonon gas interacting weakly. The two fluid hydrodynamical equations involve various kinetic coefficients which characterize the irreversible processes in phonon gas excited thermally. Therefore, in order to treat the relaxation process of pho­ non gas and to describe the dynamical behavior of two fluid, it is more fundamental to start with the kinetic stage. This situation is entirely analogous to that encoun­ tered in the classical normal fluid, apart from the existence of superfluid flow. Landau and Khalatnikov1'· 2' introduced phenomenologically the kinetic equation for phonon distribution function n (p. x) under the existence of superfluid flow. Thus conservation law for mass density p(x), equation of motion for superfluid velocity V 8 (x) and kinetic equation for n (p, x) constitute the basic set of two fluid dynamical equations. 2> The linearized and Fourier transformed expressions of these equations are written as

A mathematical model of the motion of 3-dimensional erythrocytes along fragments of capillary networks consisting of a small number of narrow capillaries is constructed. The model takes into account the lengths and inner diameters of capillaries , the form and position of an erythrocyte in a capillary, the viscosity of the plasm and the contents of an erythrocyte, the volume and surface area of an erythrocyte, the elastic characteristics of an erythr ocyte, the membrane rolling of an erythrocyte, and the intervals between the entries of erythrocytes into t he capillary network. The formulas for the dependence of pressure differential causing the blood flow i n fragments of capillary networks on the parameters of the model are obtained. The dependences of the blood flow rate through a capillary tee on the lengths of capillaries forming the tee, on the pressur e differential at the ends of the tee, and on the intervals between the entries of erythrocytes into the t ee are obtained. Experimental and theoretical studies of hemodynamics in a microcirculation system allows us to describe quantitatively the interrelated moti on of the viscous plasm and elastic cells (erythrocytes) along capillaries. In ord er to describe the motion of the plasm, we may use the basic hydrodynamic equations (equations of motion and continuity), and the form of erythrocytes is characteri zed by equations of the mechanics of a deformed solid body. The simultaneous solution of those equations subject to the corresponding boundary conditions determines the spatial distribution of the pressure in the plasm, the velocity of the plasm and erythrocytes, and the energy consumption for the blood motion in capillaries. The existing models using these relations simulate the complicated character of inte raction of blood cells in microvessels only in the first approximation (2, 4, 5, 7-12, 1 5, 16, 19, 21, 25, 26). Thus, the rheological properties of blood were considered in (26), the erythrocyte membrane rolling in wide capillaries (with the diameter exceeding the diameter of the erythrocyte) was considered in (16). An asymmetric form of an erythrocyte in a wide vessel was considered in (8), the motion of an erythrocyte having a symmetric parachute shape was considered in (15), in papers (4, 12) erythrocytes had the shape of elastic balls and in (5) were shaped as cylinders. A 2-dimensional model with a rolling membrane was constructed in (19). However, such important characteristics as the volume and surface area of erythrocytes were not taken into account in those models. The amount of oxygen transported by an erythrocyte depends on its volume and the rate of oxygen release from an erythrocyte is determined by its surface area. A three-dimensional model of the motion of an erythrocyte along a capillary was