On the weak discontinuity surfaces and tapes of equation systems in fluid mechanics

The weak discontinuity surfaces for a system of quasi-linear differential equations of higher order are developed. The classification of equation systems in fluid mechanics is based on the propagative weak discontinuity surfaces. Types of equations for different flow models are discussed. The conclusion is as follows:(a) For incompressible nonviscous flow, incompressible viscous flow and compressible viscous flow, the types of equations are all parabolic in the unsteady case and elliptic in the steady case.(b) For compressible nonviscous flow, the type of equations is hyperbolic in the unsteady case or steady supersonic case, and the type is elliptic in the steady subsonic case.     

The steady motions of a heavy ellipsoid filled with liquid on a plane with friction

The motion of a heavy uniform thin-walled ellipsoid of revolution, completely filled with an ideal incompressible liquid, performing uniform vortex motion is investigated. It is assumed that the ellipsoid is situated on a horizontal plane, from the side of which a normal reaction and a force of viscous sliding friction act on it. The equations of motion of the system, suitable both in the general case and in limiting cases of zero ellipsoid mass or zero liquid mass, are set up. Steady and periodic motions of the ellipsoid with the liquid are obtained. The conditions for uniform rotations of the ellipsoid about a vertically situated axis of symmetry to be stable are obtained.     

A Constructive Analytical Approach to the Calculation of Flows Representing the Steady Motion of a Viscous Incompressible Liquid

A constructive analytical technique is described for the calculation of the fluid velocity components satisfying the Navier-Stokes equations representing the steady motion of a viscous, incompressible liquid. These components are found in terms of the derivatives of two functions which in turn satisfy linear partial differential equations for which known solutions of general type are readily available.     

Explicit Solutions of the Two-Dimensional Navier Stokes Equations

Explicit solutions are found for the stream function satisfying the Navier Stokes equations representing the steady two-dimensional motion of a viscous incompressible liquid. The solutions contain two arbitrary analytic functions and in general are confined to certain regions of the x, y plane.     

Second approximation of the Navier-Stokes equations

Equations of motion of a viscous Newtonian fluid are derived, which, in addition to the terms of the Navier-Stokes equations, contain additional terms taking into account the relaxation effect of vorticity on the rate of strain. An independent experimental method for measuring a new parameter involved in the equations is described. As an application of the Navier-Stokes equations in the second approximation, Stokes’ hypothesis is rigorously substantiated. New similarity criteria for incompressible viscous flows are presented. The Poynting effect for viscous incompressible Newtonian fluids is theoretically explained.     

The quasistationary approximation in the problem of the motion of an isolated volume of a viscous incompressible capillary liquid

The equations of the quasistationary approximation in the problem of the motion of an isolated volume of a viscous incompressible capillary liquid are derived from the exact equations using an expansion in a small quasistationary parameter, which is equal to the ratio of the Stokes time to the capillary time. The problem contains yet another dimensionless parameter, which is proportional to the modulus of the conserved angular momentum of the liquid volume, which is also assumed to be small. Depending on the relation between these parameters, three versions of the limiting problem are obtained: the traditional version and two new versions. Asymptotic solutions of the problems which arise when the quasistationary parameter tends to zero are constructed.     

Leveling a capillary ridge generated by substrate geometry

The formation of capillary ridges is typical of thin viscous films flowing over a topographical feature. This process is studied by using a two-dimensional model describing the slow motion of a thin viscous nonisothermal liquid film flowing over complex topography. The model is based on the Navier-Stokes equations in the Oberbeck-Boussinesq approximation. The density, surface tension, and viscosity of the liquid are linear functions of temperature. For a nonisothermal flow over a planar substrate with a local heater, the influence of the heater on the free surface is analyzed numerically depending on the buoyancy effect, Marangoni stresses, and variable viscosity. The analysis shows that the film can create its own ridges or valleys depending on the heater and the dominating liquid properties. It is shown that the capillary ridges generated by the substrate features can be optimally leveled by using various types of heaters consistent with the dominating liquid properties. Numerical results for model problems are presented.     

Existence of Weak Solutions to the Problem on Three-Dimensional Steady Heat-Conductive Motions of Compressible Viscous Multicomponent Mixtures

Siberian Mathematical Journal - We consider the equations describing the three-dimensional steady heat-conductive motions of compressible viscous multicomponent mixtures. We also prove the...     

Sull'integrale di bernoulli in magnetofluidodinamica

Summary The necessary and sufficient conditions for applying, in Magnetofluid-dynamics (MFD), the Bernoulli theorem or a Bernoulli integral to a viscous, incompressible, electroconducting fluid with conservative body force in steady or unsteady motion are investigated. Some relevant MFD motions are shown.      

The evolution of the thermocapillary motion of three fluids in a plane layer

The unidirectional motion of three immiscible incompressible viscous heat-conducting liquids in a plane layer is considered. It is assumed that the motion occurs only under the action of thermocapillary forces from a state of rest. The analysis of the motion is reduced to solving linear conjugate initial boundary value problems for a system of parabolic equations. A non-stationary solution is sought by the Laplace transformation method and is obtained in the form of finite analytical expressions in transforms. It is proved that, as the time increases, the solution always reaches the steady state obtained earlier and an exponential estimate of the rate of convergence is given with an indicator which depends on the physical properties of the media and the layer thicknesses. The evolution of the velocity and temperature perturbation fields to a steady state for specific liquid media is obtained by numerical inversion of the Laplace transformation.     

Existence and zero‐electron‐mass limit of strong solutions to the stationary compressible Navier‐Stokes‐Poisson equation with large external force

In this study, we consider a viscous compressible model of plasma and semiconductors, which is expressed as a compressible Navier‐Stokes‐Poisson equation. We prove that there exists a strong solution to the boundary value problem of the steady compressible Navier‐Stokes‐Poisson equation with large external forces in bounded domain, provided that the ratio of the electron/ions mass is appropriately small. Moreover, the zero‐electron‐mass limit of the strong solutions is rigorously verified. The main idea in the proof is to split the original equation into 4 parts, a system of stationary incompressible Navier‐Stokes equations with large forces, a system of stationary compressible Navier‐Stokes equations with small forces, coupled with 2 Poisson equations. Based on the known results about linear incompressible Navier‐Stokes equation, linear compressible Navier‐Stokes, linear transport, and Poisson equations, we try to establish uniform in the ratio of the electron/ions mass a priori estimates. Further, using Schauder fixed point theorem, we can show the existence of a strong solution to the boundary value problem of the steady compressible Navier‐Stokes‐Poisson equation with large external forces. At the same time, from the uniform a priori estimates, we present the zero‐electron‐mass limit of the strong solutions, which converge to the solutions of the corresponding incompressible Navier‐Stokes‐Poisson equations.     

Stationary solutions to the equations of dynamics of mixtures of heat-conductive compressible viscous fluids

We consider the equations describing the three-dimensional steady motions of binary mixtures of heat-conductive compressible viscous fluids. An existence theorem for the boundary value problem that corresponds to flows in a bounded domain is proved in the class of weak generalized solutions.     

Zero Mach number limit to compressible quantum magnetohydrodynamic equations with vanishing viscosity coefficients

The connection between the compressible viscous quantum magnetohydrodynamic model with low Mach number and the ideal incompressible magnetohydrodynamic equations is studied in a periodic domain. More precisely, for well‐prepared initial data, we prove the convergence of classical solutions of the compressible viscous quantum magnetohydrodynamic model to the classical solutions of the incompressible ideal magnetohydrodynamic equations with a convergence rate when the Mach number, viscosity coefficient, and magnetic diffusion coefficient simultaneously tend to zero.     

Wall effects on the slow steady motion of a particle in a viscous incompressible fluid

In this paper, asymptotic expansions with respect to small Reynolds numbers are proved for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid past a single plane wall. The flow problem is modelled by a certain boundary value problem for the stationary, nonlinear Navier-Stokes equations. The coefficients of these expansions are the solutions of various, linear Stokes problems which can be constructed by single layer potentials and corresponding boundary integral equations on the boundary surface of the particle. Furthermore, some asymptotic estimates at small Reynolds numbers are presented for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid between two parallel, plane walls and in an infinitely long, rectilinear cylinder of arbitrary cross section. In the case of the flow problem with a single plane wall, the paper is based on a thesis, which the author has written under the guidance of Professor Dr. Wolfgang L. Wendland.     

Unsteady hydromagnetic motion of a rotating conducting fluid

The combined effects of stratification and magnetic field on the unsteady motion of a viscous, electrically conducting fluid between two rotating disks are analysed. Solutions are obtained for the linearized equations under Boussinesq approximation and steady state solutions are deduced from them. The results are compared with those obtained by Loper and Benton and Balanet al. Graphs are presented for the steady state velocity, magnetic field and temperature distributions.     

The Zero Mach Number Limit of the Three-Dimensional Compressible Viscous Magnetohydrodynamic Equations

This paper is concerned with the zero Mach number limit of the three-dimension- al compressible viscous magnetohydrodynamic equations. More precisely, based on the local existence of the three-dimensional compressible viscous magnetohydrodynamic equations, first the convergence-stability principle is established. Then it is shown that, when the Mach number is sufficiently small, the periodic initial value problems of the equations have a unique smooth solution in the time interval, where the incompressible viscous magnetohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Mach number goes to zero. Moreover, the authors prove the convergence of smooth solutions of the equations towards those of the incompressible viscous magnetohydrodynamic equations with a sharp convergence rate.     

Approximation of the incompressible convective Brinkman–Forchheimer equations

This paper studies the approximation of solutions for the incompressible convective Brinkman–Forchheimer (CBF) equations via the artificial compressibility method. We first introduce a family of perturbed compressible CBF equations that approximate the incompressible CBF equations. Then, we prove the existence and convergence of solutions for the compressible CBF equations to the solutions of the incompressible CBF equations.     

Simulation of Separating Flows over a Backward-Facing Step

Simulation results are reported for plane two-dimensional viscous incompressible flow in a channel with an abrupt expansion. The mathematical model is provided by the quasi-hydrodynamic equations in the incompressible fluid approximation. The computations are carried out in a range of Reynolds numbers including both laminar and turbulent flow. As the Reynolds number increases, the solution bifurcates and the steady laminar flow changes to time-dependent flow. The computation results are consistent with known experimental data. Turbulence models were not used for large Reynolds number computations.     

Hydromagnetic forced flow against a porous rotating disk

This paper deals with the steady forced flow of a viscous, incompressible and electrically conducting fluid against a porous rotating disk when a uniform magnetic field acts perpendicular to the disk surface. For small suction the equations of motion are integrated numerically by Kármán-Pohlhausen method, but for large suction a series solution in the inverse powers of the suction parameter is obtained. The effects of disk porosity and magnetic field on the various flow parameters are discussed in detail.     

Time averaging as an approximate technique for constructing quasi-gasdynamic and quasi-hydrodynamic equations

Quasi-gasdynamic and quasi-hydrodynamic equations for compressible gas flows and viscous incompressible fluid flows are constructed by averaging the corresponding Navier-Stokes equations over time with the use of certain approximations.